dune-fv a library for h-adaptive finite-volume schemes · G. Puppo (Univ. dell’Insubria, Como),...

dune-fv

a library for h-adaptive finite-volume schemes

M. SempliceDipartimento di Matematica

Universita degli Studi di Torino

Joint work with: I. Cravero (Univ. Torino), A.Coco (Univ. Bristol),G. Puppo (Univ. dell’Insubria, Como), G. Russo (Univ. Catania)

SHARK-FV, 18th of May 2015

Partly supported by GNCS-INDAM, Progetti di ricerca 2013

1 FV schemes on locally adapted meshes

2 1D reconstructions for non-uniform meshes


4 h-adaptive code and tests

5 The dune-fv library

One dimensional grids and binary trees

Locally refine cells, splitting them in two halves.A cell of level k has size hk = h02−k .

xk=1 k=2 k=3 k=3

h0

Two dimensional grids and quad-trees

locally refine cells, splitting them in two halves

a cell of level k has size hk = h02−k .

Finite volume discretization

d

dtuj +

∫∂Ωj

f (u(x(γ)))dγ = 0

boundaryquadrature

∫∂Ωj

f (u(x(γ)))dγ ' Q∂Ωj(f (u))

' Q∂Ωj(F (u; x(γ)))

numer. fluxes

F (u; x(γ)) = F(u(x(γ)in), u(x(γ)out)

)

stages

u(i)j = unj − ∆t

|Ωj |

i−1∑k=1

aikQΩj(F (u(k)))

step

un+1j = unj − ∆t

|Ωj |

ν∑i=1

biQΩj(F (u(i)))

Q: midpoint or 2-point Gauss quadrature on each edge of ∂Ωj

u(i),in/out(x(γ)) are some boundary value reconstructions.


d

dtuj +

∫∂Ωj

f (u(x(γ)))dγ = 0

boundaryquadrature

∫∂Ωj

f (u(x(γ)))dγ ' Q∂Ωj(f (u)) ' Q∂Ωj

(F (u; x(γ)))

numer. fluxes F (u; x(γ)) = F(u(x(γ)in), u(x(γ)out)

)stages

u(i)j = unj − ∆t

|Ωj |

i−1∑k=1

aikQΩj(F (u(k)))

step

un+1j = unj − ∆t

|Ωj |

ν∑i=1

biQΩj(F (u(i)))




d

dtuj +

∫∂Ωj

f (u(x(γ)))dγ = 0 R-K:c A

bt

boundaryquadrature

∫∂Ωj

f (u(x(γ)))dγ ' Q∂Ωj(f (u)) ' Q∂Ωj

(F (u; x(γ)))

numer. fluxes F (u; x(γ)) = F(u(x(γ)in), u(x(γ)out)

)stages u

(i)j = unj − ∆t

|Ωj |

i−1∑k=1

aikQΩj(F (u(k)))

step un+1j = unj − ∆t

|Ωj |

ν∑i=1

biQΩj(F (u(i)))



FV scheme on locally adapted grid

We aim at 3rd order schemes

good resolution properties

quite compact stencils(costly gathering neighbourhoods in adaptive meshes)

We need:

refinement/coarsening criteria, robust and applicable forsystems of conservation laws

time advancement scheme (global or local timestepping)

numerical flux (LLF, . . . )

a robust reconstruction procedure that

can work on locally adapted (e.g. quad-tree) meshescan be evaluated at many points on the cell boundary





We need:










We need:










We need:










We need:










We need:










We need:





can work on locally adapted (e.g. quad-tree) meshes

can be evaluated at many points on the cell boundary





We need:






Error indicator: the numerical entropy production

is defined as the residual of the entropy inequalityη(u)t +∇ · ψ(u) ≤ 0 on the numerical solution

Snj =1

|Ωj |∆t

(∫Ωj

η(u(tn+1))−∫

Ωj

η(u(tn)) +

∫ tn+1

tn

∫∂Ωj

ψ(u(t)) · n

)

