ON PARTICLE WEIGHTED METHODS AND SMOOTH PARTICLE HYDRODYNAMICS

Mathematical Models and Methods in Applied Sciencesfc World Scientific Publishing Company

ON PARTICLE WEIGHTED METHODS AND

SMOOTH PARTICLE HYDRODYNAMICS

J.P. VILA.

INSAT, Departement Genie Mathematique et Modelisation

UMR CNRS 5640, Mathematiques pour l’Industrie et la PhysiqueF 31077 TOULOUSE CEDEX

This paper deal with designing of weighted particle approximation of conservation laws.New ideas concerning the use variable smoothing length, renormalization and the use ofGodunov type Finite Difference fluxes in particle methods are introduced and discussedin connection with standard implementation of the SPH method. A detailed analysis ofboundary conditions approximation is also provided.

Keywords: weighted particle approximation, boundary condition, conservation laws

1. Introduction

The particle method known as Smooth Particle Hydrodynamics was introduced in

the end of seventies by Lucy 28 and Gingold and Monaghan16. SPH was first used

in the area of astronomy and astrophysics. Recent applications (see e.g. Ref.8) in

the field of high velocity impact have considerably enlarged the popularity of the

method.

It seems that, except in the linear case (see Mas Gallic and Raviart30) the conver-

gence of the method has not been mathematically analyzed. The aim of this paper

is to give some insight into the way of designing and analyzing SPH like methods for

conservation laws. We focus our attention on the description of new methods, i.e.

use of new formulation for variable smoothing length, renormalization, use of non

linear finite difference fluxes of Godunov type, use of weighted ghosts and boundary

particles or forces, in relation with application to Euler equations of a compressible

flow, rather than on detailed proofs of convergence. We refer to subsequent papers

(Ref.5’23 ’6) for such a detailed mathematical analysis. The paper is organized as

follows :

- After describing the classical tools in particle approximation, we give the

general form of the standard SPH approximation of a model partial differential

equation which clearly needs some amount of upwinding or artificial viscosity

to remain stable. We then present a discrete weak formulation of these particle

weighted methods. It gives a general setting for convergence analysis in the

Lax-Wendroff 26 sense (see also Ref.5 for a convergence analysis in the scalar

nonlinear case).

1

2 On Particle weighted methods and SPH

- In the third section we introduce new concepts for use of variable smoothing

length. We also provide tools for the analysis of renormalized particle weighted

methods. They lead both to consistent and conservative schemes.

- In the fourth section we then introduce a new formulation with Godunov type

finite difference numerical fluxes. We also give higher order extension of the

methods based on MUSCL techniques well known for finite difference schemes.

- The fifth section concerns application of these ideas to Euler equations. We

also review some results concerning classical SPH approximation of Euler

equations, for example entropy dissipation due to the artificial viscosity which

does not seem clearly written in the SPH literature. The subsection concerning

the use of Riemann solvers is somewhat related to previous works of Hymann

and Harten18 on self adjusting Godunov-type finite difference schemes. We

propose some new interpretation of their method which allows extension to

weighted particle methods.

- The sixth section is devoted to the description of a general setting for particle

approximation of P.D.E. on bounded domains. We develop three approaches,

the first one uses ghost and weighted ghost particles, the second one is based on

boundary particles and boundary forces, while the third one is semi-analytical.

- In the last section we apply these techniques to Euler equations. In particular

we carefully study how to define ghost particles. The conservation of energy

gives sufficient informations to determine ghost particles characteristics like

velocity correctly.

2. Smooth Particle Approximation

2.1. Basic principles of particle approximation

The design and analysis of weighted particle methods for transport equations

and Euler inviscid equations is a well documented field. Thus we briefly discuss the

main tools necessary for a comprehensive study of our results. Reference works are

quoted in this section for an exhaustive study of the different points reviewed here.

Let v a regular vector field in IRd. We consider the following model PDE in

conservation form :

Lv(Φ) + divF (x, t,Φ) = S , (2.1)

where F is the flux vector (∈ IRd) of the conservation law and Lv is the transport

operator given by : Lv :

Φ −→ Lv(Φ) =∂Φ

∂t+∑l=1,d

∂

∂xl(vlΦ) .

On Particle weighted methods and SPH 3

2.1.1. Particle approximation of functions

To get a particle approximation of the equation (2.1), let us take a set of mov-

ing particles (xi(t), wi(t))i∈P , indexed by i ∈ P , where xi(t) is the position of the

particle and wi(t) its weight. We classically move the particles along the character-

istic curves of the field ~v and also modify the weights in order to take account of

deformations due to the field ~v :

(i)d

dtxi = v(xi, t) , (ii)

d

dtwi = div(~v(xi, t))wi , (2.2)

In the simpler case of particles initially distributed on a cubic grid with uniform

spacing ∆x, the simplest choice is wi(0) = (∆x)d. The particle approximation Π(f)

of a function f is then defined as :

Π(f)(x) =∑i∈P

wi(t)f(xi(t))δ(x− xi(t)) ,

The accuracy of the approximation is connected with the quadrature formula over

IRd, given by the particles (xi(t), wi(t))i∈P :∫IRd

g(x)dx ≈∑j∈P

wj(t)g(xj(t)) . (2.3)

This formula is accurate for any t > 0 as soon as it is accurate initially and the

particles and weights move according to (2.2) (see Ref.37).

Let us introduce a regularizing kernel W (x, h) where the parameter h, the so-

called “smoothing length” characterize the regularizing scale, and W (x, h) converges

towards the Dirac measure δ(x) (in a suitable sense) when h −→ 0. This kernel

must satisfy : ∫W (x, h)dx = 1 . (2.4)

In practice we get a general kernel W from a given auxiliary function θ of a scalar

variable : W (x, h) =1

hdθ(‖x‖h

). We take usually θ as a positive function with

compact support ⊂ [0, 2], for example :

θ(y) = C ×

1− 3

2y2 +

3

4y3 if 0 ≤ y ≤ 1,

1

4(2− y)3 if 1 ≤ y ≤ 2,

where the coefficient C is2

3,

10

7π,

1

πaccording to the space dimension (1,2 or 3),

in order to satisfy the condition (2.4). We then define Πh(f) the smoothed (or

regularized) particle approximation of a function f as :

Πh(f)(x) =∑i∈P

wi(t)f(xi(t))W (x− xi(t), h) = Π(f) ∗W .


We shall use the following notations :

Wij ≡W (xi(t)− xj(t), h), ~∇Wij ≡ gradx[W (xi(t)− xj(t), h)] .

Particle approximation of derivatives is easily handled by taking direct derivation

of smoothed particle approximations which gives at the point xi :

~∇Πh(f)i =∑j∈P

wjf(xj)~∇Wij .

Remark 1 Here we use the standard notations in SPH literature. The smoothing

length is h, the kernel is W and we have chosen ∆x as the characteristic size of

the mesh. In most of the mathematical papers related to particle weighted methods,

the smoothing length is denoted ε, the kernel or cut-off function is ζε and h is the

characteristic size of the mesh. The reader has to take account of that in some of

the references quoted in this paper.

2.1.2. Basic approximation results

In classical discretization methods such as finite differences, finite volumes or

finite elements we have a unique discretization parameter which is ∆x the char-

acteristic size of the mesh. Here we get an additional parameter h , the so-called

“smoothing length”, which is the characteristic size of the regularizing kernel W .

The combined effect of these 2 parameters can be studied accurately. We refer to

the book of P.A. Raviart 37 for a detailed analysis of interpolation errors in vari-

ous Sobolev norms and semi-norms. We just recall some results of Ref.37 (see also

Ref.30) which will be useful for our analysis. We restrict ourselves to compactly

supported symmetric kernels such that W ∈ Cm+1, m ≥ 2. Then, there exists a

constant C > 0 just depending on the transport field ~v (supposed regular enough)

such that : ∑j∈P

wj |~∇Wij | ≤C

h, (2.5)

and for any u ∈Wµ,p(IRd), s ≥ 0, µ = max(r + s,m), r = 1, 2,d

m≤ p ≤ ∞

|u−Πh(u)|s,p,IRd ≤ C(hr|u|r+s,p,IRd + (1 +

∆x

h)dq

(∆x)m

hm+s||u||m,p,IRd

), (2.6)

where 1q + 1

p = 1 and the Sobolev spaces W r,l(IRd) are provided with usual norms

and semi-norms. In the following we shall examine the convergence of the method

when the discretization parameters h and ∆x go to zero. We always suppose that

such parameters go to zero in a way that the estimates of (2.6) gives convergence

(e.g. we need that the ratio (∆x)m

hm+s goes to zero).

Remark 2 It will be useful for the applications to remark that these estimates are

valid on a bounded domain Ω as soon as we consider functions with compact support

in the set Ω. This point will be detailed in Section 6.


In practical computations, these parameters are chosen so that the number of

neighbors of any particles, i.e. the number of particles located at a distance less

than a length of order h, is almost constant all other the computational domain

(≈ 25 for 2D computations, and ≈ 50 in 3D)

Additional Definition. In the following it will be interesting to consider an other

approximation of a function f related to the particle approximation Πf , but more

suitable to study weak solutions of non linear PDE.

Initially we can consider that the particles and the weights in the quadrature

formula (2.3) are such that the space IRd = ∪j∈PBj(0), where Bj(0) are distinct

sets of volume ωj = wj(0). The sets Bj(t), where Bj(t) is the image of Bj(0)

by the regular flow associated to −→v (x, t), also define a partition of IRd by sets

of volumes equal respectively to wj(t). We thus associate to a measure f∆(x) =∑i∈P

wi(t)fi(t)δ(x − xi(t)) a function f∆

(x) =∑i∈P

fi(t)χBi(t)(x) where χBi(t) is the

characteristic function of the set Bi(t).

2.2. Particle approximation of a model partial differential equation

2.2.1. Classical derivation of the approximation

Let us now consider the model PDE (2.1). By using the technics developed in

the previous paragraphs, a natural way to define a particle approximation of our

model equation should be for example :

Lv(ΠΦ) + Π(div Πh(F (x, t,Φ))

)= ΠS. (2.7)

In the second term of the left hand side of (2.1) we use a smoothed particle approx-

imation for getting an approximation of derivatives in order to avoid introducing

derivatives of Dirac which does not allow the system to be closed.

This is the approach used in Refs.30’38. Although ΠΦ in (2.7) - with Φ suppos-

edly sufficiently regular - acts directly on the model PDE, this formulation is also

connected with a weak form of the PDE. For example Lv(ΠΦ) makes sense only

against a regular test function ϕ :

< Lv(ΠΦ), ϕ >:= − < ΠΦ, L∗vϕ > ,

where −L∗v is the adjoint operator of Lv defined as :

L∗v(ϕ) =∂ϕ

∂t+∑l=1,d

vl∂ϕ

∂xl.

Let us remark that, for any ϕ sufficiently regular

d

dt(ϕi) =

d

dt(ϕ(xi(t), t)) = L∗v(ϕ)i,

d

dt(wiϕi) = wiLv(ϕ)i .


We then easily proves that (2.7) is equivalent with solving the system of o.d.e. :

(i)d

dtxi = v(xi, t), xi(0) = ξi,

(ii)d

dtwi = div(~v(xi, t))wi, wi(0) = ωi,

(iii)d

dt(wiΦi) + wi

∑j∈P

wjF (xj , t,Φj)~∇Wij = wiSi,

(iv) Φj(0) = Φ0(ξj) .

(2.8)

Remark 3 The set of equations (2.8) defines the function Φ supposed sufficiently

regular to give sense to Π(Φ) just along the characteristic curves located initially

at points (ξj)j∈P . Thus we are able to give a more general sense to (2.7) by only

considering the functions Φ∆

=∑j∈P wj(t)Φj(t)χBi(t)(x). This will be detailed in

section 2.2.3.

2.2.2. How to keep conservation

This approximation suffers a lack of conservation. The exact solution satisfies :

d

dt

(∫IRd

Φdx

)=

∫IRd

Sdx . (2.9)

Clearly the approximate solution (2.8) does not satisfy a similar requirement. This

should be the case if we add some kind of symmetry between particles in the second

term in the l.h.s. of (2.8) (iii). Generally the kernel W is symmetric (in the sense

that W (x, h) = W (−x, h)), thus its derivatives are such that :

~∇Wij = −~∇Wji . (2.10)

By integrating over IRd, this property allows symmetric interaction between par-

ticles to be cancelled. This is true for a modified particle approximation of (2.1)

defined in the following way :

Lv(ΠΦ) + Π

div[Πh(F (x, t,Φ))] +∑l=1,d

F l(x, t,Φ)∂Πh(1)

∂xl

= ΠS .

The extra term∂Πh(1)

∂xlclearly approximates 0, thus the approximation is still

consistent (see section 2.2.3 for details). We get a new system of o.d.e. :

d

dt(wiΦi) + wi

∑j∈P

wj(F (xj , t,Φj) + F (xi, t,Φi)).~∇Wij = wiSi , (2.11)

which, thanks to (2.10) satisfies :

d

dt

∑j∈P

wj(t)Φj(t)

=∑j∈P

wj(t)Sj(t) .


which is the discrete analog of the global conservation property (2.9) of our model

PDE.

We have now given the basic principles of the weighted particle approximation of

our model problem. Some comments about stability must be made before going

further in applications.

