well balanced methods for conservation laws with source …randall j. leveque applied mathematics...

Well Balanced Methods for Conservation Lawswith Source Terms

Randall J. LeVequeApplied Mathematics

University of Washington

Supported in part by NSF, ONR, NIH

Randy LeVeque, University of Washington FAN2010 Conference, Duke University, June 29, 2010

Cons. law with source terms (balance law)

In one space dimension:

qt + f(q)x = ψ(q)σx(x)

Note: if σ(x) = x then source term is just ψ(q).

Goal: Compute accurate quasi-steady solutions, smallperturbations of equilibria qe(x) satisfying

f(qe)x = ψ(qe)σx(x).

Method is well balanced if qe(x) is exactly maintained bynumerical method.

Fractional steps for a quasisteady problemAlternate between solving homogeneous conservation law

qt + f(q)x = 0 (1)

and source termqt = ψ(q). (2)

When qt � f(q)x ≈ ψ(q):

• Solving (1) gives large change in q

• Solving (2) should essentially cancel this change.

Numerical difficulties:

• (1) and (2) are solved by very different methods. Generally willnot have proper cancellation.

• Nonlinear limiters are applied to f(q)x term, not tosmall-perturbation waves. Large variation in steady state hidesstructure of waves.

Equilibrium solutionsShallow water equations with bathymetry/topography B(x):

ht + (hu)x = 0

(hu)t +(hu2 +

12gh2)x

= −ghBx(x)

Ocean-at-rest equilibrium:

ue ≡ 0, he(x) +B(x) ≡ η̄ = sea level.

The hydrostatic pressure gradient 12g(h2)x balances the source term.

Stationary atmosphere with pressure gradient balancing gravity is similar.

Flowing equilibria: Stationary solutions have hu ≡ constant and

(hu2 +

12gh2)x

+ ghBx(x) = 0.

If solution is smooth then E = 12u

2 + g(h+B) is also constant in space.

Equilibrium solutionsShallow water equations with bathymetry/topography B(x):

ht + (hu)x = 0

(hu)t +(hu2 +

12gh2)x

= −ghBx(x)

The hydrostatic pressure gradient 12g(h2)x balances the source term.

Stationary atmosphere with pressure gradient balancing gravity is similar.

Flowing equilibria: Stationary solutions have hu ≡ constant and

(hu2 +

12gh2)x

+ ghBx(x) = 0.

If solution is smooth then E = 12u

2 + g(h+B) is also constant in space.

Cross section of Indian Ocean & tsunami

Surface elevationon scale of 10 meters:

Cross-sectionon scale of kilometers:

Advection-decay example

qt + uqx = −qσx(x)

Solution advects with speed u and decays where σx > 0, growswhere σx < 0.

Ex: σ(x) = x =⇒ q(x, t) = e−t/uq0(x− ut)IBVP on x ≥ 0 with q(0, t) = µ has

Equilibrium solution: qe(x) = µe−x/u.

Decaying exponential propagates to right but decaysdownwards and the two effects balance out.

Advection-decay

Equilibrium solution

Advection-decay

Advection

Advection-decay

Exponential decay

Advection-decay equation: qt + uqx = −qσx(x)

Demos...

Outline

• Wave propagation algorithms for qt + f(q)x = 0Riemann solver:

Jump Qi −Qi−1 is split into wavesWave limiters for high resolution without oscillations.

• f-wave formulation:Jump f(Qi)− f(Qi−1) is split into waves

• Incorporate source term: qt + f(q)x = ψ(q)σx:

f(Qi)− f(Qi−1)−Ψi−1/2(σi − σi−1) is split into waves

• Choice of Ψi−1/2: Path conservative approach

Outline

Godunov’s Method for qt + f(q)x = 0

1. Solve Riemann problems at all interfaces, yielding wavesWpi−1/2 and speeds spi−1/2, for p = 1, 2, . . . , m.

Riemann problem: Original equation with piecewise constantdata.

Wave-propagation viewpoint

For linear system qt+Aqx = 0, the Riemann solution consists of

wavesWp propagating at constant speed λp.λ2∆t

W1i−1/2

W1i+1/2

W2i−1/2

W3i−1/2

Qi −Qi−1 =m∑p=1

αpi−1/2rp ≡

m∑p=1

Wpi−1/2.

