1

Eleuterio Toro Laboratory of Applied Mathematics

University of Trento, Italy www.ing.unitn.it/toro

toro@ing.unitn.it

Introduction to numerical methods for hyperbolic conservation laws:

FORCE-type schemes

2

The big picture: 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

3

Recall the integral form of the conservation laws

0)Q(FQ xt =∂+∂

in a control volume [ ] [ ]21RL t,tx,x ×

∫ ∫∫∫⎥⎥⎦

⎤

⎢⎢⎣

⎡

Δ−

ΔΔ

Δ−

Δ=

Δ

R

L

R

L

x

x

t

tL

t

tR

x

x

dttxQFt

dttxQFt

tx

dxtxQx

dxtxQx

2

1

2

1

)),((1)),((11),(1),(112

is

∫ ∫∫∫⎥⎥⎦

⎤

⎢⎢⎣

⎡−−=

R

L

R

L

x

x

t

tL

t

tR

x

x

dttxQFdttxQFdxtxQdxtxQ2

1

2

1

)),(()),((),(),( 12

[ ]2/12/11

−++ −

Δ

Δ−=⇒ ii

ni

ni FF

xtQQ

4

Conservative schemes in 1D

[ ]2/1i2/1ini

1ni FF

xtQQ −+

+ −ΔΔ

−=

0)Q(FQ xt =∂+∂

Task: define numerical flux

2/1iF+Basic property required: MONOTONICITY

5

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. Solve the Riemann problem

II: Centred approach. The numerical flux is

6

Properties required from 2-point flux )V,U(HF 2/1i =+

)U(F)U,U(HF 2/1i ==+

...)q,q,q(...Lq)q,q(hf n1i

ni

n1i

1ni

n1i

ni2/1i +−

+++ =→=

Consistency:

Definition: a monotone scheme satisfies

k0...)q,q,q(...Lq

n1i

ni

n1in

k

∀≥∂∂

+−

Monotonicity:

7

Properties required from 2-point flux

0)v,u(hv

fv

;0)v,u(hu

fu 2/1i2/1i ≤

∂∂

=∂∂

≥∂∂

=∂∂

++

Theorem: for a two-point flux, necessary conditions for monotonicity are

∑−=

++ β=

r

lk

nkik

1ni qqRemark: for a linear scheme

monotonicity requires positivity of coefficients: k0k ∀≥β

Classical centred numerical fluxes

( ) ( )nin1i

n1i

ni

LF2/1i QQ

tx

21)Q(F)Q(F

21F −

ΔΔ

−+= +++

The Lax-Friedrichs flux

Properties

1.  Linearly stable for 2.  Monotone for all CFL numbers in the stability range 3.  Largest local truncation error of all monotone schemes

1|c|0 ≤≤

x/tc ΔλΔ= the Courant number

8

9

Classical centred numerical fluxes, contin...

( ) ( ))Q(F)Q(Fxt

21QQ

21Q),Q(FF n

in1i

n1i

ni

lw2/1i

lw2/1i

LW2/1i −

ΔΔ

−+== +++++

The Lax-Wendroff flux (2 versions)

Properties

1.  Linearly stable for 2.  Non-monotone (oscillatory) 3.  Second-order accurate in space and time

( ) ( ))Q(F)Q(FAxt

21)Q()Q(F

21F n

in1i2/1i

n1i

ni

LW2/1i −

ΔΔ

−+= ++++

1|c|0 ≤≤

10

Classical centred numerical fluxes, contin...

( ) ( ))Q(F)Q(FxtQQ

21Q),Q(FF n

in1i

n1i

ni

gc2/1i

gc2/1i

GC2/1i −

ΔΔ

−+== +++++

The Godunov centred flux (1961)

Properties

1.  Linearly stable for 2. Monotone for 3. Non-monotone for

2|c|0 21≤≤

2|c| 21

21 ≤≤

21|c|0 ≤≤

11

The FORCE flux (First ORder CEntred)

Toro E F.