To analyze some stability property, we consider the simpler 1-dimensional linear

case where v = 0 and F = cΦ. We use the previous method defined by (2.11), the

particles are fixed (since v ≡ 0), we suppose that they are equally distributed on

a regular mesh of size ∆x = h so that each particle can only interact with its two

nearest neighbors. The scheme then reduces to :

d

dtΦi(t) = λ(Φi+1(t)− Φi−1(t)),

which is a centered finite difference scheme approximating our model transport

equation. When explicit in time discretization is used, it is well known that we

get generally, unconditionally unstable schemes. Note that leap-frog discretization

could make them stable. A classical remedy is to introduce some upwinding or

equivalently some artificial viscosity, this leads for our model PDE to :

d

dt(wiΦi) + wi

∑j∈P

wj(Fj + Fi + Πij).~∇Wij = wiSi, (2.12)

where Πij is an artificial viscosity term satisfying Πij = Πji , and Fi (resp. Fj)

stands for F (xi, t,Φi) (resp. F (xj , t,Φj)).

This approach is the most popular in SPH method . In section 4 we give the

basis of an alternative approach including the use of general 1D finite difference

flux.

2.2.3. Consistency and “Lax-Wendroff” like results

We deal with non linear hyperbolic conservation laws such as Euler compressible

equations, thus we must attempt to compute discontinuous solutions such as shocks.

Therefore it is necessary to introduce the notion of a weak solution of (2.1), defined

by :∀ϕ ∈ C2

0(IRd × IR+,∗),∫IRd×IR+

(Φ.L∗v(ϕ) + F (x, t,Φ).~∇(ϕ) + S.ϕ

)dxdt = 0 .

(2.13)

To get uniqueness we need to introduce the notion of entropy solution, classical

in this field, but it is not essential here since we only want to introduce the main

concepts (we refer to Ref.5 for a more detailed analysis). In order to provide a

better understanding of particle schemes like those of the previous sections, let

us now introduce a general setting for particle approximations of (2.1). It will

emphasize the importance of the weak formulation, even for the approximation of

classical solutions of (2.1).

We provide the space with the discrete scalar product :


(ϕ,Ψ)∆ :=∑i∈P

wiϕi.Ψi =

∫IRd

ϕ∆.Ψ∆dx =< Πϕ,Ψ >,

which is clearly an approximation of the scalar product in L2(IRd)m. We shall also

use

(ϕ,Ψ)t∆ :=

∫IR+

(ϕ,Ψ)∆ dt =

∫IR+

(∑i∈P

wiϕi.Ψi

)dt.

We also introduce a linear operator Dh,S which is supposed to approximate

strongly the derivative, i.e. for any ϕ regular enough

supi∈P‖Dh,Sϕi −Dϕi‖ → 0 as h and ∆x→ 0 ,

and let us define −D∗h,S as the adjoint operator of Dh,S . We thus have

(Dh,Sϕ,Ψ)∆ = −(ϕ,D∗h,SΨ

)∆. (2.14)

A discrete version of (2.1) is provided by just replacing the integration over IRd

by the discrete scalar product (., .)∆ and the derivative ~∇(ϕ) by its approximation

Dh,Sϕ :

∀ϕ ∈[C2

0(IRd × IR+,∗)]m(

Φ∆, L∗v(ϕ)

)t∆

+∑

α=1,...,d

(Fα(Φ

∆), Dα

h,Sϕ)t

∆+(S +Rh(Φ

∆), ϕ)t

∆= 0. (2.15)

Rh(Φ∆

) is an additional term which represents for example the artificial viscosity.

Making an integration by part with respect to t, we get easily that (2.15) is true

if and only if . :

(i)d

dtxi = v(xi, t), xi(0) = ξi,

(ii)d

dtwi = div (v(xi, t))wi, wi(0) = ωi,

(iii)d

dt(wiΦi) + wi

∑α=1,...,d

Dα,∗h,S(Fα)i = wi (Si +Rh(Φ)i) ,

(iv) Φj(0) = Φ0(ξj),

(2.16)

where Fi stands for F (xi, t,Φi).

Remark 4 With this general setting, it is easy to satisfy global conservation. Let

us suppose that(Rh(Φ

∆), 1)h

= 0, and that ϕ = 1 is in the kernel of Dh,S, then

(2.15) reduces to : for any g(t) ∈ C(IR+) (Φ, L∗v(1xg(t)))t∆ +(S, 1xg(t))

t∆ = 0, which

gives finally

d

dt

(∑i∈P

wiΦi

)=∑i∈P

wiSi. (2.17)


Like Lax and Wendroff 26 for finite difference schemes we are able to give suffi-

cient conditions to get a good limit weak solution :

Theorem 2.1 Let Φ∆

=∑j∈P

Φj(t)χBj (x) the function associated with the sequence

(Φj(t))j∈P of regular functions of t defined by the scheme (2.16). We suppose that

:

(i) the function Φ∆

converges boundedly almost everywhere to Φ when h and ∆x

go to zero,

(ii) ∀ϕ ∈[C2

0(IRd × IR+,∗)]m

supi∈P ‖Dh,Sϕi −Dϕi‖ → 0 as h and ∆x→ 0,

(iii) limh and ∆x→0

(Rh(Φ

∆), ϕ)t

∆= 0,

Then Φ is a weak solution of the model PDE (2.1) in the sense of Definition

2.13.

Proof. Φ∆

satisfies (2.15).Thanks to condition (ii) Dαh,Sϕ −→ ∂αϕ for the L∞

topology. :

Dαh,Sϕ −→ ∂αϕ.

Applying the Lebesgue theorem we get the result. 2

Let us see now how this result apply to the scheme (2.12). Let us take

Dh,Sϕi := D(Πhϕ)i − ϕiD(Πh1)i =∑j∈P

wj (ϕj − ϕi) ~∇Wij . (2.18)

Thanks to the error estimates (2.6) of section 2.1.2, we have the following conver-

gence results when h goes to zero (with (∆x)m

hm+1 −→ 0- ) for the L∞ topology :

Dh,Sϕi =∑j∈P

wjϕj ~∇Wij −→ Dϕi, Dh,S1i =∑j∈P

wj ~∇Wij −→ 0.

Thus condition (ii) of Theorem 2.1 is fulfilled.

A straightforward computation, using ~∇Wij = −~∇Wji proves that

D∗h,Sϕi =∑j∈P

wj (ϕj + ϕi) ~∇Wij .

Let us take (- with Πij = Πji-)

Rh(Φ∆

)i :=∑j∈P

wjΠij~∇Wij . (2.19)

It follows that the scheme (2.16) with (2.18) and (2.19) reads as (2.12). Theorem

2.1 applies to this scheme at the only additional condition that (∆x)m

hm+1 goes to zero

when h goes to zero.


Let us remark that Dh,S1 = 0, and that (since Πij = Πij) we have (Rh(Φ), 1)∆ =

0. The global conservativity of the scheme, in the sense of (2.17) follows immediately

(see Remark 4 ).

The statements (i) and (iii) differ from the standard ones in the classical Lax-

Wendroff consistency theorem :

• the hypothesis on the ratio (∆x)m

hm+1 is specific to particle methods and it is

related to the fact that we need ~∇(Πhϕ) −→ ~∇ϕ (as h −→ 0), for a regular

function ϕ. This hypothesis can be relaxed ; we have introduced in Ref.43

some renormalized weighted particle methods for which convergence can be

obtained under the only hypothesis that (∆x)h is bounded - see section 3.3 for

more details -.

• the hypothesis (iii) means that the artificial viscosity is small in some weak

sense.

We detail the last point concerning the numerical viscosity. First, let us remark

that thanks to the symmetry relations Πij = Πji and ~∇Wij = −~∇Wji, we get(Rh(Φ

∆), ϕ)

∆=

1

2

∑i,j∈P

wiwj(ϕi − ϕj)Πij~∇Wij .

Taking account of (2.5) it is easy to establish that

| (Rh(Φ), ϕ)∆ |≤ Cmeas(spt(ϕ))‖~∇ϕ‖∞‖Π‖∞,

which means that condition (iii) is true if the artificial viscosity Π goes strongly to

0.

This hypothesis can be weakened if we consider numerical viscosities such that

Πij = Qij .(Φi − Φj). (2.20)

In such a case we get

|∫IR+

|∑i,j∈P

wiwj(ϕi − ϕj)Πij~∇Wij | ≤

(

∫IR+

|∑i,j∈P

wiwjQ2ij ‖ Φi − Φj ‖2‖ ~∇Wij ‖| dt)

12

×(

∫IR+

|∑i,j∈P

wiwj | ϕi − ϕj |2‖ ~∇Wij ‖| dt)12 .

The second term in the right hand side is bounded by meas(spt(ϕ)) ‖ ~∇ϕ ‖∞√h. Thus if the first term in the r.h.s. is bounded, then condition (iii) is true.

In practise, if Qij is uniformly bounded and if we are able to get bounds on the

approximate solution such as∫IR+×IRd

‖ ~∇Φh ‖2 dxdt ≤C

hα, α < 1,


we obtain the desired estimate. These conditions are clearly weaker than the strong

convergence towards 0 of the artificial viscosity. The artificial viscosity classically

used for Euler equations (see Section 5.2) is of the type (2.20) with Qij bounded.

In the scalar nonlinear case and for explicit in time discretization we have establish

such bounds (see Ref. 5 and the comments at the end of this section- see also Ref.6

and Ref. 47 for an analysis in the case of linear symmetric hyperbolic systems).

This formulation emphasize the importance of weak formulation and its con-

nection with the consistency of the particle method. This is also the basic tool

for performing good generalization of the schemes in the case of varying smoothing

length, we refer to the next section for a discussion of this subject.

Let us also state some extensions of Theorem 2.1 which concerns equations of

the form :

Lv(Φ) + div[F (x, t,Φ)H(x, t)] = S, (2.21)

where H is a regular function (∈ W 2,∞). We define a particle approximation of

(2.21) based upon the following formulation :

Lv(ΠΦ) + Π(H(x, t)div[Πh(F (x, t,Φ))] + F (x, t,Φ).~∇(Πh(H))

)= ΠS, (2.22)

which gives an associated system of o.d.e. :

d

dt(wiΦi) + wi

∑j∈P

wj(FjHi + FiHj + Πij).~∇Wij = wiSi. (2.23)

Under the additional hypothesis that the function H ∈W 2,∞(IRd×IR+) the results

of Theorem 2.1 also apply to the scheme (2.23). We omit the detailed proof since

it is very closed to the proof of Theorem 2.1.

Strictly speaking, due to the results of Theorem 2.1, it is only possible to un-

derstand concepts of consistency in relation with particle approximation since we

are not able usually to get a priori estimates, sufficiently strong to get converging

subsequences in L1 by compactness arguments . We can modify the hypothesis (i)

in the following way :

(i)’ the functions Φ∆

(respectively F (Φ∆

)) converges in L∞w∗ towards Φ (respec-

tively G),

then we can prove in a similar way like in Theorem 2.1 that :

∀ϕ ∈ C20(IRd × IR+,∗)∫

IRd×IR+

(Φ.L∗v(ϕ) +G(x, t).~∇(ϕ) + S.ϕ

)dxdt = 0.

This result is the essential tool which allows the application of the Diperna’s11

uniqueness results concerning measured valued solutions of scalar non linear con-

servation laws to get convergence of approximate solutions. In the section 4 we

detail the construction of a large class of scheme for which we are able to prove


that hypothesis (iii) is true, and also that the approximate solutions are bounded

in L∞. We refer to Ref.5 where we establish these properties and prove convergence

of the related particle approximations.

3. Variable Smoothing Length and Renormalization

3.1. Particle approximation of functions with variable smoothing length

Variable smoothing length is a basic ingredient in performing efficient compu-

tations with weighted particle methods. The original computations made by Mon-

aghan for standard test problems such as shock tubes have proven that keeping the

smoothing length to be constant introduces numerical instabilities inside rarefaction

waves. These difficulties are clearly consequences of the decrease in the number of

neighboring particles in these area. Variable smoothing length has then been intro-

duced by Monaghan to overcome this problem, and it reveals as a very performing

and essential tool.

Some problems remain in the analysis and the designing of related schemes :

-it is not clear whether or not the variable smoothing length guarantees good

consistency with the PDE, even in the weak sense of Theorem 2.1,

-variable smoothing length is believed to be incompatible with global conservation

of the scheme (see for instance Ref.9)

In this section we introduce some new concepts related to variable smoothing

length. We thus are able to propose a new formalism which leads to weighted par-

ticles schemes with variable smoothing length, both consistent and insuring global

conservation of the physical quantities.

First let us recall two ways of introducing variable smoothing length, the so-

called “scatter” and “gather” formulation, according to Hernquist and Katz20.

They are respectively associated to the following concept of smoothing by a kernel

W (which is always supposed to satisfy (2.4) and symmetry condition W (x, h) =

W (−x, h)) :

-the “scatter” smoothing of a function :

< f(x) >s=

∫f(y)W (x− y, h(y))dy,

associated to

Πh,s(f)(x) =∑i∈P

wif(xi)W (x− xi, h(xi)) =< Π(f)(x) >s= (f(.),W (x− ., h(.))h ,

-the “gather” smoothing of a function :

< f(x) >g=

∫f(y)W (x− y, h(x))dy,


associated to

Πh,g(f)(x) =∑i∈P

wif(xi)W (x− xi, h(x)) =< Π(f)(x) >g= (f(.),W (x− ., h(x))h .

When the discretization parameters h and ∆x go to zero, we need that the

associated particle approximations Πh,s(f) and Πh,g(f) approximate the function f .

This can be mathematically proved for “gather” formulation while for the “scatter”

formulation we get an error term of order∥∥∥~∇h∥∥∥2

which does not vanish unless we

make the additional assumption that ~∇h → 0 when h → 0. We refer to Ref.23

for a new formulation of the “scatter” smoothing which turns out to be convergent

when h and ∆x go to zero as soon as ~∇h is uniformly bounded. Note that if we

suppose that h is scaled according to h = h0η(x) with η(x)a regular function, all

the formulations can be proved to be convergent. In this case ~∇h→ 0 when h→ 0.

Note also - see numerical tests in Ref.23 - that in practise it is important to achieve

an efficient smoothing of h in order to avoid numerical oscillations.