Snj is computed recycling the boundary data alreadycomputed for the evolution

it has the same size of the truncation error:

on smooth flows Snj = O(hp)on rarefaction corners Snj = O(h)on contacts Snj = O(1)on shocks Snj ∼ C/h

Puppo SIAM J. Sci. Comput., 2003

Puppo, Semplice Comm. in Comput. Phys., 2011

Puppo, Semplice Preprint arXiv:1403.4112



Snj =1

|Ωj |∆t

(∫Ωj

η(u(tn+1))−∫

Ωj

η(u(tn)) +

∫ tn+1

tn

∫∂Ωj

ψ(u(t)) · n

)









Snj =1

|Ωj |∆t

(∫Ωj

η(u(tn+1))−∫

Ωj

η(u(tn)) +

∫ tn+1

tn

∫∂Ωj

ψ(u(t)) · n

)



on smooth flows Snj = O(hp)

on rarefaction corners Snj = O(h)on contacts Snj = O(1)

on shocks Snj ∼ C/h






Snj =1

|Ωj |∆t

(∫Ωj

η(u(tn+1))−∫

Ωj

η(u(tn)) +

∫ tn+1

tn

∫∂Ωj

ψ(u(t)) · n

)







Numerical entropy production

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

shock

contact

corner

rarefaction

Sod test problem

103

N of cells

10-3

10-2

10-1

100

101

O(N)

O(1)O(1)

Numerical entropy

103

N of cells

10-8

10-7

10-6

10-5

10-4

10-3

O(1/N)O(1/N)

O(1/N2 )

O(1/N3 )

Numerical entropy

scheme of order 2 continuous linescheme of order 3 dashed line

The shallow water system

Balance law

ht + (hu)x = 0

(hu)t + (hu2 + 12gh

2)x = −ghzx

Entropy pair

η(h, u) =1

2hu2 +

1

2gh2 + gzh

ψ(h, u) = u η(h, u)

H(x)=h(x)+z(x)h(x)

z(x)

Balance laws with geometric source terms

reconstruct (h + z), q, and h obtaining point values at xj+1/2±

hydrostatic recontruction1 ⇒ states

U∗j+1/2±

=(h∗j+1/2±

, qj+1/2±

)modify fluxes:FRi = F (U∗

j+1/2−,U∗

j+1/2+) + g2 [0, (hj+1/2−)2 − (h∗

j+1/2−)2]T

FLi = F (U∗

j−1/2−,U∗

j−1/2+) + g2 [0, (hj−1/2−)2 − (h∗

j−1/2−)2]T

add the source term

∆xid

dtU i (t) + FR

i −FLi = [0, S

(p)i ]T

use a well-balanced quadrature of order p

S(1)i = 0 S

(2)i = −g

2 (hj+1/2− +hj−1/2+)(zj+1/2−− zj−1/2+)

1 Audusse et al. – SIAM J. Sci. Comput. 25 (2004), 2050–2065

Schemes of order 3 and 4

4th order evaluation of the source term via Richardson’sextrapolation of the trapezoidal quadrature2

S(4)j = −g

2

43 (hj + hj−1/2+)(zj − zj−1/2+)

+ 43 (hj+1/2− + hj)(zj+1/2− − zj)

−13 (hj+1/2− + hj−1/2+)(zj+1/2− − zj−1/2+)

⇒ need reconstructions hj , zj at cell center

2 Noelle, Pankratz, Puppo, Natvig – Well-balanced finite volume schemesof arbitrary order of accuracy for shallow water ows. JCP 213 (2006) 474–499.






WENO3 reconstruction on non-uniform mesh

Let POPT(x) be the central parabola and

PR(x) = uj +uj+1 − ujxj+1 − xj

(x−xj) PL(x) = uj +uj − uj−1

xj − xj−1(x−xj)

For general3 x ∈ Ωj ,

∃CR(x),CL(x) ∈ R : POPT(x) = CL(x)PL(x) + CR(x)PR(x)

IL =(uj−uj−1

xj−xj−1

)2αL(x) = CL(x)

(IL+ε)2 , IR = . . . , αR = . . .