On Glimm-related schemes for conservation laws. Technical Report MMU-9602, Department of Mathematics and

Physics, Manchester Metropolitan University, 1996,UK

12

Glimm’s method on a staggered mesh

13

Recall the integral form of the conservation laws

0)Q(FQ xt =∂+∂

in a control volume [ ] [ ]21RL t,tx,x ×

∫ ∫∫∫⎥⎥⎦

⎤

⎢⎢⎣

⎡

Δ−

ΔΔ

Δ−

Δ=

Δ

R

L

R

L

x

x

t

tL

t

tR

x

x

dttxQFt

dttxQFt

tx

dxtxQx

dxtxQx

2

1

2

1

)),((1)),((11),(1),(112

14

( ) ( ))Q(F)Q(Fxt

21QQ

21Q n

in1i

n1i

ni

2/1n2/1i −

ΔΔ

−+= ++++

( ) ( ))Q(F)Q(Fxt

21QQ

21Q n

1ini

ni

n1i

2/1n2/1i −−

+− −

ΔΔ

−+=

( ) ( ))()(21

21 2/1

2/12/12/1

2/12/1

2/12/1

1 +−

++

++

+−

+ −ΔΔ

−+= ni

ni

ni

ni

ni QFQF

xtQQQ

Step I

Step II

15

( )force2/1i

force2/1i

ni

1ni FF

xtQQ −+

+ −ΔΔ

−=

Question: can we write

as a one-step conservative method

with a given numerical flux

( ) ( ))()(21

21 2/1

2/12/12/1

2/12/1

2/12/1

1 +−

++

++

+−

+ −ΔΔ

−+= ni

ni

ni

ni

ni QFQF

xtQQQ

force2/1iF+

16

⎥⎦

⎤⎢⎣

⎡ −ΔΔ

−++= +++++ )QQ(

txF)Q(F2F

41F n

in1i

n1i

2/1n2/1i

ni

force2/1i

Answer: YES

The numerical flux is

( ) ( ))Q(F)Q(Fxt

21QQ

21Q),Q(FF n

in1i

n1i

ni

lw2/1i

lw2/1i

LW2/1i −

ΔΔ

−+== +++++

But recall

( ) ( )nin1i

n1i

ni

LF2/1i QQ

tx

21)Q(F)Q(F

21F −

ΔΔ

−+= +++

17

)FF(21F LF

2/1iLW2/1i

force2/1i +++ +=

The numerical flux is in fact

( ) ( ))Q(F)Q(Fxt

21QQ

21Q),Q(FF n

in1i

n1i

ni

lw2/1i

lw2/1i

LW2/1i −

ΔΔ

−+== +++++

with

( ) ( )nin1i

n1i

ni

LF2/1i QQ

tx

21)Q(F)Q(F

21F −

ΔΔ

−+= +++

18

Properties of the FORCE scheme

[ ]2/1i2/1ini

1ni

xt

ffxtqq

0qq

−++ −

ΔΔ

−=

=∂λ+∂

21

20

21

n1i1

ni0

n1i1

1ni

n1i

2ni

2force

2/1i

)c1(41b)c1(

21b)c1(

41b

qbqbqbq

)q(c4)c1()q(

c4)c1(f

−=−=+=

++=

λ−

+λ+

=

−−

+−−+

++

19

Properties of the FORCE scheme, cont.

flforce

xforcext

ccx

qqqequationModifiedMonotone

cStable

αλα

αλ

21)1(

41

1||0

2

)2(

=−

Δ=

∂=∂+∂

≤≤

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. Vol. 1 (1), pp 531-566, 2004.