Let us now detail the results for the “gather” formulation. Following the technics

in Ref.30 and Ref.37 it can be proved (see Ref.23) Lp bounds such as those of (2.6)

with s = 0:

|u−Πh,g(u)|0,p,IRd ≤ C(h0|u|1,p,IRd + (1 +

∆x

h0)dq

(∆x)m

hm0||u||m,p,IRd

), (3.1)

where we have supposed that ~∇h is uniformly bounded and there exists two positive

constants C1 and C2 such that

C1 ≤h(x)

h0≤ C2.

To get bounds on the derivative which do not need that ~∇h→ 0 we define ~∇hΠh,g,

a new approximation of the derivative. We first smooth the derivative and then we

approximate the corresponding integral :

< f(x) >~∇g:=<~∇f(x) >g =

∫~∇f(y)W (x− y, h(x))dy =

∫f(y)~∇W (x− y, h(x))dy,

~∇hΠh,g(u)(x) :=∑i∈P

wif(xi)~∇W (x− xi, h(x))

=< Π(u)(x) >~∇g=(f(.), ~∇W (x− ., h(x)

)h

~∇hΠh,g(u)(x) clearly is not the exact derivative of Πh,g(u). We split the approxi-

mation error in the following way :

~∇hΠh,g(u)− ~∇u = < Π(u)(x) >~∇g − < u(x) >~∇g + < u(x) >~∇g −~∇u

= < (Π(u)− u) (x) >~∇g + < ~∇u(x) >g −~∇u


The first term < (Π(u)− u) (x) >~∇g is controlled by the accuracy of the quadrature

formula while we have a weight of order 1hm+10

due to the regularization, the second

term < ~∇u(x) >g −~∇u is O(h0). We finally obtain the following bounds on the

derivatives :

|~∇u− ~∇hΠh,g(u)|0,p,IRd ≤ C(h0|u|2,p,IRd + (1 +

∆x

h0)dq

(∆x)m

hm+10

||u||m+1,p,IRd

),

(3.2)

which generalize those in (2.6).

3.2. Particle approximation of the model PDE with variable smoothing

length

Armed with this particle approximation with variable smoothing length together

with the general setting of section 2.2.3 we can design particle scheme which fulfill

the global conservativity property. We just need to design an operator Dh,S which

approximates strongly the classical derivative and which satisfies Dh,S1 = 0. Let

us define

Dh,Sgϕi := ~∇hΠh,g(ϕ)i − ϕi~∇hΠh,g(1)i =∑j∈P

wj (ϕj − ϕi) ~∇Wij,i, (3.3)

where ~∇Wij,i = gradxW (xi − xj , hi).Thanks to estimates (3.2), Dh,Sg satisfies the consistency condition (ii) of The-

orem 2.1 and Dh,Sg1 = 0. The resulting scheme satisfies the hypothesis of Theorem

2.1 and also the global conservation relation (2.17). A simple calculation based

upon (2.14) proves that

D∗h,Sg (Ψ)i =∑j∈P

wj(Ψi~∇Wij,i −Ψj

~∇Wji,i),

which also reads, since we have ~∇Wij,j = −~∇Wji,i (note that W (x, h) = W (−x, h)):

D∗h,Sg (Ψ)i =∑j∈P

wj(Ψi~∇Wij,i + Ψj

~∇Wij,j),

which leads to the final form of the appropriate generalization of the scheme (2.11):

d

dt(wiΦi) + wi

∑j∈P

wj(Fj ~∇Wij,j + Fi~∇Wij,i) = wiSi. (3.4)

We summarize these results in the following :


=∑j∈P


(Φj(t))j∈P of regular functions of t defined by the system of ordinary differential

equations (3.4). Then the approximate solution Φ∆

satisfies the global conservation

property (2.17).


We suppose that the function Φ∆

converges boundedly almost everywhere to Φ

when h goes to zero (the ratio (∆x)m

hm+1 also goes to zero). Then Φ is a weak solution

of the model PDE (2.1) in the sense of definition (2.13).

These results have a straightforward extension when we add a suitable artificial

viscosity term, and also when we consider tensor valued smoothing length allowing

variable resolution according to the space direction. These points are detailed in

Ref.23.

Comments It is important to notice the duality of ~∇Wij,i and ~∇Wij,j in the

formula (3.4). The first one - ~∇Wij,i - stands for the “gather” derivative as stated

in (3.3), while the second is referred as a “scatter” derivative. The adjoint operator

−D∗h,Sg naturally introduces a mixing of scatter and gather formulations. This

justify in some way the classical trick used by lot of the SPH practitioners which

consist in taking ~∇Wij := gradxW (xi − xj , hi+hi2 ) in (3.4) instead of ~∇Wij,i and

~∇Wij,j .

3.3. Renormalization

Renormalization is a technic recently appeared in SPH literature (Ref.36 and

Ref.21), it is supposed to improve accuracy of the method. We prove here that,

with the help of the general setting of section 2.2.3 it is also conservative in the

sense of (2.17). All the approximation and convergence results can be extended

by using renormalized particle weighted approximation, this is precisely studied in

Ref.43. In particular we are able to relax the assumption that the ratio ∆xh goes

to zero, and we just need that ∆xh = O(1). We also refer to the numerical tests in

Ref.46 ( due to N. Lanson) which prove that classical SPH scheme converge to a

wrong solution unless we take well chosen value of the ratio ∆xh in order to almost

satisfy condition (ii) of Theorem 2.1. We briefly resume the results.

Formally, renormalization is a tool which provides new formulae for Dh,Sf(x)

with the help of a weight matrix ( the renormalization matrix) in the following way:

Dh,Sf(x) := B(x).~∇hΠhg (f)(x)− f(x)B(x)~∇hΠh

g (1)(x).

We aim to increase the accuracy, thus instead of P0 ⊂ ker(Dh,S) we ask for Dh,Sf =~∇f for any polynomial f ∈ P1. It can be easily proved that

Proposition 3.2 We have Dh,Sf = ~∇f for any polynomial f ∈ P1, if and only if

B(x) = E(x)−1 with

E(x)αβ =∑j∈P

wj(xβj − x

β) ∂αW (x, xj).

We thus have E(x)αβ = ∂α,hΠhg (xβ) − xβ∂α,hΠh

g (1). Approximation results

(3.2) easily prove that E(x)αβ ' δαβ and consequently that B(x) makes sense, if(∆x)h −→ 0. More precisely it can be proved (see Ref.43) that:


Proposition 3.3 Let us suppose that B is uniformly bounded (with respect to h0

and ∆x) then we have :

‖Dh,Sϕ(x)−Dϕ(x)‖ ≤ Ch0‖B(x)‖‖D2ϕ‖∞,

where h0 is the characteristic scale of the smoothing length (i.e. η−h0 ≤ h ≤ η+h0

with η− and η+ two constants > 0 ).

The consistency of the method is thus satisfied at the only condition that the

smoothing length goes to zero. Moreover, it can be proved that, if ∆xh is bounded

and if the initial distribution of particles is regular enough the matrix B(x) is

uniformly bounded and that :∣∣(ϕ,Dh,S(ϕ))h∣∣ =

∣∣∣(ϕ,D∗h,S(ϕ))h

∣∣∣ ≤ C‖ϕ‖2h.This also insure stability and convergence of the method (at least in the linear

case of symmetric first order systems). The discrete operators are defined according

to :Dh,Sfi =

∑j∈P

wj(fj − fi)Bi.~∇Wij ,

D∗h,Sfi =∑j∈P

wj(fiBi.~∇Wij − fjBj .~∇Wji),(3.5)

and the scheme is :

d

dt(wiΦi) + wi

∑j∈P

wj(fiBi.~∇Wij − fjBj .~∇Wji) = wiSi.

We refer to Ref.43 for more details (see Ref.46 also for some numerical results).

The resulting method turns out to be more robust than standard methods, and

also less expensive since we can use higher values of the ratio (∆x)h . This makes

decrease the number of neighbors (a factor 2 or 3 is possible) and the cost of the

method also decrease proportionally. This is particularly true in situation with

complex physics since the additional cost due to the computation of renormalization

matrices is no more than the computation of an additional physical unknown.

4. Godunov type Particle Approximations

We develop here an alternative to artificial viscosity. Use of Riemann solvers as been

successful in the field of Finite Difference schemes and Finite Volume scheme to

increase the robustness of numerical methods. We give a new formalism of particle

weighted methods which includes such Riemann solvers ; an improved stability is

expected. We refer to Ref.35 and Ref.45 for numerical tests.

4.1. Use of Riemann solvers and finite difference fluxes

In the following we suppose that the kernel function is radial symmetric, then

we have :~∇xW (xi − xj) = −Dθijnij = ~∇Wij ,


where

nij =xj − xi‖xj − xi‖

, Dθij = Dθ(‖xi − xj‖).

We also suppose for simplicity, that S ≡ 0. Thus equation (2.11) reads as :

d

dt(wiΦi)− wi

∑j∈P

wj (Fi + Fj) .nijDθij = 0.

We thus compute the evolution of wiΦi by summing up together the interactions

of the particle located at xi with its neighboring particles xj . These interactions

are computed along the direction nij connecting xi with xj . All these features

introduces naturally at xij =xi + xj

2, the conservation law related to the direction

nij :∂

∂t(Φ) +

∂

∂x(F (xij , t,Φ).nij) = 0. (4.1)

Therefore it is natural to introduce a 1-dimensional finite difference scheme in con-

servation form associated to (4.1), which brings a sufficient numerical viscosity. Such

a scheme consists in replacing the centered approximation (F (Φi) + F (Φj)).nij by

the numerical flux of a Finite Difference scheme 2g(nij ,Φi,Φj), which is required

to satisfy :(i) g(n, u, u) = F (u).n,(ii) g(n, u, v) = −g(−n, v, u).

The numerical viscosity Q(n, u, v) is classically defined in the scalar case (i.e.

Φ ∈ IR) as :

Q(n, u, v) =F (u).n− 2g(n, u, v) + F (v).n

v − u.

In the case of an explicit particle scheme the connection with finite 1D difference

schemes is rather precise. We can establish in a way similar to the one used for finite

volume schemes (see for example Ref.4) that the particle scheme is a convex combi-

nation of 1D finite difference schemes at which we add an error term corresponding

with the residual term ~∇(Πh1). These results are detailed in Ref.5 where we ana-

lyze with B. Ben Moussa, the convergence of explicit in time, scalar, discretization

of (4.2). We emphasize that regularity assumptions on the transport field v are

essential ingredients in the convergence results together with the condition relating

the mean distance ∆x between particles and the characteristic size h of the kernel

(cf. section 2.1.2).

There is a lot of numerical flux well suited for such upwinding, among them we

can quote the Lax Friedrichs and the Godunov schemes. They are connected with

the notion of Riemann problem associated to the conservation law (4.1). For exam-

ple the Godunov scheme is such that g(n, u, v) = F (w(0, u, v)).n, where w(xt , u, v)

is the solution of the Cauchy problem for (4.1) with the initial data :

Φ(x, 0) =

u if x < 0,v if x > 0.


In the scalar case, they are monotone finite difference schemes (see Crandall and

Majda 14 and Kuznetsov and Volosin 22) , they also belong to the widest class of

E-schemes (Osher 34).

By introducing the numerical viscosity our numerical scheme, which consists in

finding functions t ∈ IR+ −→ ui(t) ∈ IR, i ∈ P solutions of the differential system :

d

dt(wiΦi)− wi

∑j∈P

wl2g(nij ,Φi,Φj)Dθij = 0, Φi(0) = Φ0(ξi) (4.2)

also reads as :

d

dt(wiΦi) + wi

∑j∈P

wj(F (xij , t,Φi) + F (xij , t,Φj)).nij +Q(nij ,Φi,Φj)(Φi − Φj)Dθij = 0,

Φi(0) = Φ0(ξi).

This form is very closed to the one classically used in SPH literature (for example,

artificial viscosity in the momentum equation - see section 5.2 for details -)

4.2. Higher order version of the method

In the field of Particle methods a classical tool to increase accuracy is to increase

the smoothness of the kernel together with making equal to zero its momentum.

This approach has been used in SPH calculations, for example by using a “super

Gaussian” kernel. To overcome numerical difficulties due to the use of artificial

viscosity - for example the sensitivity of the numerical results to the value of the

coefficients α and β -, we have introduced in the previous section an alternative to

the classical artificial viscosity. The increase of accuracy by modifying the kernel

introduce some instabilities due to the non positiveness of the kernel. Thus, we pro-

pose a different method based upon the well known techniques of MUSCL schemes

developed by Van Leer41 for finite difference schemes in the end of seventies, and

extended more recently to finite volume methods (see Ref.42 for presentation of the

method and Ref.4 for a mathematical analysis).

The idea is to take account of informations given by ~∇Πh(Φ) to compute a nu-

merical flux which increases the accuracy of the method :

We replace the flux g(nij ,Φi,Φj) which approximates the flux of the conser-

vation law (4.1) located at xij by g(nij ,Φij ,Φji). Φij is an approximation of Φ at

xij given by a first order Taylor expansion from the point xi :

Φij = Φi + ~∇(Πh(Φ))i.(xij − xi).

It is well known in the field of MUSCL finite difference schemes that such an approx-

imation leads to unstable schemes unless we introduce a limitation of the derivative~∇(Πh(Φ))i of the unknown used to compute the values at xij . We propose to start

from the value :~∇h(Φl)i = ~∇(Πh(Φl))i =

∑j∈P

wjΦlj~∇Wij ,


and then to make a loop over all the neighboring particles such that each component

of the gradient is reduced in a way that for all neighboring particles j :Φlij − Φli = λlij(Φ

lj − Φli),

with 0 ≤ λlij ≤ 1,

where the interface values at xij have been computed with help of the limited

gradient according to :

Φlij = Φli + ~∇h(Φl)i.(xij − xi).

