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HighHigh--order ADER schemes for order ADER schemes for hyperbolic equations: a reviewhyperbolic equations: a review

Eleuterio TOROLaboratory of Applied Mathematics

University of Trento, ItalyWebsite: www.ing.unitn.it/toro

Email: toro@ing.unitn.it

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I am indebted to my collaborators:

Michael Dumbser and Svetlana Tokareva (Trento, Italy)Arturo Hidalgo (Madrid, Spain)

Carlos Pares, Manuel Castro and Alberto Pardo (Malaga, Spain)

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Part A:

FORCE-type centred, monotone schemes for hyperbolic equations in multiple space dimensions on general meshes

Part B:

ADER high-order methods

Part C:Sample applications: the 3D Euler equations, the 3D Baer-Nunziato equations for compressible multiphase flows on unstructured meshes

CONTENTS

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We are interested in developing numerical methods to solve

)()()()()( QDQSQHQGQFQ zyxt

)()()()()( QDQSQQCQQBQQAQ zyxt

Source terms S(Q) may be stiff

Advective terms may not admit a conservative form(nonconservative products)

Meshes are assumed unstructured

Very high order of accuracy in both space and time

May use upwind or centred approaches for numerical fluxes

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Design constraints:

We want the schemes to be

Conservative (in view of the Lax-Wendroff theorem)

Non-linear(in view of the Godunov’s theorem)

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In view of Godunov’s theorem our numerical approach relies on:

1.A monotone numerical flux (at most first order accurate)

2.A framework to construct non-linear numerical methods of high-order of accuracy in both space and time in which

The monotone flux is the bluilding block

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Part A:

FORCE-type centred, monotone schemes for hyperbolic equations in multiple space dimensions solved on general meshes

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Conservative schemes in 1D

2/1i2/1ini

1ni FF

xtQQ

0)Q(FQ xt

Task: define numerical flux

2/1iF

Basic property required: MONOTONICITY

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2/1in

1i

ni

xt

F0x if Q

0x if Q)0,x(Q

0)Q(FQ

There are two approaches:

)Q,Q(HF n1i

ni2/1i

I: Upwind approach (Godunov). Solve the Riemann problem

II: Centred approach. The numerical flux is

(Toro E F. Riemann solvers and numerical methods for fluid dynamics. Springer, 3° edition 2009)

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The FORCE flux (First ORder CEntred)

Brief review of 1D case

TORO E F, BILLETT S J. (2000). Centred TVD schemes for hyperbolic conservation laws. IMA JOURNAL OF NUMERICAL

ANALYSIS. vol. 20, pp. 47-79 ISSN: 0272-4979.

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Glimm’s method on a staggered mesh

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Recall that the integral form of the conservation laws

0)Q(FQ xt

in a control volume 21RL t,tx,x

R

L

R

L

x

x

t

tL

t

tRxx

x

xx dttxQFdttxQFdxtxQdxtxQ

2

1

2

1

)),(()),((),(),( 11

12

1

is

FORCE: replace random state by averaged state

Averaging operator

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)()(21

21

112/1

2/1ni

ni

ni

ni

ni QFQF

xtQQQ

)Q(F)Q(Fxt

21QQ

21Q n

1ini

ni

n1i

2/1n2/1i

)()(21

21 2/1

2/12/1

2/12/1

2/12/1

2/11

ni

ni

ni

ni

ni QFQF

xtQQQ

Step I

Step II

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force2/1i

force2/1i

ni

1ni FF

xtQQ

with numerical flux

Step III: One-step conservative form

)(21

2/12/12/1LF

iLW

iforce

i FFF

)Q(F)Q(Fxt

21QQ

21Q),Q(FF n

in

1in

1ini

lw2/1i

lw2/1i

LW2/1i

ni

n1i

n1i

ni

LF2/1i QQ

tx

21)Q(F)Q(F

21F

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)FF(21F LF

2/1iLW

2/1iforce

2/1i

The numerical flux is in fact

)Q(F)Q(Fxt

21QQ

21Q),Q(FF n

in

1in

1ini

lw2/1i

lw2/1i

LW2/1i

with

ni

n1i

n1i

ni

LF2/1i QQ

tx

21)Q(F)Q(F

21F

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Properties of the FORCE scheme

flforce

xforcext

ccx

qqqequationModifiedMonotone

cStable

21)1(

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1||0

2

)2(

Proof of convergence of FORCE scheme in:

Chen C Q and Toro E F.Centred schemes for non-linear hyperbolic equations.

