ADER schemes for linear advection-diffusion problems with ...berthon/conferences/castro_200… ·...

ADER schemes for linear advection-diffusionproblems with nonlinear source term using

several flux reconstructions.

R. Castedo, A. HidalgoDpto. de Matemática Aplicada y Métodos Informáticos,

E.T.S.I. Minas, Univ. Politécnica de [email protected], [email protected]

Hyperbolic systems with source term – Castro Urdiales ´09 07/09/2009

R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 2

AIM.

To compare the behaviour of ADER scheme with Cauchy- Kowaleski procedure (E.F. Toro & R.C. Millington, 2001) and ADER with local space-time discontinuous Galerkin (M. Dumbser, C. Enaux & E.F. Toro, 2008), when solving problems with stiff source terms.

07/09/2009

R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 3

Contents.

1. The numerical scheme.

2. ADER approach.2.1. ADER-FV.2.2. ADER-DG.

3. Numerical results.3.1. Linear advection problem.3.2. Linear advection-reaction problem.3.3. The scalar model problem of LeVeque and Yee.3.4. Advective-reactive-diffusive linear equation: convergence and practical application.

4. Conclusions and future research.

07/09/2009

4

1. The numerical scheme.

∂∂t

U(x,t) +∂∂x

F(U(x,t)) = S(x,t,U(x,t))

Finite Volume scheme for hyperbolic systems in non-conservative form:

[ ]

1/2

1/2

1

11/2

1/2

11/2 1/2 1/2 1/2

1 ( , ) ,

1 ( ( , )) ,

1 1 ( ( , )) .

i

i

n

n

ni

ni

xn ni x

in tn n n n

i i i i i i in ti

t x

i in t xi

x t dxx

t t x t dtx t

x t dxdtx t

+

−

+

++

−

++ − + +

⎧ =⎪ Δ⎪Δ ⎪

= − − + Δ =⎨Δ Δ⎪⎪ =⎪ Δ Δ⎩

∫

∫

∫ ∫

U U

U U F F S F F U

S S U

along with an initial and boundary conditions. Integrating this equation in the following space-time control volume

R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009

Qi = xi−1/ 2 ,xi+1/ 2⎡⎣ ⎤⎦ × t n ,t n+1⎡⎣ ⎤⎦.

5

Fci+1/ 2 =1Δt

v(xi+1/ 2 ,τ )U (xi+1/ 2 ,τ )dτtn

tn+1

∫

Fdi+1/ 2 = −1Δt

K(xi+1/ 2 ,τ )∂∂x

U (xi+1/ 2 ,τ ) dτtn

tn+1

∫

⎫

⎬⎪⎪

⎭⎪⎪

Fi+1/ 2 = Fci+1/ 2 + Fdi+1/ 2


2. ADER approach.2.1. ADER-FV. 2.3. ADER-DG.

ADER-FV of E.F. Toro and V.A. Titarev, the time evolution is achieved via a Taylor series expansion (solution of the DRP in each cell interface) where the time derivatives are computed by repeated differentiation of the governing PDE with respect to space and time (Cauchy-Kovalewski procedure) Not able to handle stiff source term.

Si =1Δxi

1Δt n S(U i (x,t)) dx dt

xi−1/2

xi+1/2∫tn

tn+1

∫ .

Source term expression inside each volume:

We discretize the space integral with N-point Gaussian rule, then we reconstruct values in each Gaussian point in analogous procedure to the flux evaluation (WENO) and finally for each point we perform the C-K procedure and replace time derivatives by space derivatives.

6R. Castedo & A. Hidalgo – Hyperbolic systems with source term ´09 07/09/2009

2. ADER approach.2.1. ADER-FV. 2.2. ADER-DG.

ADER-DG of M. Dumbser et. al., the time evolution is obtained via a local space-time Discontinuous Galerkin (DG) finite element scheme Able to handle stiff source term.

In order to describe the ADER-DG approach we will consider the simple scalar model equation:

f (u) = vu and S(u) = −qu, v > 0,q > 0,

Basis functions are defined by the space-time tensor product of the Legendre polynomials

Φk = Φk (ξ,τ ) = Ψ i (ξ) ⋅Ψ i (τ ).kΦ ( ), ( ).i iξ τΨ Ψ

∂∂τ

u + v *∂∂ξ

u = −q *u, with v* = Δtξx ⋅ v and q* = Δt ⋅q.

1

ˆ ˆ( , ) ( , ) : ( , )dN

i ii i l l l l

lu u u uξ τ ξ τ ξ τ

=

= = Φ ⋅ = Φ ⋅∑The local numerical solution inside each space-time control volumeiu :iQ

Modified governing PDE

Multiplication of the new PDE with basis functions, and integration over the reference element:

[ ] [ ]1 0

, , , * , * , .k i k i k i k i k iu w u v u q uτ τ τ ξ

∂ ∂Φ − Φ − Φ + Φ = − Φ

∂ ∂

7

3. Numerical Results.3.1. Linear advection problem. 3.2. Linear advection-reaction problem.

3.3. Scalar problem of LeVeque and Yee. 3.4. Advective-reactive-diffusive linear equation.

∂∂t

u x,t( )+ ∂∂x

u x,t( )= 0, x∈ (-1,1), t > 0,

u(x,0) =

exp(−ln2(x +0.7)2 /0.0009), -0.8≤ x ≤ −0.6,1, -0.4 ≤ x ≤ −0.2,1− 10x −1 , 0.0≤ x ≤ 0.2,

(1−100(x −0.5)2)1/2, 0.4 ≤ x ≤ 0.6,0, otherwise.