ωL(x) = αL(x)αL(x)+αR(x) ωR(x) = αR(x)

αL(x)+αR(x)

Reconstrauction in Ωj at x = ωL(x)PL(x) + ωR(x)PR(x)

3 Gerolymos – Representation of Lagrange reconstructing polynomial bycombination of substencils JCAM 236 (2012), 2763–2794.


Let POPT(x) be the central parabola and

PR(x) = uj +uj+1 − ujxj+1 − xj

(x−xj) PL(x) = uj +uj − uj−1

xj − xj−1(x−xj)

For general3 x ∈ Ωj ,

∃CR(x),CL(x) ∈ R : POPT(x) = CL(x)PL(x) + CR(x)PR(x)

IL =(uj−uj−1

xj−xj−1

)2αL(x) = CL(x)

(IL+ε)2 , IR = . . . , αR = . . .

ωL(x) = αL(x)αL(x)+αR(x) ωR(x) = αR(x)

αL(x)+αR(x)

Reconstrauction in Ωj at x = ωL(x)PL(x) + ωR(x)PR(x)

3 Gerolymos – Representation of Lagrange reconstructing polynomial bycombination of substencils JCAM 236 (2012), 2763–2794.


For example, for right cell boundary:

CL(xj+1/2) =hj−1

hj−1 + hj + hj+1CR(xj+1/2) =

hj + hj+1

hj−1 + hj + hj+1

for left cell boundary:

CL(xj−1/2) =hj + hj+1

hj−1 + hj + hj+1CR(xj−1/2) =

hj−1

hj−1 + hj + hj+1

Each reconstruction point requires its set of weights

For uniform meshes, CL(xj+1/2) = 13 ,CR(xj+1/2) = 2

3 , etc

Weights depend on the mesh geometry in the neighborhoodand should be recomputed after grid adaption

It is impossible to find weights for cell center x = xj ,which is a problem for well-balanced quadratures



CL(xj+1/2) =hj−1

hj−1 + hj + hj+1CR(xj+1/2) =

hj + hj+1

hj−1 + hj + hj+1


CL(xj−1/2) =hj + hj+1

hj−1 + hj + hj+1CR(xj−1/2) =

hj−1

hj−1 + hj + hj+1



3 , etc





CL(xj+1/2) =hj−1

hj−1 + hj + hj+1CR(xj+1/2) =

hj + hj+1

hj−1 + hj + hj+1


CL(xj−1/2) =hj + hj+1

hj−1 + hj + hj+1CR(xj−1/2) =

hj−1

hj−1 + hj + hj+1



3 , etc





CL(xj+1/2) =hj−1

hj−1 + hj + hj+1CR(xj+1/2) =

hj + hj+1

hj−1 + hj + hj+1


CL(xj−1/2) =hj + hj+1

hj−1 + hj + hj+1CR(xj−1/2) =

hj−1

hj−1 + hj + hj+1



3 , etc



1D CWENO third order reconstruction

P(2)opt(x)


P(2)opt(x)

Pl (x)

Pr (x)

2/3 =hj−1 + hj

hj−1 + hj + hj+1

1/3 =hj+1

hj−1 + hj + hj+1

1/3

2/3

WENO3: mesh-dependent, point-dependent coefficients


P(2)opt(x)

Pl (x)

Pr (x)

Pc (x)

∀x : P(2)opt(x) =

1

2Pc(x) +

1

4Pr (x) +

1

4Pl(x)


P(2)opt(x)

Pl (x)

Pr (x)

Pc (x)

∀x : P(2)opt(x) =

1

2Pc(x) +

1

4Pr (x) +

1

4Pl(x)

1/4

1/4

1/2


Generalize the construction of CWENO4 to non-uniform mesh:POPT fitting uj−1, uj , uj+1

Choose Cξ > 0, CC + CR + CL = 1 and define PC by∀x : POPT (x) = CCPC (x) + CRPR(x) + CLPL(x)