Journal of Hyperbolic Differential. Equations 1 (1), pp 531-566, 2004.

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The FORCE flux for the scalar case:more general averaging.

10,F)1(FF LF2/1i

LW2/1i2/1i

(GFORCE) c1

1(FORCE) 2/1

Wendroff)-(Lax 1)Friedrichs-(Lax 0

Special cases:

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Related works on centred schemes include

H Nessyahu and E Tadmor. Non-oscillatory central differencing for hyperbolic conservation Laws. J. Computational Physics, Vol 87, pp 408-463, 1990.

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Recent extensions of FORCE

• Multiple space dimensions

• Unstructured meshes

• Path-conservative version

• High-order non-oscillatory (FV, DG, PC)

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Toro E F, Hidalgo A and Dumbser M.FORCE schemes on unstructured meshes I:

Conservative hyperbolic systems.(Journal of Computational Physics, Vol. 228, pp 3368-3389, 2009)

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Illustration in 2D

Consider, for example, triangular meshes in 2 space dimensions

Compact stencil

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Definition of secondary mesh for intercell fluxes

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Averaging operator applied on edge-base control volume gives

Initial condition: integral averages at time n niQ

j

j

j

j

nSV

V

Portion of j edge-base volume inside cell i

Portion of j edge-base volume outside cell i

Area of face j (between cells i and j)

Unit outward normal vector to of face j

Stage I

),,( HGFF

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Averaging operator applied on primary mesh gives

Initial condition: integral averages at time n+1/22/1n2/1jQ

Stage II

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Stage III: one-step conservative scheme

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Numerical flux in conservative scheme

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The Cartesian case

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Three space dimensions

0)Q(H)Q(G)Q(FQ zyxt

2/1k,j,i2/1k,j,ik,2/1j,ik,2/1j,ik,j,2/1ik,j,2/1in

k,j,i1nk,j,i HH

ztGG

ytFF

xtQQ

Task: to define numerical fluxes

Conservative one-step scheme:

2/1k,j,ik,2/1j,ik,j,2/1i H;G;F

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)Q(F)Q(Fxt

21QQ

21Q

),Q(FF

nk,j,i

nk,j,1i

nk,j,1i

nk,j,i

lwk,j,2/1i

lwj,2/1i

lwj,2/1i

Lax-Wendroff type flux

nk,j,i

nk,j,1i

nk,j,1i

nk,j,i

lfk,j,2/1i QQ

tx

21)Q(F)Q(F

21F

3

)FF(21F lf

k,j,2/1ilw

k,j,2/1iforce

k,j,2/1i

The FORCE flux in 3D is

Lax-Friedrichs type flux

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)()(21

21

)(

112/1

2/12/1

ni

ni

ni

ni

lwi

lwi

lwi

QFQFxtQQQ

QFF

ni

n1i

n1i

ni

lf2/1i QQ

tx

21)Q(F)Q(F

21F

parameter:

)FF(21F lf

2/1ilw

2/1iforce

2/1i

One-dimensional interpretation

0)Q(FQ xt

2/1i2/1ini

1ni FF

xtQQ

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One-dimensional interpretation

0)t,x(q)t,x(q xt

2/1i2/1ini

1ni ff

xtqq

The scheme is monotone and linearly stable under the condition

12c

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Numerical resultsfor the 1D Euler equations

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35

35.0c

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Analysis of the FORCE schemes in 1,2 and 3 space

dimensions

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Consider model hyperbolic equation