Qn+1i = Qni −

∆t∆x[λ2W2

i−1/2 + λ3W3i−1/2 + λ1W1

Upwind wave-propagation algorithm

Qn+1i = Qni −

∆t∆x

m∑p=1

(spi−1/2)+Wpi−1/2 +

m∑p=1

(spi+1/2)−Wpi+1/2

s+ = max(s, 0), s− = min(s, 0).

Note: Requires only waves and speeds.

Applicable also to hyperbolic problems not in conservation form.

For qt + f(q)x = 0, conservative if waves chosen properly,e.g. using Roe-average of Jacobians.

Great for general software, but only first-order accurate (upwindmethod for linear systems).

Wave-propagation form of high-resolution method

Qn+1i = Qni −

∆t∆x

m∑p=1

(spi−1/2)+Wpi−1/2 +

m∑p=1

(spi+1/2)−Wpi+1/2

− ∆t

∆x(F̃i+1/2 − F̃i−1/2)

Correction flux:

F̃i−1/2 =12

Mw∑p=1

|spi−1/2|(

1− ∆t∆x|spi−1/2|

)W̃pi−1/2

where W̃pi−1/2 is a limited version ofWp

i−1/2 to avoid oscillations.

(Unlimited waves W̃p =Wp =⇒ Lax-Wendroff for a linearsystem =⇒ nonphysical oscillations near shocks.)

Summary of wave propagation algorithms

For qt + f(q)x = 0, the flux difference

A∆Qi−1/2 = f(Qi)− f(Qi−1)

is split into:

left-going fluctuation: A−∆Qi−1/2, updates Qi−1,right-going fluctuation: A+∆Qi−1/2, updates Qi,Waves: Qi −Qi−1 =

∑αprp =

Often take A±∆Qi−1/2 =∑

(sp)±Wp.

f-wave formulation: Bale, RJL, Mitran, Rossmanith, SISC 2002

f-waves: f(Qi)− f(Qi−1) =∑βprp =

(sgn(sp))±Zp.

In either case, limiters are applied to waves or f-waves for usein high-resolution correction terms.

A∆Qi−1/2 = f(Qi)− f(Qi−1)

is split into:

∑αprp =

(sp)±Wp.

(sgn(sp))±Zp.

A∆Qi−1/2 = f(Qi)− f(Qi−1)

is split into:

∑αprp =

(sp)±Wp.

(sgn(sp))±Zp.

Incorporating source term in f-waves

Concentrate source at interfaces: Ψi−1/2(σi − σi−1)

Split f(Qi)− f(Qi−1)− (σi − σi−1)Ψi−1/2 =∑

pZpi−1/2

Use these waves in wave-propagation algorithm.

Steady state maintained:

If f(Qi)−f(Qi−1)∆x = Ψi−1/2

(σi−σi−1)∆x then Zp ≡ 0

Near steady state:

Deviation from steady state is split into waves and limited.

Incorporating source term in f-waves

qt + f(q)x = ψ(q)σx(x) =⇒ Ψi−1/2(σi − σi−1)

Question: How to average ψ(q) between cells to get Ψi−1/2?

For some problems (e.g. ocean-at-rest) can simply usearithmetic average.

Ψi−1/2 =12

(ψ(Qi−1) + ψ(Qi)).

Shallow water equations with bathymetry B(x)

ht + (hu)x = 0

(hu)t +(hu2 +

12gh2)x

= −ghBx(x)

UsingΨi−1/2 = −g

2(hi−1 + hi)

gives exactly well-balanced method, but only because hydrostatic pressure isquadratic function of h:

f(Qi)− f(Qi−1)−Ψi−1/2(Bi −Bi−1) =

i −12gh2

2(hi−1 + hi)(Bi −Bi−1)

2(hi−1 + hi)((hi +Bi)− (hi−1 +Bi−1))

= 0 if hi +Bi = hi−1 +Bi−1 = η̄.

ht + (hu)x = 0

(hu)t +(hu2 +

12gh2)x

= −ghBx(x)

2(hi−1 + hi)

f(Qi)− f(Qi−1)−Ψi−1/2(Bi −Bi−1) =

i −12gh2

2(hi−1 + hi)(Bi −Bi−1)

2(hi−1 + hi)((hi +Bi)− (hi−1 +Bi−1))

= 0 if hi +Bi = hi−1 +Bi−1 = η̄.

ht + (hu)x = 0

(hu)t +(hu2 +

12gh2)x

= −ghBx(x)

2(hi−1 + hi)

f(Qi)− f(Qi−1)−Ψi−1/2(Bi −Bi−1) =

i −12gh2

2(hi−1 + hi)(Bi −Bi−1)

2(hi−1 + hi)((hi +Bi)− (hi−1 +Bi−1))

= 0 if hi +Bi = hi−1 +Bi−1 = η̄.