In the scalar case it can be proven that the method is L∞ stable and convergent

with suitable assumptions on the numerical flux.

5. Application to Euler equations

5.1. Basic principle

Thanks to the set of approximation rules described in the previous section we are

now able to design the SPH approximation of Euler equations of a compressible

fluid. We start by the simplest case of a single compressible gas. Such a fluid

satisfy the following Euler system of equations :

Lv(Φ) +∑l=1,d

∂

∂xl(F l(Φ)) = 0, (5.1)

where Φ, the vector of conservative variables and the fluxes F l are given by :

(i) Φ =

ρρv1

ρv2

E

, (ii) F 1(Φ) =

0p0v1p

, (iii) F 2(Φ) =

00pv2p

.

We have supposed for simplicity that the dimension of space is d = 2. The equation

of state of the fluid gives the pressure as a function p(ρ, u) of the density and the

internal energy. The total energy E is defined by : E = ρ(u+1

2‖~v‖2).

This system as the same form that the model PDE (2.1). We use the following

particle approximation :

Lv(ΠΦ) + Π

∑l=1,d

F l(Φ)∂Πh(1)

∂xl+∂Πh(F l(Φ))

∂xl

= 0,

to get the system of ordinary differential equations defined for i ∈ P by :


(i)d~xidt

= ~vi, (ii)d

dt(wiρi) = 0,

(iii)d

dt(wiρi~vi) + wi

∑j∈P

wj(pi + pj)~∇Wij = 0,

(iv)d

dt(wiEi) + wi

∑j∈P

wj(pi~vi + pj ~vj).~∇Wij = 0.

(5.2)

Note that we need to add suitable initial conditions to this system. The equation

(ii) in (5.2) gives :

wi(t)ρi(t) = cst ≡ mi.

The quantity mi is constant with the time, its dimension is a mass, thus it is

natural to call it the mass of the particle i. Taking account of that in the two

others equations, we get finally :d

dt(~vi) = −

∑j∈P

mj(pi + pjρiρj

)~∇Wij ,

d

dt(ui) = −

∑j∈P

mjpjρiρj

(~vj − ~vi).~∇Wij .(5.3)

This set of equation is not the standard set of equation used in SPH codes.

Indeed, it works generally well. Furthermore Theorem 2.1 insures consistency with

weak solutions of Euler equations for limit solution obtained with this scheme under

reasonable hypothesis. The classical formulation of SPH ( Refs.31’7’38) is recovered

in a slightly different fashion. We first remark that momentum and energy conser-

vation equations could be written as : Lv(ρ~v) = −~∇p = −ρ~∇(p

ρ)− p

ρ~∇ρ,

Lv(ρu) = −p div(~v) = −pρ

(div(ρ~v)− ~v.~∇ρ).

Thus, we define a particle approximation of these 2 equations together with the

conservation of mass, by :Lv(Π(ρ)) = 0,

Lv(Π(ρ~v)) = Π

(−ρ~∇Πh(

p

ρ)− p

ρ~∇Πh(ρ)

),

Lv(Π(ρu)) = −Π

(p

ρ(div(Πh(ρ~v))− ~v.~∇Πh(ρ)

).

We then, obtain the system of o.d.e. :d

dt(~vi) = −

∑j∈P

mj(piρ2i

+pjρ2j

)~∇Wij ,

d

dt(ui) = − pi

ρ2i

∑j∈P

mj(~vj − ~vi).~∇Wij .


This is the standard form of SPH given in most of papers. We obtain here again,

consistency of limit solutions with weak solutions of Euler equations by using the

extension of Theorem 2.1 to the equation (2.21) with H = ρ, F i = 1ρF

i . Note that

we need additional assumption on the regularity of the function ρ (ρ ∈W 2,∞)).

To close the system of equations it remains to give an equation for the evolution

of the density ρ. We find in the SPH literature two formulae :

• the first one consists in using the smoothed particle approximation of the

density function ρ(x) ≈∑j∈P

wjρjW (x−xj) which leads, taking account of the

mass conservation, to :

ρi =∑j∈P

mjWij , (5.4)

• the second one consists in solving the o.d.e. :

d

dt(ρj) = −

∑i∈P

mi(~vj − ~vi).~∇Wij ,

which can be seen as an approximation of the mass conservation written as :

d

dt(ρ) = −ρdiv~v, (5.5)

whered

dtis the Lagrangian derivation.

Looking more carefully at these two kind of approximations we are able to

establish a rather precise connection between the two approaches. Let us recall

that according to the basis developed in the first section we must make change the

weight wi of the particles according to the equation (2.8)(ii). Taking account that

the mass of particle is constant, we get :

d

dt(ρj) = −ρjdiv~v. (5.6)

Thus we have a kind of discretization of the mass conservation equation written

in the form (5.5). We need to define an approximation of ρdiv~v at point xj . One

possible choice is the following :

ρjdiv~vj = (divΠh(ρ~v)(xj)− ~vj .~∇Πh(ρ(xj))) =∑i∈P

mi(~vj − ~vi).~∇Wij . (5.7)

We finally obtain :d

dt(ρj) = −

∑i∈P

mi(~vj − ~vi).~∇Wij . (5.8)

Let us recall that the velocity and the coordinates of the particles moves accord-

ing tod~xidt

= ~vi .Taking account of this equation into (5.8), we obtain :

d

dt(ρi) =

d

dt

∑j∈P

mjW (xi(t)− xj(t), h)

, (5.9)


which leads by direct integration to :

ρi(t) = ρi(0) +∑j∈P

mj(Wij −W (xi(0)− xj(0))), (5.10)

which reduces to (5.4) when the initial distribution of density is chosen so that :

ρi(0) =∑j∈P

mjW (xi(0)− xj(0)).

In practical codes we have to make a time discretization. SPH codes use classi-

cal explicit in time discretization of ordinary differential equation, thus we should

obtain different result by using formula (5.4) which is a primitive of the o.d.e. (5.9),

and consequently could be related to a kind of implicit discretization.

It generally appears that formulation (5.4) is more robust than (5.9), in particular

in situation with strong shocks. Nevertheless (5.9) is generally used in situation

with free surfaces, since it avoids the automatic smoothing of the density towards

the value 0 produced by the formula (5.4). We remark that such an implicit dis-

cretization could be done even in the case of free surfaces by using (5.10).

5.2. Artificial viscosity and entropy condition

As we have previously said it is necessary to add some artificial viscosity, mainly

when we want to deal with shocks. The more popular way to add such contribution

in SPH is to take a pseudo-viscous pressure of the type proposed by Von Neumann

and Richtmeyer39, p is modified in p+ Πv, with :

Πv =

βρl2(div(~v))2 − αρlc div(~v) if div(~v) < 0,0 elsewhere,

(5.11)

where α and β are nondimensional coefficients whose value is of order 1, c is the

sound velocity and l is the characteristic width of the shock we want to compute

( of order ∆x or h). We refer to the paper of Noh33 for an exhaustive study of

different types of artificial viscosity, and to the paper by Monaghan and alii 25 for

some comparison in a frame specific to SPH, see also the recent paper by Cottet13

where some related artificial viscosity terms are presented.

We investigate here the effect of introducing in SPH formalism this “pseudo-

viscous pressure” type artificial viscosity. At the continuous level, this approach

consists in solving modified equations :Lv(ρ) = 0,

Lv(ρ~v) = −~∇(p+ Πv),Lv(ρu) = −(p+ Πv) div(~v),

where the viscous pressure Πv is given by a (5.11) type formula. The effective

introduction of the artificial viscosity is provided by substituting to pi (resp. pj)


pi + 12ρ

2iΠij (resp. pj + 1

2ρ2jΠij) and modifying equations (5.3) as it follows :

d

dt(~vi) = −

∑j∈P

mj(piρ2i

+pjρ2j

+ Πij)~∇Wij ,

d

dt(ui) = − pi

ρ2i

∑j∈P

mj(~vj − ~vi).~∇Wij −1

2

∑j∈P

mjΠij(~vj − ~vi).~∇Wij .(5.12)

In these equations Πij is the artificial viscosity term. It is defined as :

Πij =

µij(βµij − αc)1/2(ρi + ρj)

if (~vi − ~vj).(~xi − ~xj) < 0,

0 elsewhere,(5.13)

where c is the mean sound velocity and µij given by :

µij =h(~vi − ~vj).(~xi − ~xj)|~xi − ~xj |2 + εh2

,

with ε 1. We thus have build an approximation of (5.11). This is perfectly rigor-

ous in the one-dimensional case. Such a formula which has been initially proposed by

Monaghan25 is also perfectly rigorous when applied to compute the second deriva-

tive of isotropic scalar functions. This has been used by Monaghan31 for thermal

diffusion computations, and Raviart and Mas-Gallic29, (see also Degond and Mas-

Gallic10) have provided a mathematical justification of these formulae in such scalar

isotropic cases. A more detailed study of the approximation of this pseudo-viscous

pressure leads to the introduction of non diagonal diffusion tensor (see Ref.13).

Although it is not possible to prove that the usual choice of Πij provides us a

consistent approximation of the pseudo-viscous pressure, nevertheless this approxi-

mation has a very nice property which concerns entropy production :

Proposition 5.1 Let us suppose that the kernel W satisfies W ′(x) ≤ 0 for x ≥ 0,

then the continuous field (Φ) associated with the solution of (5.12) satisfy the second

principle of the thermodynamics Lv(Π(ρs)) ≥ 0 or equivalently Tidsidt≥ 0 where s

is the specific entropy.

Proof. We combine the energy equation in (5.12) with the definition of the specific

entropy s :

Tds = du− p

ρ2dρ,

to get (see Gingold and Monaghan 15 for related computations) :

Tidsidt

= −1

2

∑j∈P

mjΠij(~vj − ~vi).~∇Wij .

Now taking a kernel W (as the B-spline of the first section) such that W ′(x) ≤ 0

for x ≥ 0, we get easily that with the choice (5.13) for Πij we have :

Πij(~vj − ~vi).~∇Wij ≤ 0,

and the desired result follows.2


Comments. As a consequence of this result we have the analogous of Theorem

2.1, in the sense that if the numerical solution (ρs)∆ as a limit (ρs), this limit satisfy

the entropy inequality.

A similar result still holds for the formulation (5.3) with artificial viscosity

(pi + pjρiρj

+ Πij). To get it, we need to modify the approximation of div~v in (5.8) in

the following way :

d

dt(ρj) = −

∑i∈P

mipjρipiρj

(~vj − ~vi).~∇Wij .

In some sense the hypothesis on the kernel limits the accuracy of the method

since “super Gaussian” kernels and in a general manner any kernel with a second

momentum equal to zero does not satisfy this requirement. It is also a numerical

evidence (see numerical tests in Ref.15) that the use of such kernels may lead to

development of instabilities.

5.3. Some conservation properties

As we have previously said, taking symmetric interaction between particles al-

lows automatically the mass, momentum and energy to be conserved. Let us take

the following approximation of integrals :∫IRd

ρdx ≈∑i∈P

wiρi =∑i∈P

mi,

∫IRd

ρ~vdx ≈∑i∈P

wiρi~vi =∑i∈P

mi~vi,∫IRd

(1

2ρ‖~v‖2 + ρu)dx ≈

∑i∈P

wi(1

2ρi‖~vi‖2 + ρiui) =

∑i∈P

mi(1

2‖~vi‖2 + ui).

We easily verify, taking account of the identity (2.10) into the system (5.12) includ-

ing artificial viscosity terms, that we have conservation of quantities approximating

globally the mass , the momentum, the angular momentum and the total energy of

the system :

d

dt

∑i∈P

mi = 0,d

dt

∑i∈P

mivi = 0,

d

dt

∑i∈P

mivi × ri = 0,d

dt

∑i∈P

mi(1

2‖~vi‖2 + ui) = 0.

5.4. Use of Riemann solvers

5.4.1. Exact Riemann solver

Instead of using artificial viscosity of section 5.2, we can use techniques of section

4.1 based upon Godunov type schemes and Riemann solvers. This approach is

connected with the notion of Arbitrary Lagrange Euler approximation (A.L.E.), it


is also in some sense a generalization of ideas develop by Harten and Hymann 18 in

their work on self adjusting grid methods for conservation laws.

Let us give a regular transport field v0(x, t). We then consider the following

conservative form of Euler equations in 2D (for simplicity)

Lv0(Φ) +∑i=1,d

∂

∂xl(F lE(Φ)− v0,lΦ) = 0,

where the fluxes F lE are given by :

(i) F 1E(Φ) =

ρv1

p+ ρ(v1)2

ρv1v2

v1(p+ E)

, (ii) F 2E(Φ) =

ρv2

ρv1v2

p+ ρ(v2)2

v2(p+ E)

.

We thus have to solve between each particle i and j, the Riemann problem :∂

∂t(Φ) +

∂

∂x((FE(Φ).nij − v0(xij , t).nijΦ)) = 0,

Φ(x, 0) =

Φi if x < 0,Φj if x > 0

,(5.14)

Let us consider the classical Riemann problem for Euler equations :∂

∂t(Φ) +

∂

∂x(FE(Φ).nij) = 0,

Φ(x, 0) =

Φi if x < 0,Φj if x > 0

,(5.15)

and let us denote by ΦE(x

t; Φi,Φj) the solution of this problem. An easy calculation

proves that the solution of ( 5.14) is given by :Φ = ΦE(

x+X0(t)

t; Φi,Φj),

X0(t) =

∫ t

0

v0(xij , τ).nijdτ.(5.16)

It follows that a reasonable choice for the flux gE(nij ,Φi,Φj) of the Godunov

scheme associated with our Smooth Particle approximation is :λ0ij = v0(xij , t).nij ,

Φij(λ0ij) = ΦE(λ0

ij ; Φi,Φj),GE(Φi,Φj) = FE(Φij(λ

0ij))− v0(xij , t)⊗ Φij(λ

0ij),

gE(nij ,Φi,Φj) = GE(Φi,Φj).nij .