;q)q(h;q)q(g;q)q(f0)q(h)q(g)q(fq

321

zyxt

k,j,2/1ik,j,2/1ik,j,2/1ik,j,2/1ik,j,2/1ik,j,2/1in

k,j,i1nk,i,i hh

ztgg

ytff

xtqq

Monotonicity, linear stability and numerical viscosity

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Stability and monotonicity results

FORCE-type fluxes

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Part A. Summary:

I have presented a new conservative, centred monotone scheme for hyperbolic systems in three space dimensions, solved on general unstructured meshes

The flux can be used in the following schemes:

• Finite volume• Discontinuous Galerkin FE methods (ADER, RK)• Path-conservative methods (Pares et al. 2006)

The flux is the building block of high-order non- linear methods

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Part B:

High-Order Methods

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The ADER approach

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Toro E F, Millington R and Nejad L (1999). Towards Very High-Order Godunov Schemes. In Godunov Methods: Theory and

Applications, Edited Review, E F Toro (Editor), Kluwer/Plenum Academic Publishers, 2001, pp 907-940.

First written account of schemes for linear problems in 1D, 2D and 3D on structured meshes.

Further developments:

Toro, Titarev, Schwartzkopff , Munz, Dumbser, Kaeser, Takakura, Castro, Vignoli, Hidalgo, Enaux,Grosso, Russo,

Felcman,…

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High accuracy.But why ?

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CPU Time106 107 108 109 1010 101110-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

O2O3O4O5O6O15O16O23O24

iso-error

iso

-CP

UT

ime

0.8 orders

3.8

orde

rs

3.3

ord

ers

Collaborators: Munz, Schwartzkoppf (Germany), Dumbser (Trento)

Test for acoustics ADER

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t x

xixti

t

iti

x

xx

ni

i

i

i

i

dxdttxQSS

dQFF

dxxQQ

0

11

02/1

12/1

1

2/1

2/1

2/1

2/1

)),((

))((

)0,(

Exact relation between integral averages

i2/1i2/1ini

1ni tSFF

xtQQ

Exact relation

)Q(S)Q(FQ xt

],0[],[ 2/12/1 txx ii Integration in space and time

on control volume

Integral averages

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t

LRti dQFF0

12/1 ))((

Godunov’s finite volume scheme in 1D

)(QLR

2/1i2/1ini

1ni FF

xtQQ

Conservative formula

Godunov’s numerical flux

: Solution of classical Riemann problem

x

x

t

q(x,0)

qL

qR

QL QR

x=0

x=0

Classical Riemann Problem

0 0

)0,(

0)(

1 xifQxifQ

xQ

QFQ

ni

ni

xt

)/()0( txQ

))0(())0(( )0(

0

)0(12/1 QFdQFF

t

ti

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t x

xixti

t

iti

x

xx

ni

i

i

i

i

dxdttxQSS

dQFF

dxxQQ

0

11

02/1

12/1

1

2/1

2/1

2/1

2/1

)),((

))((

)0,(

Illustration of ADER finite volume method

i2/1i2/1ini

1ni tSFF

xtQQ

Update formula

)Q(S)Q(FQ xt

],0[],[ 2/12/1 txx ii

Integral average at time n

Control volume in computational domain

Numerical flux

Numerical source

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ADER on 2D unstructured meshes

The numerical flux requires the calculation of an integral in space alongThe volume/element interface and in time.

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Local Riemann problems from high-order representation of data

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Key ingredient:

the high-order (or generalized)

Riemann problem

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The high-order (or derivative, or generalized) Riemann problem:

KGRP 0)(0)(

)0,(

)()(

xifxQxifxQ

xQ

QSQFQ

R

L

xt

Initial conditions: two smooth functions

)x(Q),x(Q RL

For example, two polynomials of degree K

The generalization is twofold: (1) the intial conditions are two polynomials of arbitrary degree(2) The equations include source terms

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53

x

Tim

e

-10 0 100

3

6

9

12

15

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Four solvers:

C E Castro and E F Toro. Solvers for the high-order Riemann problem for hyperbolic balance laws. Journal Computational Physics Vol. 227, pp 2482-2513,, 2008

M Dumbser, C Enaux and E F Toro. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws . Journal of Computational Physics, Vol 227, pp 3971-4001, 2008.