Indicators IC , IR , IL and weights ωC , ωR , ωL as in usual WENO

ωC , ωR , ωL do not depend on geometry nor evaluation point:

Pj(x) = ωCPC (x) + ωRPR(x) + ωLPL(x)

is uniformly accurate and can be evaluated where needed(cell boundaries, cell centers, etc)

4 Levy, Puppo, Russo – Compact Central WENO schemes formultidimensional conservation laws – SIAM J. Sci. Comput. (2001)

















Accuracy test on the reconstruction (1D)

Tests on fixed nonuniform meshesErrors of the values reconstructed from cell averages

Smooth test, quasi-uniform gridN ‖E‖1 rate ‖E‖∞ rate

20 5.63e-03 2.17e-0240 8.65e-04 2.70 3.85e-03 2.5080 1.08e-04 3.01 5.26e-04 2.87

160 1.26e-05 3.10 5.36e-05 3.30320 1.47e-06 3.09 7.00e-06 2.94640 1.75e-07 3.08 8.85e-07 2.98

1280 2.11e-08 3.05 1.11e-07 3.002560 2.59e-09 3.03 1.39e-08 3.00

Smooth test, random gridN ‖E‖1 rate ‖E‖∞ rate

20 5.36e-03 1.87e-0240 4.97e-04 3.43 1.35e-03 3.7980 5.07e-05 3.29 1.35e-04 3.32

160 5.46e-06 3.22 1.67e-05 3.02320 6.25e-07 3.13 2.08e-06 3.01640 7.46e-08 3.07 2.64e-07 2.98

1280 9.11e-09 3.03 3.15e-08 3.072560 1.12e-09 3.02 4.19e-09 2.91

Shallow water equations, trancritical flow with a shock

Third order (CWENO) scheme with 200 points:

8 9 10 11 12 13 14 150.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28Water height

VIJIVIRGI

WSPYXMSRSRRSRYRMJSVQQIWL

WSPYXMSRSRYRMJSVQQIWL

Shallow water equations, trancritical flow with a shock

Third order (CWENO) scheme with 200 points:

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45Water height

XSTSKVETL]

QIWLWM^ISJRSRYRMJSVQQIWL

XVERWGVMXMGEPWLSGO

RSSWGMPPEXMSRWLIVI

A note on the choice of ε

Computation of the nonlinear weigts (both WENO and CWENO):

αξ =Cξ

(Iξ + ε)2ωλ =

αλ∑ξ∈R,L,0

αξ

ε originally introduced to avoid zero denominator

ε affects convergence rates in both WENO and CWENO

on uniform meshes, ε ∼ h2 can restore WENO convergence 5,or use ‘mapped WENO’ 6, modify indicators, . . .

on uniform meshes, for CWENO both ε ∼ h and ε ∼ h2 work 7

5 Arandiga et al., SINUM 20126 Henrick et al., JCP 20057 Kolb, SINUM 2014

A primer on the choice of ε


αξ =Cξ

(Iξ + ε)2ωλ =

αλ∑ξ∈R,L,0

αξ

The Jiang-Shu regularity indicators:

Ij ,ξ =∑r≥1

(hj)2r−1

∫Ωj

[P

(r)j ,ξ (x)

]2dx .

The relative value of ε and hj biases (or not) the reconstructiontowards the central one, because

Ij ,ξ =

O(h2

j ) in general

O(1) when there is a discontinuity in the stencil ofPj ,ξ

O(h4j ) when there is an extremum in the stencil ofPj ,ξ

A primer on the choice of ε


αξ =Cξ

(Iξ + ε)2ωλ =

αλ∑ξ∈R,L,0

αξ

The Jiang-Shu regularity indicators:

Ij ,ξ =∑r≥1

(hj)2r−1

∫Ωj

[P

(r)j ,ξ (x)

]2dx .