The resulting particle approximation is given by :d

dt(xi) = v0(xi, t),

d

dt(wi) = widiv(v0(xi, t)),

d

dt(wiΦi) + wi

∑j∈P

wj2GE(Φi,Φj)~∇iWij = 0, Φi(0) = Φ0(ξi).


The detailed equations for mass, momentum and total energy conservation are :

d

dt(wiρi) + wi

∑j∈P

wj2ρ0E,ij(v

0E,ij − v0(xij , t)).~∇Wij = 0,

d

dt(wiρivi) + wi

∑j∈P

wj2[ρ0E,ijv

0E,ij ⊗ (v0

E,ij − v0(xij , t)) + p0E,ij

].~∇Wij = 0,

d

dt(wiρiEi) + wi

∑j∈P

wj2[E0E,ij(v

0E,ij − v0(xij , t)) + p0

E,ijv0E,ij

].~∇Wij = 0,

where (ρ0E,ij , ρ

0E,ijv

0E,ij , E

0E,ij)

T = Φij(λ0ij).

5.4.2. Approximate Riemann solver

Instead of using the Godunov scheme we could use approximate Riemann solvers

such as those developed by Roe40, see also Ref.1 and Ref.44 for generalization to

real gases, Harten - Lax - Van Leer17, Osher34.

Let Φa(xt ; Φi,Φj) the approximate solution of the Riemann problem (5.15) given

by such a solver. Associated with this approximate Riemann solver we generally

have two real functions, σ−(Φi,Φj) and σ+(Φi,Φj) such that :

Φa(x

t; Φi,Φj) =

Φi if x

t ≤ σ−(Φi,Φj),Φj if x

t ≥ σ+(Φi,Φj).

It follows from (5.16) that Φa(x+X0(t)

t; Φi,Φj) is also a good approximation of

the “moving” Riemann problem (5.14). The resulting numerical flux is thus given

by :

ga(nij ,Φi,Φj) =

[F (Φi)− σ−(Φi,Φj)Φi −

∫ λ0ij

σ−(Φi,Φj)

Φa(s; Φi,Φj)ds

].nij

=

[F (Φj)− σ+(Φi,Φj)Φj +

∫ σ+(Φi,Φj)

λ0ij

Φa(s; Φi,Φj)ds

].nij

Comments Even if we take the transport field equal to the velocity of the par-

ticles, the mass conservation do not keep wiρi = mi constant as in the standard

method developed in the previous paragraph. Although we loose this nice property

of the method, we keep global conservation of mass, momentum and energy. We

also expect more robustness since we have convergence results in the scalar case

(see section 4.1).

We believe that it is sometimes quite essential to move the particles with a

smoother velocity field than the exact velocity together for theoretical and compu-

tational reasons (the XSPH variant of the method introduced by Monaghan31 moves

the particle with smooth velocity, see also Ref.13 for connection of the smoothing

of the velocity with the artificial viscosity). In this sense our approach is different

from the one developed by Bicknell9.


Numerical experiments with these schemes, including 1D an 2D comparisons

with standard SPH codes can be found in Ref.45 and Ref.35.

6. Particle Formulation of Boundary Conditions

6.1. Particle approximation on a bounded set

We deal here with the particle approximation of a function f defined over an open

bounded set Ω in IRd. Let us denote by P (Ω), the set of index of the particles of Ω.

The weights associated with each particle need to define a good quadrature formula

over Ω. We define the approximation ΠΩ in the following way :

ΠΩ(f)(x) =∑

i∈P (Ω)

wi(t)f(xi(t))δ(x− xi(t)).

The consistency of this approximation is satisfied if and only if :∑i∈P (Ω)

wi(t)f(xi(t)) ≈∫

Ω

f(x)dx.

This property will remain true if we move the particles with the field ~v together

with modifying the weights according to :

d

dt(wj) = wj div(~v).

From a practical point of view we need to insure initially the accuracy of the quadra-

ture formula. In most of practical computations the particles are initially distributed

on a regular grid (for instance cubic grids) and it is quite easy to find suitable weights

and positions. A minimum rule could be the following : find some control volume

for each particle close to the boundary, take the mid point formula for quadrature

and move the particle at this point. This choice is not very accurate but sufficient

in most of cases.

As in the unbounded case it is useful to introduce the discrete scalar product

(ϕ,Ψ)h,Ω :=∑

i∈P (Ω)

wiϕi.Ψi =

∫Ω

ϕ∆.Ψ∆dx, (6.1)

which approximates the standard one in L2(Ω).

To define a Smoothed Particle approximation on a bounded domain requires a

convolution by a regularizing kernel :

ΠhΩ(f)(x) =

∑i∈P (Ω)

wi(t)f(xi(t))W (x− xi(t)) = ΠΩ(f) ∗W (x).

Different approaches are then possible to define particle approximations which takes

account of boundary conditions, we shall detail three solutions :


- the classical approach of ghost particles. This very well known approach for

plane boundaries is generalized to the case of general curved boundaries. We

thus propose new treatments of polyhedral boundaries as those we encounter

in industrial problems. This will be detailed in the section 6.3.1 .

- a technique based on boundary particles and boundary forces, which is related

to the approach proposed by Monaghan 31. Our general tool deals with the

consistency of the method at the boundary.

- a semi-analytic approach which uses approximation of integrals of the kernel

and its derivatives. This approach is detailed in the section 6.3.3. It was

first proposed by Benz 7, (see also Herand 19), we give here a more general

treatment which in particular deals with some difficulties due to the boundary

conditions at free surfaces.

We finally refer to Ref.6 where B. Ben Moussa studies in the scalar nonlinear

case the convergence of the particle approximation on bounded domains developed

according to the ideas of the following sections. In the next sections, when it is not

ambiguous we omit the dependance in Ω of the set P (Ω) and we use P instead of

P (Ω).

Before going further we need to analyze the extension of approximation results on

unbounded domains to bounded domains. As a summary we can say that estimates

(2.6) remains true on a bounded domain as soon as we consider function with a

support strictly included in the domain. Let us clarify this point :

Let us first suppose that we have a local coordinate system (x, y) in the neigh-

borhood of ∂Ω such that x = x− yn(x) with (x, y) ∈ ∂Ω× [0, δ0], for some δ0 > 0.

We then introduce for δ0 > δ > 0, χδ(x(x, y)) ∈ D(Ω) a regularization of the

characteristic function of the set Ω such that :

χδ(x(x, y)) =

0 0 ≤ y ≤ δ2 ,

Ψ(y) δ2 ≤ y ≤ δ,

1 y ≥ δ,(6.2)

where Ψ is an increasing function which satisfies 0 ≤ Ψ(y) ≤ 1. We can also

consider χδ as a function of D(IRd). We recall that the kernel W is supposed having

a compact support, precisely we state that spt(W ) ⊂ B(0, C0h) where B(0, R) =

x ∈ IRd; ||x|| ≤ R and C0 is some positive constant.

The estimates (2.6) are direct consequences of basic results stated in Ref.37 :

||u−Π(u)||−m,p,IRd ≤ C(∆x)m||u||m,p,IRd ,

|u− u ∗W |0,p,IRd ≤ Chr|u|r,p,IRd ,

||f ∗ g||Lp(IRd) ≤ C||f ||−m,p,IRd ||g||m,1,IRd .

The proof of the first one relies on the fact that the particles xi of weight wiprovide a good quadrature formula over IRd. This is still true over a bounded


domain Ω. Arguing as in the proof of Lemma 8 in Ref.30 it follows that for any

u ∈Wm,p0 (Ω), m > d, 1 ≤ p ≤ ∞ :

||u−ΠΩ(u)||−m,p,Ω ≤ C(∆x)m||u||m,p,Ω.

For u ∈Wm,p(Ω), considering uχδ as a function of Wm,p(IRd) we get :

|uχδ − uχδ ∗W |0,p,Ω = |uχδ − uχδ ∗W |0,p,IRd ≤ Chr|uχδ|r,p,IRd ≤ C(δ)hr||u||r,p,Ω.

Let us now split uχδ −ΠhΩuχ

δ as :

uχδ −ΠhΩuχ

δ = uχδ − uχδ ∗W + uχδ ∗W −ΠΩuχδ ∗W

= uχδ − uχδ ∗W + (uχδ −ΠΩuχδ) ∗W.

Since we have that spt(W ) ⊂ B(0, C0h), then spt(ΠhΩuχ

δ) ⊂ Ω\Ω[0,δ−C0h)] where

Ω]α,β] = x = x− yn(x) ∈ IRd;α < y ≤ β,

and

|uχδ −ΠhΩuχ

δ|m,p,Ω = |uχδ −ΠhΩuχ

δ|m,p,IRd≤ |uχδ − uχδ ∗W |m,p,IRd + |(uχδ −ΠΩuχ

δ) ∗W |m,p,IRd≤ C(δ)hr||u||m+r,p,Ω + ||uχδ −ΠΩuχ

δ||−m,p,IRd ||W ||m,1,IRd≤ C(δ)hr||u||m+r,p,Ω + C(∆x)m||uχδ||m,p,IRd ||W ||m,1,IRd

≤ C(δ)hr||u||m+r,p,Ω + C(δ)(∆x)m

hm+1||u||m,p,Ω.

We thus get the desired estimate when m > d. We obtain the general result for

m ≤ d by arguing as in the proof of Theorem 5.1 of Ref.37 (taking account of that

W as a compact support). Finally we get with the same hypothesis as in (2.6) :

For any u ∈Wµ,p(Ω), s ≥ 0, µ = max(r + s,m), r = 1, 2,d

m≤ p ≤ ∞,

|uχδ −ΠhΩuχ

δ|s,p,Ω ≤ C(δ)(hr||u||r+s,p,Ω + (1 +∆x

h)dq

(∆x)m

hm+s||u||m,p,Ω).

(6.3)

6.2. Particle approximation of a model PDE on a bounded domain

We consider the model PDE of section 2.2, on a bounded domain Ω. To get

uniqueness of solution we need suitable boundary conditions on the boundary ∂Ω.

In the general non linear case the mathematical problem of defining such a boundary

condition is a difficult task. For scalar conservation laws uniqueness is achieved

through a suitable entropy condition at the boundary (see Ref.3, and Ref.4 for a

convergence proof in the case of Finite Volume schemes), for more general equations

we refer to the work of Leroux 27 and the more recent works by Audounet and Mazet


2 and Lefloch and Dubois 12. All these techniques lead to weak formulations of the

boundary condition. Here we are just interested in the designing of particle methods

suitable to approximate such a weak solution, thus we limit ourselves to a model

situation where the weak solution of the problem :Lv(Φ) + divF (x, t,Φ) = S for x ∈ Ω,F (x, t,Φ).n = g(x, t,Φ).n for x ∈ ∂Ω,

(6.4)

is defined as :

∀ϕ ∈ C2(Ω× IR+,∗),∫Ω×IR+

(Φ.L∗v(ϕ) + F (x, t,Φ).~∇(ϕ) + S.ϕ

)dxdt

−∫∂Ω×IR+

g(x, t,Φ).nϕdσ(x)dt = 0

(6.5)

We have supposed here that the transport field v is such that v.n = 0 at the

boundary ∂Ω, otherwise, additional terms must be needed in the weak formulation

(6.5), we do not detail these points, we just point out that all the technics developed

here can be extended to these situation.

In the unbounded case, the unknown Φ, evolves at each particle according to the

derivative of the field F which acts as a volume source term. To take account of any

boundary condition on the flux F we need to compute a specific volume source term.

To this ends, let us first suppose that we get a regularized approximation, Gh,Ω, of

the boundary flux g.n in the sense that (when h −→ 0) : ∀ϕ ∈ C2(Ω× IR+,∗) :

(Gh,Ω, ϕ)∆,Ω −→∫∂Ω

g(x, t,Φ).n ϕdx. (6.6)

This property means that we replace a measure supported by ∂Ω with a regularized

measure supported by Ω. Three different technics are proposed in section 6.3, in

order to achieve that.

Following the general setting of section 2.2.3 it is convenient to design Φ∆

, a

particle approximation of the new system in the following way : ∀ϕ ∈ C2(Ω×IR+,∗),(Φ

∆, L∗v(ϕ)

)t∆,Ω

+(F (Φ

∆), Dh,Sϕ

)t∆,Ω

+(S +Rh(Φ

∆), ϕ)t

∆,Ω− (Gh,Ω, ϕ)

t∆,Ω = 0,

(6.7)

which yieldsd

dt(wiΦi) + wiD

∗h,S(F )i = wi(Si +Rh(Φ)i −Gi). (6.8)

We expect to deal with operators Dh,S similar to those used in the unbounded

case. For sake of generality ( we include both the standard, the gathered and the

renormalized cases) we consider Dh,S given by

Dh,Sϕi :=∑j∈P

wj(ϕj − ϕi)Aij , (6.9)


where Aij is given by ~∇Wij , ~∇Wij,i or Bi.~∇Wij,i according to the case considered.

In all cases, it can be proved ( see Ref.43) that there exists C > 0 such that

(i)∑j∈P

wj‖Aij‖ ≤C

h0, (iii)‖Aij +Aji‖ ≤

C

(h0)d,

(ii)‖∑j∈P

wjAij‖ ≤ C, (iv)Aij = 0 if ‖xi − xj‖ ≤ Ch0,(6.10)

where h0 is the characteristic scale of the smoothing length.