Advection-decay example

qt + uqx = −qσx(x).

on 0 ≤ x ≤ 10 with

σ(x) = Ae−(x−5)2 +B(tanh(x− 5) + 1)

Equilibrium solution:

q(x, 0) = qe(x) = e−σ(x)/u.

Note: σ is nearly flat at boundaries, increasing and thendecreasing in domain.

So solution advects with speed u and decays to nearly zero,then grows again.

Demos...

Use path conservative approach

Rewrite qt + f(q)x = ψ(q)σx(x) as

qt + f(q)x − ψ(q)σx = 0σt = 0

This is a nonconservative hyperbolic system wt +B(w)wx = 0with:

w =[qσ

], B(w) =

[f ′(q) −ψ(q)

Eigenvalues of B are those of A and λ = 0.

Assume “nonresonant” case: 0 is not an eigenvalue of A.

qt + f(q)x − ψ(q)σx = 0σt = 0

w =[qσ

], B(w) =

[f ′(q) −ψ(q)

qt + f(q)x − ψ(q)σx = 0σt = 0

w =[qσ

], B(w) =

[f ′(q) −ψ(q)

Nonconservative hyperbolic problems

Now considerqt +A(q)qx = 0

where A(q) is not the Jacobian of a flux function f(q).Still hyperbolic if A is diagonalizable with real eigenvalues.

Correct discontinuous (weak) solutions harder to define.

Vanishing viscosity approach: add diffusive term of order ε,

qt +A(q)qx = εDqxx

For conservation law, there is generally a unique limit as ε→ 0.

Not true for nonconservative system.Limit can depend on D, or perhaps other higher order terms.

Note that if q is discontinuous then qx is a delta function withsupport at points where A(q) is discontinuous.

qt +A(q)qx = εDqxx

Dal Maso, LeFloch, Murat (DLM) theory

qt +A(q)qx = 0.

Define nonconservative product in terms of Borel measuresrelated to family of paths in state space.

q(s) = Φ(ql, qr, s)

is a path from ql to qr parameterized by s ∈ [0, 1].

Jump conditions for a discontinuity propagating with speed ξ:

ξ(qr − ql) =∫ 1

0A(q(s))q′(s) ds.

If A(q)qx = f(q)x (conservative) then∫ 1

0A(q(s))q′(s) ds = f(qr)− f(ql)

independent of the choice of path.

qt +A(q)qx = 0.

ξ(qr − ql) =∫ 1

0A(q(s))q′(s) ds.

independent of the choice of path.

qt +A(q)qx = 0.

ξ(qr − ql) =∫ 1

0A(q(s))q′(s) ds.

independent of the choice of path.Randy LeVeque, University of Washington FAN2010 Conference, Duke University, June 29, 2010

Path conservative numerical methods

Given Qi−1, Qi, want to determine A−∆Qi−1/2, A+∆Qi−1/2 toincrement cells on either side:

Qn+1i = Qni −

∆t∆x

(A+∆Qi−1/2 +A−∆Qi+1/2)

For a conservation law: Conservative method if

A−∆Qi−1/2 +A+∆Qi−1/2 = f(Qi)− f(Qi−1)

Path conservative method (for a given family of paths Φ):Choose A−∆Qi−1/2, A+∆Qi−1/2 so that

A−∆Qi−1/2 +A+∆Qi−1/2 =∫ 1

0A(q(s))q′(s) ds.

where q(s) = Φ(Qi−1, Qi; s).