The relative value of ε and hj biases (or not) the reconstructiontowards the central one, because

Ij ,ξ =

O(h2

j ) in general

O(1) when there is a discontinuity in the stencil ofPj ,ξ

O(h4j ) when there is an extremum in the stencil ofPj ,ξ

Choosing ε on non-uniform meshes

Also on non-uniform meshes (both WENO and CWENO):

αξ =Cξ

(Iξ + ε)2Ij ,ξ =

O(h2

j ) in general

O(1) if discontinuous in stencil(j , ξ)

O(h4j ) if extremum in stencil(j , ξ)

ε acts as a threshold

ε should shade extrasmooth values O(h4j )

we may have cells of very different size in the mesh

choosing ε as a function of the (local) mesh size, e.g. ε = h2j ,

is of paramount importance in h-adaptive schemes.

Cravero, Semplice On the accuracy of WENO and CWENOreconstructions of third order on nonuniform meshes SubmittedPreprint http://arxiv.org/abs/1503.00736



αξ =Cξ

(Iξ + ε)2Ij ,ξ =

O(h2

j ) in general











αξ =Cξ

(Iξ + ε)2Ij ,ξ =

O(h2

j ) in general











αξ =Cξ

(Iξ + ε)2Ij ,ξ =

O(h2

j ) in general









Accuracy test on linear transport, smooth solution

Third order scheme on random grids:

101 102 103 10410−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−11−norm error

10−6 weno

10−6 cwenoh wenoh cweno

h2 weno

h2 cweno

Accuracy test on linear transport, discontinuous solution

101 102 103 10410−3

10−2

10−1

1001−norm error

10−6 WENO3

10−6 CWENO3h WENO3h CWENO3

h2 WENO3

h2 CWENO3

101 102 103 1041.85

1.9

1.95

2

2.05

2.1

2.15

2.2

2.25Total Variation






First neighbours in quad-tree meshes (examples)

1 2 3

4

567

8

1

2 3 45

1 2 3

10

11

“Dimensional splitting” is non-trivial. . . At second order, see e.g.

Puppo, Semplice Finite volume schemes on 2D non-uniform gridsProceedings of HYP2012

Second degree optimal polynomial in 2D

In a locally adapted mesh, the number of neighbours is always ≥ 5:

Ωi

Choose P2OPT(x , y) such that

exact cell average in Ωi :

1|Ωi |

∫Ωi

POPT = ui

least squares in theneighbourhood Ni of Ωj , i.e.minimize∑j∈Ni

∣∣∣∣∣ 1|Ωj |

∫Ωj

POPT − uj

∣∣∣∣∣2

First degree polynomials in 2D

Let NNEi be the neighbours in the NORTH-EAST sector:

Ωi

Choose P1NE(x , y) such that

exact cell average in Ωj :

1|Ωi |

∫Ωj

PNE = ui

least squares in theneighbourhood Ni of Ωj , i.e.minimize∑

j∈Ni

∣∣∣∣∣ 1|Ωj |

∫Ωj

PNE − uj

∣∣∣∣∣2

There are always at least 2 cells in each of NNEi ,NNW

i ,N SWi ,N SE

i .

First degree polynomials in 2D

Let NNEi be the neighbours in the NORTH-EAST sector:

Ωi

Choose P1NE(x , y) such that

exact cell average in Ωj :

1|Ωi |

∫Ωj

PNE = ui

least squares in theneighbourhood Ni of Ωj , i.e.minimize∑

j∈Ni

∣∣∣∣∣ 1|Ωj |

∫Ωj

PNE − uj

∣∣∣∣∣2

There are always at least 2 cells in each of NNEi ,NNW

i ,N SWi ,N SE

i .