Note that we have added an artificial viscosity term (S → S +Rh(Φ)) as in the

scheme (2.16) of Theorem 2.1. Although we have a weak discrete formulation it is

not obvious at all that we get consistency even if condition (ii) of Theorem 2.1 is

true. The main point is to prove that

(F (Φ), Dh,Sϕ)∆,Ω →∫

Ω×IR+

F (v)~∇(ϕ)dxdt.

Let us see, at least formally and in the simple case of the standard approximation

(Aij = ~∇Wij) how it works. We have

(F (Φ), Dh,Sϕ)∆,Ω ≈∫IRd×IR+

F.(~∇(χΩϕ)− ϕ~∇(χΩ))dxdt.

The Dirac distributions in ~∇(χΩϕ) and ϕ~∇(χΩ) cancel each other in the continuous

case, and that gives the result.

We shall prove the following :


=∑j∈P


(Φj(t))j∈P of regular functions of t defined by the system of ordinary differential

equations (6.8).We suppose that :

(i) the function Φ∆

converges boundedly almost everywhere to Φ when h goes to

zero (the ratio (∆x)m

hm+1 also goes to zero) ,

(ii) Dh,S is given by (6.9), satisfies conditions (6.10) and

∀ϕ ∈[C2

0(Ω× IR+,∗)]m

supi∈P‖Dh,Sϕi −Dϕi‖ → 0 as h and ∆x→ 0,

(iii) ∀ϕ ∈ C2(Ω× IR+,∗) limh and ∆x→0

(Rh(Φ

∆), ϕ)t

∆= 0,

(iv) ∀ϕ ∈ C20(Ω× IR+) limh and ∆x→0 (Gh,Ω, ϕ)∆,Ω =

∫∂Ω

g(x, t,Φ).n ϕdx .

Then Φ is a weak solution of our model PDE in the sense of Definition (6.5).


Proof. As we have pointed it previously, the key point is the study of

(F (Φ), Dh,Sϕ)∆,Ω in (6.7). We introduce the function χδ, and we evaluate

R :=(F (Φ), Dh,Sϕ+ ϕDh,Sχ

δ −Dh,Sϕχδ)

∆,Ω.

Taking Dh,S satisfying (6.10) a simple calculation leads to :

R =∑i∈P

wiFi

∑j∈P

wj(ϕj − ϕi)(1− χδj)Aij

=∑i∈P

wiFiEi.

From (6.10)(iv) Aij = 0 if ‖xi − xj‖ ≥ C0h, and (1 − χδj) = 0 unless yj ≤ δ. We

thus have :

Ei =∑j∈P

wj(ϕj − ϕi)(1− χδj)Aij = 0,

unless 0 ≤ yi ≤ δ+C0h. In this case, we also have ‖xi−xj‖ ≤ C0h and consequently

|ϕi − ϕj | ≤ C(ϕ)h ; it follows that :

‖Ei‖ ≤ C(ϕ)h∑j∈P

wj‖Aij‖.

Taking account of the estimate (6.10)(i) we get ‖Ei‖ ≤ Ch and :

‖R‖ ≤ C∑

i∈P,0≤yi≤δ+C0h

wi‖Fi‖ ≤ Cmeas(∂Ω)(δ + h)‖F‖∞. (6.11)

We then split R in R = (F (Φ), Dh,Sϕ)∆,Ω + I2 + I3 with

I2 =(F (Φ)ϕ,Dh,Sχ

δ)

∆,Ω, I3 =

(−F (Φ), Dh,Sϕχ

δ)

∆,Ω.

ϕ and ϕχδ are regular with compact support in Ω, we thus have (as a consequence

of estimates (6.3)) Dh,Sχδ → Dχδ and Dh,Sϕχ

δ → Dϕχδ in L∞(Ω) when h and

∆x −→ 0, and consequently :

I2 −→∫

Ω

F (Φ)ϕDχδdx, I3 −→ −∫

Ω

F (Φ)Dϕχδdx,

I2 + I3 −→ −∫

Ω

F (Φ)χδDϕdx.

Taking account of the estimate (6.11) and making δ → 0, we get finally :

(F (Φ), Dh,Sϕ)∆,Ω −→ −∫

Ω

F (Φ)~∇(ϕ)dx. (6.12)

The other terms are dealt as in the proof of Theorem 2.1.2

As in section 2.2.3 we state extensions to Theorem 6.1 which concern approxi-

mations of the equation :Lv(Φ) + div[F (x, t,Φ)H(x, t)] = S for x ∈ Ω,HF (x, t,Φ).n = g(x, t).n for x ∈ ∂Ω,

(6.13)


given by :

d

dt(wiΦi) + wi

∑j∈P

wj(F (xj , t,Φj)Hi + F (xi, t,Φi)Hj + Πij).~∇Wij

= wiSi − wighi .(6.14)

Under the additional hypothesis that the function H ∈ W 2,∞, the results of

Theorem 6.1 also apply to the equation (6.13) and the class of particle schemes

(6.14).

6.3. Approximation of the boundary term

As a consequence of the Theorem 6.1 it is sufficient to provide an approximation

of the boundary term in the weak formulation according to hypothesis (iii). In the

general non linear case the task could appear difficult since we have the dependency

of g(x, t,Φ) in Φ. We refer to Ref.6, (Ch. IV) for such an analysis. Here we address

to a simplified situation, let g = g(x, t). We emphasize that all the recipes also

works in the nonlinear case. Thus we investigate different means of regularizing the

measure associated to a surface integral with help of particle approximations.

We focus ourselves on a model situation where we have a local system of co-

ordinate (x, y) in the neighborhood of ∂Ω such that x = x − yn(x) with (x, y) ∈∂Ω × [0, ε], for some ε > 0, and we want to design particle approximation of the

integral : ∫∂Ω

g(x(x, 0), t).n(x)ϕ(x(x, 0), t)dσ(x). (6.15)

We shall analyze three technics which allow to perform such approximations :

- Ghost Particles

- Boundary Particles and Forces

- Semi-Analytic Approach

6.3.1. Ghost particle approach

Classical Ghost Particles In case of plane boundaries the techniques of ghost

particles allows us to compute boundary conditions in an interesting way. For

simplicity let us suppose that our computational domain is the half space x1 < 0.

Let us define the symmetry F0 with respect to our boundary (the hyperplane x1 = 0)

by :

F0

x1

x2

...xd

=

−x1

x2

...xd

.


Let us suppose that the set P (Ω) of particles (xj , wj) gives a consistent quadra-

ture formula in Ω. Clearly the set of particles P (Ω) ∪ F0(P (Ω)) where particles of

F0(P (Ω)) are (F0(xj), wj), defines now a consistent quadrature formula over the set

Ω ∪ F0(Ω) = IRd.

To design a particle approximation of (6.4) we consider it as the restriction of a

more general problem over IRd. We suppose that we can construct g ∈ W 2,∞(IRd)

an extension of g, with g(0, x2, . . . , xd) = g(x2, . . . , xd) and we propose the following

approximation of (6.4) : Lv(ΠΩΦ) + ΠΩ

(div [Πh

Ω(F (x, t,Φ)) + ΠhF0(Ω)g]

+F (x, t,Φ).~∇(ΠhΩ(1)) + g(x, t).~∇(Πh

F0(Ω)(1)))

= ΠΩS,(6.16)

instead of Lv(ΠΩΦ) + ΠΩ

(div [(Πh

Ω + ΠhF0(Ω))(F (x, t,Φ))]

+F (x, t,Φ).~∇(ΠhΩ(1) + Πh

F0(Ω)(1)))

= ΠΩS,

that must be used if the equation :

Lv(Φ) + divF (x, t,Φ) = S,

be satisfied all over IRd. The terms involving g, the extension of the function giving

the flux at the boundary, can be understood as forcing terms. A simple calculation

proves that (6.16) is satisfied if and only if :

d

dt(wiΦi) + wi

∑j∈P

wj(F (xj , t,Φj) + F (xi, t,Φi)).~∇Wij = wiSi − wighi ,

where

ghi =∑

j∈F0(P )

wj(gi + gj).~∇Wij

It follows that this approach is equivalent with the following approximation of

the boundary integral :∫∂Ω

g(x).n(x)ϕ(x(x, 0), t)dσ(x) ≈∑

i∈P,j∈F0(P )

wiwjϕi(gi + gj)~∇Wij .

We shall prove at Proposition 6.2 the corresponding approximation result.

Weighted ghost particles To generalize the previous techniques of ghost parti-

cles, a natural feature is to use a more general transformation than the symmetry

F0. Let us then consider a bounded domain Ω ⊂ IRd. We suppose that there exists

a local system of coordinate (x, y) over ∂Ω× [−ε, ε] such that we can construct an

extension Ω = Ω ∪ Ω]−ε,0] of the set Ω (we suppose that ε is fixed and that h is

sufficiently small). In practise that could be done by using a finite element mapping


to the reference element, associated with a crude triangulation by suitable polyhe-

dra of the area closed to the boundary. In order to simplify we prefer to use the

diffeomorphism which maps any point x = x − yn(x) of local coordinate (x, y) in

Ω, to the point of IRd defined by :

F (x) = x+ yn(x).

To any particle xi of P , sufficiently close to the boundary ∂Ω we associate a

ghost particle located at F (xi). We get a quadrature formula valid over the set Ω

by taking the weight of the ghost as the weight of the particle multiplied by J(xi(t))

the Jacobean determinant | det(DF ) | at point xi(t). We thus have :∫Ω

f(x)dx ≈∑

i∈P∪Gh

ωi(t)f(xi(t)), (6.17)

where Gh = F (Ω[0,ε]) and ωi(t) =

ωi(t) if xi ∈ Ω,ωF−1(i)(t)J F−1(xi(t)) if xi ∈ Ω]−ε,0[.

The construction of the previous paragraph is then possible by introducing

g ∈ W 2,∞(Ω) an extension of g, with g(x(x, 0)) = g(x) and defining the new

approximation by :

Lv(ΠΩΦ) + ΠΩ

(div [Πh

Ω(F (x, t,Φ)) + ΠhΩ]−ε,0[ g]

+F (x, t,Φ).~∇(ΠhΩ(1)) + g(x, t).~∇(Πh

Ω]−ε,0[(1)))

= ΠΩS,(6.18)

which leads exactly as previously to the following approximation of the boundary

integral :

(Gh,Ω, ϕ)∆,Ω :=∑

i∈P,j∈Gh

wiwjϕi(gi + gj)~∇Wij . (6.19)

Let us now use the general set up of Theorem 6.1 together with the operator

Dh,S of (6.9). We define a modified operator, let Dh,S as

Dh,Sϕi :=∑j∈P

wj(ϕj −ϕi)Aij +∑j∈G

wj(ϕj −ϕi)Aij = Dh,Sϕi+∑j∈G

wj(ϕj −ϕi)Aij .

The additional term is such that for any ϕ compactly supported in Ω we have

Dh,Sϕ→ Dϕ.

Let us also define D∗h,Sϕ as :

D∗h,Sϕi =∑

j∈P∪Gwj(ϕiAij − ϕjAji).

We thus have D∗h,Sϕ → Dϕ in D′(Ω). Note anyway that −D∗h,S is not the adjoint

of Dh,S with respect to the scalar product (6.1), it is in fact the restriction to Ω of

the adjoint of Dh,S with respect to the scalar product

(ϕ,Ψ)h,Ω

:=∑

i∈P∪Gwiϕi.Ψi.


Let us first remark that the formula (6.19) can be written as

(Gh,Ω, ϕ)∆,Ω :=∑j∈G

wj(giAij − gjAji) =(ϕ, D∗h,S g −D∗h,S g

)∆,Ω

. (6.20)

We shall prove the

Proposition 6.2 Let g(x) a function defined on the boundary ∂Ω such that there

exists g ∈ H(div; Ω) an extension of g, with g(x(x, 0)).n(x) = g(x). then for any

test function ϕ ∈ C2(Ω× IR+) we have :

limh and ∆x→0 (Gh,Ω, ϕ)∆,Ω =

∫∂Ω

g(x).n(x)ϕ(x(x, 0), t)dσ(x),

where (Gh,Ω, ϕ)∆,Ω is defined according to (6.20) and Dh,S satisfies condition (ii)

of Theorem 6.1

Proof. Due to the hypothesis, we have∫∂Ω

g(x, t).nϕdσ =

∫Ω

ϕdivg(x, t)dx+

∫Ω

g(x, t)~∇(ϕ)dx.

Since we have D∗h,Sϕ→ Dϕ in D′(Ω), it follows that (when h and ∆x→ 0)(ϕ, D∗h,S g

)∆,Ω−→

∫Ω

ϕdivg(x, t)dx.

From the proof of Theorem 6.1 we also have

(g, Dh,Sϕ)∆,Ω −→∫

Ω

g(x, t)~∇(ϕ)dx.

Thus we get :

(Gh,Ω, ϕ)∆,Ω =(ϕ, D∗h,S g −D∗h,S g

)∆,Ω

−→∫

Ω

ϕdivg(x, t)dx+

∫Ω

g(x, t)~∇(ϕ)dx =

∫∂Ω

g(x, t).nϕdσ.

which is the desired result. .

In section 7.1.1 we detail a natural way to introduce such a type of formula

in the case of Euler equations. To this ends it is also useful to give the following

extension of Proposition 6.2, suitable for use with formulation (6.13) (we omit the

proof) :

Proposition 6.3 Let g(x), l(x) functions defined on the boundary ∂Ω such that

there exists g (resp. l) ∈W 2,∞(Ω) an extension of g (resp. l), with g(x(x, 0)) = g(x)

(resp. l(x(x, 0)) = l(x), then for any test function ϕ ∈ C2(Ω× IR+) we have :

limh and ∆x→0

∑i∈P,j∈Gh

wiwjϕi li lj(gi + gj)~∇Wij

=

∫∂Ω

l2(x)g(x).n(x)ϕ(x(x, 0), t)dσ(x).