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⎫

⎬

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

and periodic boundary conditions.

Problem presented by E.F. Toro and V.A. Titarev in 2005.


WAF-SUPERBEE

RUSANOV

G-FORCE




Problem presented by V.A. Titarev in 2005.

The exact solution is:

9

∂∂t

u x,t( )+ ∂∂x

u x,t( )= 5u x,t( ), x ∈ (-1,1), t > 0,

u(x,0) = sin4 π x( ),

⎫

⎬⎪

⎭⎪

and periodic boundary conditions. u(x,t) = sin4 π (x − t)( )e5t .




Problem presented by LeVeque and Yee in 1990, is a scalar linear advection problem, with a nonlinear reaction term, which can be stiff.

The initial condition is:

10

∂∂t

u +∂∂x

u = −qu(u −1) u −12

⎛⎝⎜

⎞⎠⎟

, x ∈[0;1], t > 0,

with transmissive boundary conditions.

1 if 0.3,( , )

0 if 0.3.x

u x tx≤⎧

= ⎨ >⎩




With this initial condition, the source term is zero, and the analytical solution of the problem is:

u(x,t) = u(x − t,0).


x

u

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Exact sol.ADER3-DGADER4-DGADER3-FVADER4-FV

q = 10

x

u

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1


q = 100

Parameters:Courant = 0.75; Time = 0.3;

Cells = 100; Values of q: q = 10 and q = 100.




x

u

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1


q = 1000

Parameters:Courant = 0.75.

Time = 0.3.

Cells = 500; Values of q: q = 1000.



13

∂∂t

u x,t( )+1.5∂∂x

u x,t( )− 0.8∂2

∂x2 u x,t( )= 3u x,t( ), -10 < x < -10, t > 0,

u(x,0) = e− x2

,

⎫

⎬⎪

⎭⎪

and periodic boundary conditions.

u(x,t) =

e3t

2 0.8πte−(ξ2 )e

(−(x−1.5t−ξ)2

4⋅0.8t)

−10

10

∫ dξ.

The exact solution of this problem is the following equation:

Test problem in order to know the order of the method ADER-FV:

ADER-FV

Convective flux WENO-GFORCE 5 th order.

Diffusive flux Centered reconstruction 6 th order.

Reactive term Centered reconstruction 6 th order.


3. Numerical Results.3.1. Linear advection problem. 3.2. Linear advection-reaction problem. 3.3. Scalar problem of

LeVeque and Yee. 3.4. Advective-reactive-diffusive linear equation.

14

( ) ( ) ( ) ( )2

2, 0.1 , , 0.1 , , [-10,10],

5 si x 0,( ,0)

0.5 si x > 0.L

R

u x t u x t K u x t u x t xt x x

uu x

u

⎫∂ ∂ ∂+ − = ∈ ⎪∂ ∂ ∂ ⎪

⎬= ≤⎧ ⎪= ⎨ ⎪=⎩ ⎭

u(x,t) =

e0.1t

2 Kπt5e

(−(x−0.1t−ξ)2

4⋅Kt)

−10

0

∫ dξ+ 0.5e(−

(x−0.1t−ξ)2

4⋅Kt)

0

10

∫ dξ.

The exact solution is

We can assume that u(x,t) is a solute concentration that flows inside a porous and saturated geological unit, due to advective-reactive-diffusive mechanisms. In order to describe this, we have the following transport equation, in which we want to observe the effect of variation in the Diffusion coefficient:

u(−10,t) = 5e0.1t ,u(10,t) = 0.5e0.1t .

⎧⎨⎪

⎩⎪Dirichlet boundary

conditions

Parameters:

Courant = 0.4; Dif. Coef. (α) = 0.4; Time = 10;

Δx = 1/15; Δt = min(coΔx/0.1, αΔx2/K).







Spatial Coordinate (x)

Solu

tion:

u(x

,10)

Exact sol. K=0Num. sol. K=0Exact sol. K=0.1Num. sol K=0.1Exact sol. K=0.25Num. sol. K=0.25 Exact sol. K=0.5Num. sol. K=0.5

16

4. Conclusions and future research.

The results obtained for the ADER-FV and ADER-DG suggest that is possible to reach high order for smooth solutions and avoid spurious oscillations near discontinuities. We also observe improvements in the solution when we move from the lower to higher order schemes.

It can be noted that we reach the theoretical first order accuracy in the integral L1 norm, for discontinuous solutions.

It is difficult to obtain accurate solutions near discontinuities.

When dealing with stiff source terms, classical ADER-FV may not work, while the new ADER-DG does a good job for these cases. Also we avoid the use of Taylor expansion in time and we do not need to use the C-K procedure (especially cumbersome in the case of nonlinear problems).

The new ADER-DG can be considered as more general, since it can be implemented for a general physical flux so that, if we change the problem we are solving, we just have to plug in the scheme the new flux.

Further research will include the extension to 2D and 3D, as well as resolution of Geological Engineering problems.


R. Castedo, A. HidalgoDpto. de Matemática Aplicada y Métodos Informáticos,

E.T.S.I. Minas, Univ. Politécnica de [email protected], [email protected]


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