20

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

10,F)1(FF LF2/1i

LW2/1i2/1i <ω<ω−+ω= ++

ω+

(GFORCE) c1

1(FORCE) 2/1

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

+=ω

=ω

=ω

=ω

Special cases:

21

Monotonocity

1||1

121

||110 maxmax ≤

+≡≤

+≡≤≤

ccωωω

0 0.5 10

0.5

1

MONOTONE

SCHEMES

Godunov Centred

Godunov Upwind

Courant number

WeightW

LF

FORCE

LW

22

FORCE’s friends and relatives

•  The composite schemes of Liska and Wendroff (friend)

Liska R and Wendroff B. Composite schemes for conservation laws. SIAM J. Numerical Analysis, Vol. 35, pp 2250-2271, 1998

•  The centred scheme of Nessyahu and Tadmor (relative)

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

23

Numerical results for 1D Euler equations

24

25

26

27

28

29

How about extensions of FORCE ?

•  High-order non-oscillatory extensions •  Source terms •  Multiple space dimensions •  Unstructured meshes

30

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)

FORCE schemes on unstructured meshes

31

Illustration in 2D

Triangular primary mesh Primary and secondary mesh

32

Averaging operator applied on edge-based 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

Step I

),,( HGFF ==

33

Averaging operator applied on primary mesh gives

Initial condition: integral averages at time n+1/2 2/1n2/1jQ

++

Step II

34

Step III: one-step conservative scheme

35

( ) ( ))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

−ΔΔα

−+=

=

++α

+

α+

α+

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 −

ΔαΔ

−+= ++α+

)FF(21F lf

k,j,2/1ilw

k,j,2/1iforce

k,j,2/1iα+

α+

α+ +=

The FORCE flux in α space dimensions on Cartesian meshes

Lax-Friedrichs type flux

36

Stability and monotonicity results

FORCE-type fluxes

37

( ) ( ))()(21

21

)(

112/1

2/12/1

ni

ni

ni

ni

lwi

lwi

lwi

QFQFxtQQQ

QFF

−Δ

Δ−+=

=

+++

++

αα

αα

( ) ( )nin1i

n1i

ni

lf2/1i QQ

tx

21)Q(F)Q(F

21F −

ΔαΔ

−+= ++α+

)FF(21F lf

2/1ilw

2/1iforce

2/1iα+

α+

α+ +=

One-dimensional interpretation

0)Q(FQ xt =∂+∂

[ ]2/1i2/1ini

1ni FF

xtQQ −+

+ −ΔΔ

−=

α: parameter

38

Numerical results for the 1D Euler equations

39

Numerical results for the 1D Euler equations

40

Numerical results:

Euler equations in 2D and 3D

41

2D Euler equations: reflection from triangle

42

3D Euler equations: reflection from cone

43

Numerical results:

The Baer-Nunziato equations in 2D and 3D

44

Application of ADER to the 3D Baer-Nunziato equations

11 non-linear hyperbolic PDES stiff source terms: relaxation terms

45

EXTENSION TO NONCONSERVATIVE SYSTEMS:

Manuel Castro, Carlos Pares, Eleuterio Toro, Michael Dumbser and Arturo Hidalgo.

FORCE schemes on unstructured meshes II: non- conservative hyperbolic systems.

Computer Methods in Applied Mechanics and Engineering. Vol. 199, Issue 9-12, pages 625-647, 2010.

Also published (NI09005-NPA) in pre-print series of the

Newton Institute for Mathematical Sciences University of Cambridge, UK.

It can be downloaded from

http://www.newton.ac.uk/preprints2009.html

M Castro, A Pardo, C Pares and E F Toro On some fast well- balanced first order solvers for

nonconservative systems Mathematics of Computation. Vol. 79, pages 1427-1472, 2009.

.

46

Double Mach reflection for the 2D Baer-Nunziato equations

47

Double Mach reflection for the 2D Baer-Nunziato equations

48

Summary on FORCE

Ø  A centred scheme Ø  One-step scheme Ø  In conservative form, with a numerical flux Ø  Monotone Ø  Linearly stable up to CFL =1, 1/2, 1/3 Ø  Very simple to use, applicable to any system (useful for

complicated systems) Ø  High-order extensions (TVD, WENO, DG, ADER)

Ø  Further reading: Chapter 18 of:

Toro E F. Riemann solvers and numerical methods for fluid dynamics. Springer, Third Edition, 2009.

49

Thank You