E F Toro and V A Titarev. Soloution of the generalized Riemann problem for advection-reaction equations. Proc. Royal Society of London, A, Vol. 458, pp 271- 281, 2002.

E F Toro and V A Titarev. Derivative Riemann solvers for systems of conservation laws and ADER methods. Journal Computational Physics Vol. 212, pp 150-165,2006

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K

1k

k)k(

tLR !k)0,0(Q)0,0(Q)(Q

The leading termand

higher-order terms

Solver 1Toro E. F. and Titarev V. A. Proc. Roy. Soc. London. Vol. 458, pp 271-281, 2002

Toro E. F. and Titarev V. A. J. Comp. Phys. Vol. 212, No. 1, pp. 150-165, 2006.

(Based on work of Ben-Artzi and Falcovitz, 1984, see also Raviart and LeFloch 1989)

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Initial conditions

q

q

q

q

q

q

q

q

q

(3)

(2)

(K)

(1)

(2)

(3)

(K)

q (1)L

L

L

L

R

R

R

R

R

L(x)

(x)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

x=0x

q(x,0)

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0x if )0(Q0x if )0(Q

)0,x(Q

0)Q(FQ

R

L

xt

)t/x(D )0(Solution:

)0(D)0,0(Q )0(Leading term:

Computing the leading term:Solve the classical RP

Take Godunov state at x/t=0

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)Q,....,Q(G)t,x(Q )k(x

)0(x

)k()k(t

Computing the higher-order terms:

First use the Cauchy-Kowalewski (*) procedure yields

q)(qq)(q

qq0qq

)m(x

m)m(t

)2(x

2)2(t

xt

xt

Example:

Must define spatial derivatives at x=0 for t>0

(*) Cauchy-Kowalewski theorem. One of the most fundamental results in the theory of PDEs. Applies to problems in which all functions involved are analytic.

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Then construct evolution equations for the variables:

)t,x(Q)k(x

)Q,...,Q,Q(H)Q()Q(A)Q( )k(x

)1(x

)0(x

)k()k(xx

)k(xt

For the general case it can be shown that:

Neglecting source terms and linearizing we have

0)Q())0,0(Q(A)Q( )k(xx

)k(xt

Computing the higher-order terms

0q)(q)(0qq xxxtxt Note:

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0x if )0(Q0x if )0(Q

)0,x(Q

0)Q())0,0(Q(A)Q(

R)k(

x

L)k(

x)k(x

)k(xx

)k(xt

)t/x(D )k(Similarity solution

)0(D)0,0(Q )k()k(x

All spatial derivatives at x=0 are now defined

Evaluate solution at x/t=0

Computation of higher-order termsFor each k solve classical Riemann problem:

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))0,0(Q),....,0,0(Q(G)0,0(Q )k(x

)0(x

)k()k(t

Computing the higher-order termsAll time derivatives at x=0 are then defined

!k)0,0(Q)0,0(Q)(Q

kK

1k

)k(tLR

Solution of DRP is

GRP-K = 1( non-linear RP) + K (linear RPs)

Options: state expansion and flux expansion

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t x

xixti

t

iti

x

xx

ni

i

i

i

i

dxdttxQSS

dQFF

dxxQQ

0

11

02/1

12/1

1

2/1

2/1

2/1

2/1

)),((

))((

)0,(

Illustration of ADER finite volume method

i2/1i2/1ini

1ni tSFF

xtQQ

Update formula

)Q(S)Q(FQ xt

],0[],[ 2/12/1 txx ii

Integral average at time n

Control volume in computational domain

Numerical flux

Numerical source

The latest solverM Dumbser, C Enaux and E F Toro. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. Journal of Computational Physics, Vol 227, pp 3971-4001, 2008.