6.3.2. Boundary particles and boundary forces

Here, we start directly from the boundary integral

(G,ϕ)∂Ω :=

∫∂Ω

g(x(x, 0), t)ϕ(x(x, 0), t)dσ(x).

Let θδ a regular function (∈ C2) of the real variable y, such that :(i) 0 ≤ θδ(y),(ii) θδ(y) = 0, for y ≥ δ,

(iii)

∫ δ

0

θδ(y)dy = 1.

(6.21)

It follows that :

(G,ϕ)∂Ω =

∫Ω

g(x(x, 0), t)θδ(y)ϕ(x(x, 0), t)J∂Ω(x)J(x(x, y))dx,

where J∂Ω and J are the Jacobian associated with the changes of coordinates. ϕ is

regular, thus for δ sufficiently small :

(G,ϕ)∂Ω ≈∫

Ω

g(x(x, 0), t)θδ(y)ϕ(x, t)J∂Ω(x)J(x(x, y))dx. (6.22)

To get an approximation of the integral in the r.h.s. of (6.22), we first consider a

finite element type triangulation T∂Ω of the boundary ∂Ω. To this triangulation we

associate a finite element interpolation rT (g) of the function g(x) (we omit the time

dependence) :

rT (g)(x) =∑i∈NT

g(xi)Ψi(x),

where the summation is over the degrees of freedom NT (respectively located at xi ∈∂Ω, i ∈ NT ) of the Finite Element, associated with the basis polynomial functions

Ψi(x). These degrees of freedom, located on the boundary will be considered as

boundary particles. We thus propose as an approximation of the boundary integral

the following formula :

(G,ϕ)∂Ω ≈ (Gh,Ω, ϕ)∆,Ω :=∑

i∈P (Ω),j∈NT

wiϕiJ(xi)J∂Ω(xj)g(xj)Ψj(xi)θδ(yj),

which leads according to (6.8) at :

ghi = J(xi)∑j∈NT

g(xj)J∂Ω(xj)Ψj(xi)θδ(yj).

This formula can be interpreted by associating to each boundary particle j a force

field fj(x) defined by :

fj(x) = J(x)J∂Ω(xj)g(xj)Ψj(x)θδ(y).


The resulting force on each fluid particle i ∈ P (Ω) is then :

ghi =∑j∈NT

fj(xi).

By using standard approximation results, together with the hypothesis (6.21)(iv)

we can prove the following result (we omit the detailed proof) :

Proposition 6.4 Let g(x) ∈ L∞(∂Ω) and θδ a regular function (∈ C2) which sat-

isfy conditions (6.21), then for any test function ϕ ∈ C2(Ω× IR+) we have

limh and ∆x→0,δ−→0

∑i∈P (Ω),j∈NT

wiJ(xi)J∂Ω(xj)ϕig(xj)Ψj(xi)θδ(yj)

=

∫∂Ω

g(x).n(x)ϕ(x(x, y), t)dσ(x).

6.3.3. Semi-analytic approach

Basic Principles. The idea of the Semi-Analytic Approach is to modify the ap-

proximation given by (6.18) in a way that avoids to compute the “forcing terms”

involving g with help of particle approximation, and therefore propose an analytic

approximation of these terms. We proceed as follows by substituting to (6.18) the

following equation :Lv(ΠΩΦ) + ΠΩ

(div [Πh

Ω(F (x, t,Φ)) + gχΩ]−ε,0[ ∗W ]

+F (x, t,Φ).~∇(ΠhΩ(1)) + g(x, t).~∇(χΩ]−ε,0[ ∗W )

)= ΠΩS.

(6.23)

This leads to a voluming term ghi given by :

ghi =

∫Ω]−ε,0[

(g(z) + g(xi))~∇W (xi − z)dz,

and to the corresponding approximation of the boundary integral defined by :∫∂Ω

g(x).n(x)ϕ(x(x, 0), t)dσ(x) ≈∑

i∈P (Ω)

wiϕi

∫Ω]−ε,0[

(g(z) + g(xi))~∇W (xi − z)dz.

We finally remark that the following approximations could also be used :∫∂Ω

g(x)ϕ(x(x, 0), t)dσ(x) ≈ 2∑

i∈P (Ω)

wiϕi

∫Ω]−ε,0[

g(z)~∇W (xi − z)dz,

∫∂Ω

g(x)ϕ(x(x, 0), t)dσ(x) ≈ 2∑

i∈P (Ω)

wiϕig(xi)

∫Ω]−ε,0[

~∇W (xi − z)dz.


Approximation of kernel dependent integrals. We address here the problem

of computing the integrals :∫IRd/Ω

W (x− y)dy,

∫IRd/Ω

~∇xW (x− y)dy.

In case of plane boundaries these integrals can be computed exactly for polynomial

Kernels, we refer for instance to the paper of Herand 19 where such results are given

by means of formal calculus computer codes. In some situation we have to deal with

a free surface, thus the exact location of the boundary can’t be really computed.

Indeed SPH gives us a mean to compute some approximation of this boundary and

related integrals. The computational domain is precisely defined by the particles,

which gives us a good mean to evaluate integrals over the set Ω according to :∫Ω

f(y)dy ≈∑

i∈P (Ω)

wifi.

Considering that∫IRd

W (x− y)dy = 1 and

∫IRd

~∇xW (x− y)dy = 0,

therefore we get : ∫IRd/Ω

W (x− y)dy = 1−∫

Ω

W (x− y)dy,∫IRd/Ω

~∇xW (x− y)dy = −∫

Ω

~∇xW (x− y)dy,

which leads to the following approximations :

∫IRd/Ω

W (xi − y)dy ≈ 1−∑

j∈P (Ω)

wjWij ,∫IRd/Ω

~∇xW (xi − y)dy ≈ −∑

j∈P (Ω)

wj ~∇Wij .(6.24)

As previously it can be proved under the assumption that g is regular enough

convergence results such as those in Proposition 6.4.

7. Particle Approximation of Boundary Conditions for Euler Equations

We consider the case of Euler equations on a bounded set. We deal with a moving

boundary Γ, its outward normal is denoted by ~n. We shall consider two situations :

- a rigid boundary. The boundary conditions are then :

~v.~n = ~vnb, (7.1)

where vnb is the normal velocity of the boundary (a given function),


- a free surface boundary

~v.~n = vnb,σ.~n.~n = −p = −patm,

where patm is a given outside pressure (for example the atmospheric pressure).

We can write the weak form of Euler equation as :∫Ω×IR+

ρL∗v(ϕ)dxdt = 0,∫Ω×IR+

(ρvL∗v(ϕ) + p~∇(ϕ)

)dxdt−

∫∂Ω×IR+

pout(x, t)ϕ.ndσ(x)dt = 0,∫Ω×IR+

(EL∗v(ϕ) + pv.~∇(ϕ)

)dxdt−

∫∂Ω×IR+

pout(x, t)v.nϕdσ(x)dt = 0,

where pout is the pressure at the boundary (to be determined).

7.1. Ghost particles

7.1.1. General set-up

For simplicity we restrict ourselves here to the case of a rigid fixed boundary, so

that the boundary condition reduces to (7.1) with vnb = 0. Following the ideas of

section 6.3.1 we propose as a particle approximation :

Lv(ΠΩρ) = 0,

Lv(ΠΩρv) + ΠΩ

(ρdivΠh

Ω(p

ρ) +

p

ρ~∇(Πh

Ω(ρ)

),

+ΠΩ

(ρdivΠh

Ω]−ε,0[(p

ρ) +

p

ρ~∇(Πh

Ω]−ε,0[(ρ)

)= 0,

Lv(ΠΩE) + ΠΩ

(ρv~∇Πh

Ω(p

ρ) +

p

ρdiv(Πh

Ω(ρv)

),

+ΠΩ

(ρv ~∇ Πh

Ω]−ε,0[(p

ρ) +

p

ρdiv(Πh

Ω]−ε,0[(ρv))

)= 0.

To close the problem we need to define the extension (ρ, p, v) of the data at the

boundary. We proceed as follows :

ρ(x, y) =

ρ(x, y) if y ≥ 0,ρ(x,−y) if y < 0

, p(x, y) =

p(x, y) if y ≥ 0,p(x,−y) if y < 0

,

v(x, y) =

v(x, y) if y ≥ 0,v(x,−y)− 2[v(x,−y).n(x)]n(x) if y < 0

.

This technique produces additional terms for the momentum and Energy conser-

vation equations, which according to results of Proposition 6.3 (with l = ρ, g =p

ρ


and l = ρ, g =p

ρv) satisfy the following approximation results (if the field are

sufficiently regular) :∑i∈P,j∈Gh

mimjϕi(piρ2i

+pjρ2j

)~∇Wij ≈∫∂Ω

p(x)ϕ(x(x, 0), t)dσ(x),∑i∈P,j∈Gh

mimjϕi(piρ2i

vj +pjρ2j

vi)~∇Wij ≈∫∂Ω

p(x)v.nϕ(x(x, 0), t)dσ(x).(7.2)

We recall the basic features of the classical “ghosts” method. To any inner

particle we associate a ghost particle located at a point given by the symmetry with

respect to the boundary. The physical properties (density, internal energy (and

consequently pressure)) of the ghost are the same as those of the inner particle. The

velocity of the ghost is also given by the symmetry with respect to the boundary of

the inner particle velocity.

This technique is also used in Finite Volumes computer codes to deal with similar

rigid boundaries : symmetrical nodes are introduced to compute fluxes on boundary

edges of the finite volume mesh.

The second integral in (7.2) is clearly equal to 0 (since v.n = 0 on ∂Ω). Thus

it could be possible to construct an approximation of the energy equation without

taking account of ghost particles. Such a choice should allow to keep automatically

global conservation of the energy. We have here additional terms, nevertheless we

shall prove that it allows to keep consistency with global conservation equations,

and naturally justify the choice of symmetry for the velocities of ghost particles.

We may consider moving boundary. The analysis with ghosts is possible, again.

Note that we had to take account of the velocity of the boundary in order to evaluate

the velocity of the ghosts, this will not be detailed.

When we consider a bounded set Ω with a boundary ∂Ω along which we want

to satisfy a rigid fixed boundary condition, we get from conservation equations of

momentum and energy (at least when the volume source terms are equal to 0) that :

d

dt

∫Ω

ρ~vdx = −∫∂Ω

p~ndσ, (7.3)

d

dt

∫Ω

(ρu+1

2ρ‖v‖2)dx = 0. (7.4)

In the following sections we shall suppose for simplicity that the computational

domain is Ω = x ∈ IRd, x1 < 0 in such a way that we have a plane boundary

located at x1 = 0.

We will consider ghost of a particle i ∈ P (Ω) located at xi. The set of ghosts will

be indexed by i ∈ P (Ω), the ghost g(i) will be located at xg(i) = F0(xi).

The values of the density and the energy for the ghost are chosen according to :ρg(i) = ρi,pg(i) = pi.

(7.5)


Since the Jacobean of F0 is equal to 1, it follows that in the case of classical ghosts,

the mass mg(i) of the ghost will be equal to the mass mi of the particle. The value

of the ghost velocity ~vi will be determined later on.

We will also consider more general transformation than the symmetry F0, which

allows us to deal with weighted ghosts. The general case should include the possi-

bility to account of multiple ghosts. The main results of the section remain true in

this case.

7.1.2. Pressure and Conservation of momentum

The global momentumQT =

∫Ω

ρ~vdx in the set Ω can be approximated by QhT =∑i∈P (Ω)

mi~vi. Thus, we get :

d

dtQhT = −

∑i∈P (Ω),j∈Gh(Ω)

mimj(piρ2i

+pjρ2j

+ Πij)~∇Wij .

We first neglect the artificial viscosity terms (which are of order h), and consequently

we just consider the terms involving the pressure.

d

dtQhT ≈ −


wiwjρiρj(piρ2i

+pjρ2j

)~∇Wij .

We apply Proposition 6.3 with l = ρ, g =p

ρ2to obtain when h −→ 0 that :

d

dtQhT −→ −

∫Ω

~∇pdx = −∫∂Ω

p~ndx,

which is the desired result. Thus the discrete global momentum satisfy a simi-

lar equation as the exact global momentum, involving the pressure forces at the

boundary.

Remark 1 These results remains true for weighted ghosts and more general situ-

ations involving multiple ghosts (i.e. when a particle interact with several bound-

aries).

Remark 2 We can notice that this result concerning the conservation of momen-

tum is independent of the velocity of the ghosts. Thus, we have to take account of

conservation of the energy to find a suitable rule for choosing this velocity.

7.1.3. Velocity and conservation of energy

Let us define the discrete global Energy EhT =∑

i∈P (Ω)

mi(1

2‖~vi‖2 + ui). The

exact global energy is conserved, this is also true for the discrete global energy

on unbounded domains, we shall prove that under a reasonable choice of ghost

velocities this is also true in the case of bounded domains.


Classical Ghosts. We first consider the case of classical ghosts, a straightforward

calculus proves that :

d

dtEhT = −


mimj

(~vipjρ2j

+ ~vjpiρ2i

+1

2Πij(~vi + ~vj)

).~∇Wij .

We then, separate the viscous and non viscous part of this rate of energy production :

d

dtEhT = ∆EhT + ∆EΠh

T ,

where

∆EhT = −∑

i∈P (Ω),j∈Gh(Ω)

mimj

(~vipjρ2j

+ ~vjpiρ2i

).~∇Wij .

We thus obtain :

∆EhT = −∑

i∈P (Ω),l∈P (Ω)

mimg(l)

(~vipg(l)

ρ2g(l)

+ ~vg(l)piρ2i

).~∇Wig(l)

= −∑

i,l∈P (Ω)

miml~viplρ2l

.~∇Wig(l) −∑

i,l∈P (Ω)

miml~vg(l)piρ2i

.~∇Wig(l)

= −∑

i∈P (Ω)

mi

~vi. ∑l∈P (Ω)

mlplρ2l

.~∇Wig(l)

+ ~vg(i).