CWENO third order reconstruction in 2D

Pc defined by

POPT = 12Pc + 1

8PNE + 18PNW + 1

8PSE + 18PSW

non-linear weights are defined with the usual regularityindicators and the reconstruction is

Pj(x , y) = ωcPc + ωNEPNE + . . .+ ωSWPSW

P is later evaluated where needed (quadrature points on ∂Ωj)

Semplice, Coco, Russo Adaptive Mesh Refinement for HyperbolicSystems based on Third-Order Compact WENO Reconstruction J.Sci. Comput. (2015)


Pc defined by

POPT = 12Pc + 1

8PNE + 18PNW + 1

8PSE + 18PSW






Pc defined by

POPT = 12Pc + 1

8PNE + 18PNW + 1

8PSE + 18PSW





Accuracy test on the reconstruction (2D)

Reconstruction of boundary extrapolated data for

sin(πx) cos(πy)

Example of adapted grid based on 8× 8 coarse grid:

-1 1

-1

1

Adaptive grid, ε = 10−6

NC ‖E‖1 rate ‖E‖∞ rate

82 6.19e-02 3.11e-01

162 1.11e-02 2.48 9.61e-02 1.69

322 1.75e-03 2.67 2.53e-02 1.93

642 2.33e-04 2.91 6.40e-03 1.98

Adaptive grid, ε = hjNC ‖E‖1 rate ‖E‖∞ rate

82 4.48e-02 1.90e-01

162 7.00e-03 2.68 3.74e-02 2.34

322 8.16e-04 3.10 3.79e-03 3.30

642 8.91e-05 3.20 2.55e-04 3.89






Loop of the h-adaptive algorithm

compute un+1j and

indicator Snj

∃j :

Snj > Sref

hj > hmin

?

coarsen to parent

if all sons have

Snj < Scoa

locally

refine

(locally)

recompute

NO

YES

(nexttimestep)

Local recomputation with a 3 stages Runge-Kutta

Split the shaded cell and set the subcells averages at time tn byaveraging the reconstruction on each subcell.

(first computation)

x

xtn

tn+1

(local recomputation of timestep)

1d linear transport of a smooth solution

solution, indicator, cell levels errors vs number of cells

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

Linear transport test 2

so

luti

on

exact

numerical

10−8

10−6

10−4

10−2

100

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−3

−2

−1

0

x

ce

ll l

ev

el

101

102

103

104

10−5

10−4

10−3

10−2

10−1

100

no

rm−

1 e

rro

rs

average N cells

Linear transport test 2

r=2, L=1

r=2, L=4, s=4

r=3, L=1

r=3, L=4, s=8


2d smooth problem (solution and error)

ut +∇ · ([−yx ]ϕ(r)u ) = 0 ϕ(r) = tanh(r)

cosh2(r)1

0.385r

on Ω = [−4, 4]2

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

Solution at t=4.0

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

101

102

103

10−4

10−3

10−2

10−1

100

101

no

rm−

1 e

rro

rs

average N cells per direction

2D smooth test

r=2, L=1

r=2, L=4, s=4

r=3, L=1

r=3, L=4, s=8

2d smooth problem (grids)

(Darker cells are smaller)

Second order Third order

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

Cell level (r=2)

0

1

2

3

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

Cell level (r=3)

0

1

2

3

Work-precision diagrams

For comparing different h-adaptive numerical schemes:

choose proxy of CPU time, e.g. time average of number ofcells N

choose norm of error, e.g. 1-norm

On uniform meshes, we expect:

E ∼ Hp = N−p/d in Rd , for smooth flows, i.e.H → H/2⇒ E → E/2p.

E ∼ H = N−1/d , for shocked solutions

For h-adaptive meshes, H and N are not linked any more

On shocks, can we do better than E ∼ N−1/d?Can we hope to observe E ∼ N−p/d?

Work-precision diagrams

For comparing different h-adaptive numerical schemes:

choose proxy of CPU time, e.g. time average of number ofcells N

choose norm of error, e.g. 1-norm

On uniform meshes, we expect:

E ∼ Hp = N−p/d in Rd , for smooth flows, i.e.H → H/2⇒ E → E/2p.

E ∼ H = N−1/d , for shocked solutions

For h-adaptive meshes, H and N are not linked any more

On shocks, can we do better than E ∼ N−1/d?Can we hope to observe E ∼ N−p/d?