Extends Harten’s method (1987)**

•Evolves data left and right prior to “time-interaction”•Evolution of data is done numerically by an implicit space- time DG method•The solution of the LOCAL generalized Riemann problem is therefore implicit•The scheme remains globally explicit•Stiff source terms can be treated adequately•Reconciles stiffness with high accuracy in both space and time

**C E Castro and E F Toro. Solvers for the high-order Riemann problem for hyperbolic balance laws. Journal Computational Physics Vol. 227, pp 2482-2513,2008

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Main features of ADER schemes

)(

)()()()(

QSQCQBQAQ

QSQHQGQFQ

zyxt

zyxt

One-step fully discrete schemesReconstruction done once per time step

Accuracy in space and time is arbitrary

Unified framework

Finite volume, DG finite element and Path-conservativeformulations

General meshes

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Part C:

APPLICATIONS

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Main applications so far

1, 2, 3 D Euler equations on unstructured meshes3D Navier-Stokes equationsReaction-diffusion (parabolic equations)Sediment transport in water flows (single phase)Two-phase sediment transport (Pitman and Le model)Two-layer shallow water equationsAeroacoustics in 2 and 3DSeismic wave propagation in 3DTsunami wave propagationMagnetohydrodynamics3D Maxwell equations3D compressible two-phase flow, etc.

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Two and three-dimensional Euler equations

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2D Euler equations on unstructured triangular meshes

Two space dimensions

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70

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2D Euler equations: reflection from triangle

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3D Euler equations: reflection from cone

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Baer-Nunziato equations

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)()()( QSQTQFQ xxt

7

6

5

4

3

2

1

2

2

)( 0

0

)(

)ˆˆˆˆˆ(ˆˆˆˆˆˆ

ˆˆˆ0

)()(

ˆˆˆˆˆˆ

ˆˆ

sssssss

QS

VPP

VVPP

QT

pEupu

u

pEupu

u

QF

Eu

Eu

Q

ii

i

i

ii

i

The Baer-Nunziato equations in 1D

:Q Solid phase :Q̂ Gas phase

puVi ˆP i

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Application of ADER to the 3D Baer-Nunziato equations

12 PDESStiff source terms: relaxation terms

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EXTENSION TO NONCONSERVATIVE SYSTEMS:Path-conservative schemes

DUMBSER M, HIDALGO A, CASTRO M, PARES C, TORO E F. (2009).FORCE Schemes on Unstructured Meshes II: Nonconservative Hyperbolic Systems.

(Under review)

Also published (NI09005-NPA) in pre-print series of the Newton Institute for Mathematical Sciences

University of Cambridge, UK.

It can be downloaded fromhttp://www.newton.ac.uk/preprints2009.html

CASTRO M, PARDO A, PARES C, TORO E F (2009).ON SOME FAST WELL-BALANCED FIRST ORDER SOLVERS FOR

NONCONSERVATIVE SYSTEMS. MATHEMATICS OF COMPUTATION. ISSN: 0025-5718. Accepted.

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Three space dimensions

Unstructured meshes

Path-conservative method

Centred non-conservative FORCE is bluilding block

ADER: high-order of accuracy in space and time

(implemented upto 6-th order in space and time)

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Reference solutions to the BN equations

Exact Riemann solver of Schwendemann et al. (2006) (1D)

Exact smooth solution the 2D BN equations to be used in convergence rate studies (Dumbser et al. 2009)

Spherically symmetric 3D BN equations reduced to 1D system with geometric source terms. This is used to test 2 and 3 dimensional solutions with shocks (Dumbser et al. 2009)

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Convergence rates study in 2D unstructured meshes

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BN equations: spherical explosion test

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Double Mach reflection for the 2D Baer-Nunziato equations

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Double Mach reflection for the 2D Baer-Nunziato equations

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Summary:

Part A: FORCE scheme in 2 and 3D on general meshes.

Part B: ADER high-order schemes

Part C: Application to the 3D Euler and Baer- Nunziato two-phase flow equations

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Thank you

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