∑l∈P (Ω)

mlplρ2l

~∇Wlg(i)

.

It follows that a sufficient condition to get conservation of the non viscous part of

the energy is that the velocity of the particle and the velocity of its ghost satisfy :

~vi.

∑l∈P (Ω)

mlplρ2l

.~∇Wig(l)

+ ~vg(i).

∑l∈P (Ω)

mlplρ2l

~∇Wlg(i)

= 0. (7.6)

For symmetric kernel, the condition (7.6) is satisfied by the method of the image in

which we take :

~vg(i) = F0(~vi). (7.7)

Indeed, in this case we have ~nig(j) = aij~n+~bij ,

~njg(i) = aij~n−~bij ,Dθig(j) = Dθjg(i) = cij ,

where ~n is the outward normal to the boundary and ~bij .~n = 0 . Then (7.6) reduces

to :

(~vi + ~vg(i)).~naij + (~vi − ~vg(i)).~bij = 0. (7.8)

(7.8) is clearly true for the choice (7.7). With this choice we can also prove that the

contribution of artificial viscosity is equal to zero. Similar computation as in the


previous paragraph prove easily that :

∆EΠhT =

−1

2

∑i∈P (Ω)

mi

~vi. ∑l∈P (Ω)

mlΠig(l).~∇Wig(l) + ~vg(i).∑

l∈P (Ω)

mlΠlg(i)~∇lWlg(i)

.

Taking account carefully of the transformation F0 we get that the term µij in the

artificial viscosity is such that µig(l) = µlg(i) which gives Πig(l) = Πlg(i) = dil.

It follows that we have also ∆EΠhT = 0 and consequently

d

dtEhT = 0.

Weighted Ghosts. Similar computations are possible in this case which leads to

similar results at least for the non-viscous part of the rate of production, we can

establish that under the condition :

~vi.

∑l∈P (Ω)

mg(l)plρ2l

.~∇Wig(l)

+ ~vg(i).

∑l∈P (Ω)

mlplρ2l

~∇lWlg(i)

= 0, (7.9)

the non-viscous rate of production of energy is zero. Unfortunately, although it is

easy to produce a ghost velocity which satisfy this requirements, we are not able to

prove that the associated condition for the artificial viscosity term is satisfied. It

remains that it is certainly possible to chose an artificial viscosity at the boundary

so that the total rate of energy production is zero.

7.2. Boundary particles and boundary forces

For momentum equation it is sufficient to provide an approximation of the pres-

sure at the boundary to introduce according to ideas of section 6.3.2 suitable bound-

ary forces. This can be achieved by computing locally approximation of p (for any

i ∈ NT ) as :

p(xi) =

∑j∈P

wjpjWij∑j∈P

wjWij

.

This produces repulsive forces near the boundary and has the same nice property to

keep perfect equilibrium for moving particles at a speed parallel to a plane bound-

ary in a field of constant pressure with equidistributed particles. This approach

as similar characteristics as the forces proposed by Monaghan 32. We also have

repulsive forces (not necessarily bounded locally) but the conditions on θδ allows

consistency with the pressure forces at the boundary. The corresponding term in

Ref.32 could leads to unbounded integrals in the analysis of the consistency.


7.3. Semi-analytical approach

We apply the semi-analytical approach of section 6.3.3. For simplicity we con-

sider in a first time the non-standard form of SPH, and we thus define the particle

approximation of Euler equations as :Lv(ΠΩ(ρ)) = 0,

Lv(ΠΩ(ρ~v)) = −ΠΩ

(~∇Πh

Ω(p)− p~∇(ΠhΩ(1))

)+ (pn)h,

Lv(ΠΩ(E)) = −ΠΩ

(div(Πh

Ω(p~v))− p~v.~∇(ΠhΩ(1))

)+ (pv.n)h,

where the terms (pn)h and (pv.n)h are given, according to section 6.3.3 by :

(pn)hi = patm

∫IRd/Ω

~∇W (xi − z)dz,

(pv.n)hi = patmvi.

∫IRd/Ω

~∇W (xi − z)dz,

and computed by using (6.24) as :

(pn)hi = −patm∑j∈P

wj ~∇Wij ,

(pv.n)hi = −patmvi.∑j∈P

wj ~∇Wij .

Straightforward computations leads consequently to :d

dt~vi = −

∑j∈P

mj(pi − patm) + (pj − patm)

ρiρj~∇iWij ,

d

dtui = −

∑j∈P

mjpiρiρj

(~vj − ~vi).~∇iWij .

(7.10)

These formulae have a nice interpretation, since we can understand the mo-

mentum equation as an equation with a modified vacuum level (p = patm) which

automatically handle for equilibrium at the free surface.

Similar computations are possible for standard SPH and leads to :d

dt~vi = −

∑j∈P

mj(pi − patm

ρ2i

+pj − patm

ρ2j

)~∇iWij ,

d

dtui = − pi

ρ2i

∑j∈P

mj(~vj − ~vi).~∇iWij .

(7.11)

Consistency has to be understood in a different way. By using the extension of

Theorem 6.1 to the scheme (2.23) with F =patmρ

and H = ρ we see that :

∑i,j∈P

wjwipatmϕi(ρiρj

+ρjρi

)~∇iWij −→ patm

∫Ω

~∇(ϕ)dx.


It follows that :∑i,j∈P

mjmipatmϕi(1

ρ2j

+1

ρ2i

)~∇iWij −→∫∂Ω

patmnϕdσ(x),

which is the desired result.

7.4. Comments about convergence and stability

Here we have just given some basic rules to design efficient schemes at the

boundary. From a practical point of view the engineer involved in computations

with SPH and boundary conditions as to satisfy an additional requirement which

practically is :

insure discrete stability of equilibrium situations.

In particular for initial data in which where the pressure and the velocity fields

are uniform, we expect that the discrete equation for velocity keeps this equilibrium.

This is not true for a general distribution of particle, nevertheless this is true in

the case of particles distributed on regular grids, and that gives conditions on the

initial locations of the particles with respect to the boundary. For an example of an

industrial problem with boundary conditions we refer to Ref.24 where we present

computations - with the SPH computer code SMFI of XRS and Simulog - of a

liquid jet disintegration in a gas stream and some comparisons with experiments

performed at the CORIA in Rouen.

These considerations also have some links with the theoretical framework of the

previous section. Extension of Theorem 6.1 with measure-valued solution is the

essential tool to deal with convergence, nevertheless we need a priori estimates. To

obtain these a priori estimates we need to satisfy rules similar to the equilibrium

condition for a uniform field, this is detailed in Ref.6.

Acknowledgment

The author thanks Professor W. Benz of University of Berne who gave him its

interests for SPH method. He also thanks Professor J.J. Monaghan for interesting

discussions which where possible during the few days of a summer school organized

at Kaiserslautern university by Professor H.Neuntzert. Thanks also to Professor

G.H. Cottet for its helpful comments.

References

1. Abgrall R., Generalization of the Roe scheme for computing flows of mixed gases withvariable concentration, La Recherche Aerospatiale, 6, (1988), pp. 31-43.

2. Audounet J., Mazet P., Seminaire Toulouse III, (1983-1984).3. Bardos C.,Leroux A.Y., Nedelec J.C., First order quasilinear equations with boundary

conditions, Com. Part. Diff. Eq., 4, 9, (1979), pp.1017-1034.4. Benharbit S., Chalabi A., Vila J.P., Numerical viscosity, and convergence of finite vol-

ume Methods for conservation laws with Boundary conditions, S.I.A.M Journal of Num.Analysis, (1995).


5. Ben Moussa B., Vila J.P., Convergence of SPH method for scalar nonlinear conservationlaws, S.I.A.M Journal of Num. Analysis, to appear.

6. Ben Moussa B., PHD Thesis INSAT, (1998).7. Benz W., Smooth Particle Hydrodynamics : a Review., Harvard-Smithsonian Center for

Astrophysics preprint 2884, (1989).8. Benz W., Asphaug A., Impact Simulations with Fracture : I. Methods and Tests, Icarus,

(1993).9. Bicknell G.V., The equations of motion of particles in smoothed particle hydrodynam-

ics,SIAM J. Sci. Stat. Comput., 12, 5, (1991), pp.1198-1206.10. Degond P.,Mas-Gallic S., The weighted particle method for convection-diffusion equa-

tions, Math. of Comp., (1989)11. Diperna R.J., Measure-valued solution to conservation laws Arch. Rat. Mech. Anal., 88,

(1985), pp.223-270.12. Dubois F., Lefloch P., Boundary conditions for nonlinear hyperbolic systems of conser-

vation laws, J.D.E., 71, 1, (1988), pp. 93-122.13. Cottet G.H., Artificial viscosity for vortex and particle methods, J.C.P., (1996).14. Crandall M., Majda A., Monotone Difference Approximations for Scalar Conservation

Laws, Math. of Comp., 34, 149, (1980), pp.1-21.15. Gingold R.A., Monahghan J.J., Shock simulation by the particle method S.P.H. J.C.P.,

52, (1983), pp.374-389.16. Gingold R.A., Monahghan, MNRAS, (1977), pp.181-375.17. Harten A.,Lax P.D.,Van Leer B., On upstream differencing and Godunov type schemes

for hyperbolic conservation laws, ICASE 82-5, (1982).18. Harten A., Hyman J.M. ,Self adjusting grid methods for one-dimensional hyperbolic

conservation laws, J. Comp. Phys, (1983).19. Herant, Dirty Tricks for SPH, Mem. S. A. It., Vol. XX, (1993).20. Hernquist L., Katz N., TREESPH : a unification of SPH with the hierarchical tree

method, The Astr. J. S.S., 70, (1989), pp. 419-446.21. Johnson G.R., Beissel S.R., Normalized Smoothing functions for impact computations,

Int. Jour. Num. Methods Eng., (1996)22. Kuznetsov N.N., Volosin S.A.., On monotone difference approximations for a first order

quasilinear equation, Soviet Math. Dokl., 17, (1976) , pp. 1203-1206.23. Lacome J.L., PHD Thesis INSAT, (1998).24. Larrey E., Vich G., Merlo A., Ledoux M., Rompteaux A., Vila J.P. ,Liquid Jet disin-

tegration in a gas stream, ICLAS, Seoul, (1997).25. Lattanzio J.C.,Monaghan J.J., Pongracic H., Schwarz M.P., Controlling penetration,

SIAM J. Sci. Stat. Comput., 7 , 2, (1986).26. Lax P.D.,Wendroff B., Systems of Conservation Laws CPAM, 13, (1960), pp.217-237.27. Le Roux A.Y., Approximation de quelques problemes hyperboliques non lineaires, These

universite de Rennes, (1979).28. Lucy .L., Astrono. J., 82 ,1013, (1977).29. Mas-Gallic S., Raviart P.A., Particle Approximation of Convection-Diffusion Problems,

preprint LAN Paris VI, (1984).30. Mas-Gallic S., Raviart P.A., A Particle Method for First-order Symmetric Systems,

Numer. Math. , 51, (1987), pp. 323-352.31. Monaghan, Smooth Particle Hydrodynamics , Annu. Rev. Astron. Astrop., 30, (1992),

pp. 543-74.32. Monahghan J.J., lecture notes, Kaiserslautern University, (1995).33. Noh W.F., Errors for calculations of strong shocks using an artificial viscosity and an

artificial heat flux, JCP 72, (1978), pp.78-120.34. Osher S., Riemann solvers, the entropy condition and difference approximations, Siam


Jour. Num. Anal. , 21, 2, (1984), pp. 217-235.35. Paris L., Rapport de Fin d’etude INSA GMM, (1996).36. Randles R.W., Libertsky L.D., Smoothed Particle Hydrodynamics, Some recent im-

provements and Applications, Comp. Meth. Appli. Mech. Eng, 139, (1996), pp. 375-408.37. Raviart P.A. An analysis of particle methods, Num. method in fluid dynamics, F. Brezzi

ed.. Lecture Notes in Math., 1127, Berlin, (Springer, 1985).38. Raviart P.A. Particle approximation of first order systems, Journal of Comp. Mathe-

matics, 4, 1,(1986).39. Richtmeyer R.D., Morton K.W., Difference methods for initial value problems, (Inter-

science Publishers, New York, 1967).40. Roe P.L., Approximate Riemann solvers, parameter vectors and difference

schemeJ.C.P., 43, (1981), pp.357-372.41. Van Leer B., Towards the ultimate conservative difference scheme V, J.C.P. , 32, (1979),

pp.101-136.42. Vijayasundaram G., Transonic Flow Simulations Using an upstream Centered scheme

of Godunov in Finite Elements, J.C.P. , 63 , (1986), pp 416-433.43. Vila JP., Meshless Methods for Conservation Laws, submitted Math. of Comp.44. Vila JP., Construction of Roe type Matrices, a new approach. Application to real gas

dynamic, Third Int Conf on Hyp. Probl. Uppsala, Engquist B, Gustafsson Edt. (1991).45. Vila JP., Weighted Particle-Finite Volume Hybrid schemes, Finite Volumes for Complex

Applications ed. F. Benkhaldoun, R. Vilsmeier, (HERMES, Paris, 1996).46. Vila JP., Developpements recents et Applications des Methodes particulaires

regularisees, ESAIM Proceeding CANUM 97, (1997).47. Vila JP., Villedieu P., Convergence de la methode des volumes finis pour les systemes

de Friedrichs, Note au CRAS, (1997).

Date post:	20-Nov-2023
Category:	Documents
Upload:	insa-toulouse
View:	0 times
Download:	0 times

ON PARTICLE WEIGHTED METHODS AND SMOOTH PARTICLE HYDRODYNAMICS

Documents