A simple scaling argument

p order of the scheme (on smooth solution)

H coarse cell size

` refinement levels: fine cells are of size h = H/2`

O(Hp) typical error on smooth part, i.e. (1/H)d = Nd cells

O(h) typical error on shock, located on (d−1)-dimensionalsubmanifold

In order to reduce the total error by a factor 2p, we have to:

use coarse cells of size H/2

let the finest cells be of size h/2p, by increasing the availablenumber of refinement levels by p−1.

E ∼ (Ntot)−p/d may be observed for d = 1 or d = 2, p ≤ 2, since

then the number of small cells does not increase too much.



H coarse cell size











H coarse cell size









Burgers equation: formation of standing shock

101

102

103

10−4

10−3

10−2

10−1

no

rm−

1 e

rro

rs

average N cells

Burgers equation with standing shock

r=2, unif

r=2, L=3, s=2

r=2, L=3+, s=2

Burgers equation: formation of standing shock

101

102

103

10−4

10−3

10−2

10−1

no

rm−

1 e

rro

rs

average N cells

Burgers equation with standing shock

r=3, unif

r=3, L=3, s=2

r=3, L=3+, s=2

r=3, L=3++, s=2

r=3, L=3+, s=4

Burgers equation: solution with moving shocks

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

Burgers equation with moving shocks

t=0

t=0.25

Burgers equation: solution with moving shocks

102

103

10−4

10−3

10−2

10−1

no

rm−

1 e

rro

rs

average N cells

Burgers equation with moving shocks

r=2, unif

r=2, L=3+, s=2

r=3, unif

r=3, L=3++, s=2

1d Euler equations: shock-acoustic interaction

0.2 0.3 0.4 0.5 0.6 0.7 0.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

Shock−acoustic interaction

reference

N=1024

N=512


0.2 0.3 0.4 0.5 0.6 0.7 0.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8


referenceN

0=256, L=12, S

ref=1e−2 (<N>=656)


0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

2

4

6

8

10

Adaptive solution with r=3, N0=256, L=12, ε=1e−2

x

ce

ll l

ev

el


102

103

104

10−3

10−2

10−1

no

rm−

1 e

rro

rs

average N cells


r=2, L=1

r=2, L=6+, s=2

r=2, L=6+, s=4

r=3, L=1

r=3, L=6++, s=2

r=3, L=6++, s=4

2d Riemann problem

102

103

10−3

10−2

no

rm−

1 e

rro

rs

(average N cells)1/2

2D Riemann problem

r=2, L=1

r=2, L=4, s=2

r=2, L=4+, s=2

r=3, L=1

r=3, L=4+, s=2

r=3, L=4++, s=2

2d Riemann problem

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cell level (r=2)

0

1

2

3

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cell level (r=3)

0

1

2

3

2d Euler equations: shock-bubble interaction


Comparison of the solutions (Schlieren plots)

The third order solution is much sharper.


Grids (blue) and solutions (gray levels)

The third order solution is much sharperand it is computed on a much coarser grid!






The dune-fv library

Download source code from my webpage (GPL licence)After download you can integrate

on locally adapted quad-tree meshes

linear advection, Burgers’, uniform and non-uniform rotationin 2D, Euler equations

with spatial reconstructions up to order 3

with Runge-Kutta up to order 3

with numerical entropy as error indicator

Design principles:

DUNE library (www.dune-project.org) for grid management

coded in C++

make (heavy) use of polymorphism for maximum flexibility

all polymorphism is static (CRTP) for efficiency

easy to provide your own “plug-in”

The dune-fv library

Download source code from my webpage (GPL licence)After download you can integrate

on locally adapted quad-tree meshes

linear advection, Burgers’, uniform and non-uniform rotationin 2D, Euler equations

with spatial reconstructions up to order 3

with Runge-Kutta up to order 3

with numerical entropy as error indicator

Design principles:

DUNE library (www.dune-project.org) for grid management

coded in C++

make (heavy) use of polymorphism for maximum flexibility

all polymorphism is static (CRTP) for efficiency

easy to provide your own “plug-in”

Compile-time options

Compiles for linear advection, 1D, second order

dune-fv/src$ make dune_fv

Compiles with reconstructions of order n = 1, 2, 3

dune-fv/src$ make CPPFLAGS=’-D FV_REC=n’ dune_fv

or Runge-Kutta of order m = 1, 2, 3

dune-fv/src$ make CPPFLAGS=’-D FV_RK=m’ dune_fv

Compiles for 2D with quad-tree mesh of quadriraterals.

dune-fv/src$ make GRIDDIM=2 GRUDTYPE=ALUGRID_CUBE dune_fv

Structure of the main loop

main.cc#include "dune/fv/claw/claw_linear.hh"

typedef CLawLinear<GRIDDIM> CLaw;

#include "dune/fv/flux/llfe.hh"

typedef LLFE<CLaw> F;

#include "dune/fv/rec/reccweno.hh"

typedef RecP2<G,M,t_Urk,flagVec> Reconstruction;

typedef HeunTableaux Tableaux;

typedef ERK<Tableaux,G,M,F,Reconstruction,flagVec> RungeKutta;

RungeKutta RK(grid,mapper,Ui,Fi,Uf,Ff,Icons/*,Idiff*/);

while (t<tfin)

dt = RK.advance(t);

t += dt;

Conservation laws: the base class

clawbase.hh

//System of m conservation laws in dim spatial dimensions

template<class CLaw, int D, int M>

class CLawBase

public:

enum dim=D;

enum m=M;

typedef DoF<M> t_u;

typedef typename Dune::FieldVector<double,D> t_c;

CLawBase(t_u& _u):u(_u) //constructor from DoF of cell

t_u getF(t_c n, t_c x, double t) return ReferToDerived().getF(u,n,x,t);

double getVmax(t_c n, t_c x, double t)return ReferToDerived().getVmax(u,n,x,t);

double getVnormal(t_c n, t_c x, double t)return ReferToDerived().getVnormal(u,n,x,t);

double getEnt()return ReferToDerived().getEnt(u);

double getEntF(t_c n, t_c x, double t)return ReferToDerived().getEntF(u);

protected:

t_u &u;

private:

const CLaw& ReferToDerived() constreturn static_cast<const CLaw&>(*this);

;

Conservation laws

Defining a new scalar conservation law in 2Dmynewclaw.hh

class MyNewCLaw : public CLawBase<MyNewCLaw,2,1>

public:

MyNewCLaw(t_u _u):t_CLawBase(_u) //constructor

t_u getF(t_c n, t_c x, double t); //exact flux in direction n

double getEnt(); //entropy function

double getEntF(t_c n, t_c x, double t);

private:

//private data if needed (e.g. pressure for Euler eqns.)

//note that CLawBase already stores a reference to _u

;

Conclusions

FV scheme, AMR on quad-tree mesh, no virtual cells

numerical entropy production driving AMR

CWENO reconstructions in 1D and 2D nonuniform grids

3rd order stencil is very small: first neighbours

3rd order scheme employs much coarser grids

coding based on DUNE library (www.dune-project.org)

my code (the dune-fv module) is available athttp://www.personalweb.unito.it/matteo.semplice/

future work: MOOD, local timestepping. . .

Thanks for the attention!

Conclusions

FV scheme, AMR on quad-tree mesh, no virtual cells

numerical entropy production driving AMR

CWENO reconstructions in 1D and 2D nonuniform grids

3rd order stencil is very small: first neighbours

3rd order scheme employs much coarser grids

coding based on DUNE library (www.dune-project.org)

my code (the dune-fv module) is available athttp://www.personalweb.unito.it/matteo.semplice/

future work: MOOD, local timestepping. . .

Thanks for the attention!

Date post:	20-Sep-2020
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dune-fv a library for h-adaptive finite-volume schemes · G. Puppo (Univ. dell’Insubria, Como),...

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