Path conservative numerical methods

Given Qi−1, Qi, want to determine A−∆Qi−1/2, A+∆Qi−1/2 toincrement cells on either side:

Qn+1i = Qni −

∆t∆x

(A+∆Qi−1/2 +A−∆Qi+1/2)

For a conservation law: Conservative method if

A−∆Qi−1/2 +A+∆Qi−1/2 = f(Qi)− f(Qi−1)

Path conservative method (for a given family of paths Φ):Choose A−∆Qi−1/2, A+∆Qi−1/2 so that

A−∆Qi−1/2 +A+∆Qi−1/2 =∫ 1

0A(q(s))q′(s) ds.

where q(s) = Φ(Qi−1, Qi; s).

qt + f(q)x − ψ(q)σx = 0σt = 0

w =[qσ

], B(w) =

[f ′(q) −ψ(q)

qt + f(q)x − ψ(q)σx = 0σt = 0

w =[qσ

], B(w) =

[f ′(q) −ψ(q)

qt + f(q)x − ψ(q)σx = 0σt = 0

w =[qσ

], B(w) =

[f ′(q) −ψ(q)

For any path w(s) in state space we obtain∫ 1

B(w(s))w′(s) ds =∫ 1

[f ′(q(s)) −ψ(q(s))

] [q′(s)σ′(s)

=[ ∫ 1

0f ′(q(s))q′(s)− ψ(q(s))σ′(s) ds

No increment to σ as expected. Increment to Q is:

A∆Qi−1/2 = f(Qi)− f(Qi−1)−∫ 1

ψ(q(s))σ′(s) ds.

If σ(s) = σi−1 + s(σi − σi−1), then σ′(s) = σi − σi−1.

=⇒ Recover f-wave modification with particular average of ψ.

Can do better: Choose path in state space to simplify integral and guaranteewell balanced.

B(w(s))w′(s) ds =∫ 1

[f ′(q(s)) −ψ(q(s))

] [q′(s)σ′(s)

=[ ∫ 1

0f ′(q(s))q′(s)− ψ(q(s))σ′(s) ds

A∆Qi−1/2 = f(Qi)− f(Qi−1)−∫ 1

B(w(s))w′(s) ds =∫ 1

[f ′(q(s)) −ψ(q(s))

] [q′(s)σ′(s)

=[ ∫ 1

0f ′(q(s))q′(s)− ψ(q(s))σ′(s) ds

A∆Qi−1/2 = f(Qi)− f(Qi−1)−∫ 1

B(w(s))w′(s) ds =∫ 1

[f ′(q(s)) −ψ(q(s))

] [q′(s)σ′(s)

=[ ∫ 1

0f ′(q(s))q′(s)− ψ(q(s))σ′(s) ds

A∆Qi−1/2 = f(Qi)− f(Qi−1)−∫ 1

Equilibrium path

Consider path (Qi−1, σi−1) −→ (Q̂i, σi) −→ (Qi, σi)parameterized by s going from 0 to 1/2 to 1, say.

First part: σ varies, q remains in equilibrium(Source term balances f ′(q(s))q′(s))

Second part: σ constant.

0f ′(q(s))q′(s)− ψ(q(s))σ′(s) ds

=∫ 1

1/2f ′(q(s))q′(s)− ψ(q(s))σ′(s) ds

=∫ 1

1/2f ′(q(s))q′(s) ds

= f(Qi)− f(Q̂i)

Equilibrium path

Consider path (Qi−1, σi−1) −→ (Q̂i, σi) −→ (Qi, σi)parameterized by s going from 0 to 1/2 to 1, say.

First part: σ varies, q remains in equilibrium(Source term balances f ′(q(s))q′(s))

Second part: σ constant.

0f ′(q(s))q′(s)− ψ(q(s))σ′(s) ds

=∫ 1

1/2f ′(q(s))q′(s)− ψ(q(s))σ′(s) ds

=∫ 1

1/2f ′(q(s))q′(s) ds

= f(Qi)− f(Q̂i)

Equilibrium path

Example: qt + uqx = −qσx(x).

For u > 0 the Riemann solution for Wi−1,Wi has two waves:speed 0: jump from (Qi−1, σi−1) to (Q̂i, σi),speed u: jump from (Q̂i, σi) to (Qi, σi).

where Q̂i = Qi−1e−∆x/u.

So ∫ 1

0f ′(q(s))q′(s)− ψ(q(s))σ′(s) ds = f(Qi)− f(Q̂i)

= u(Qi −Qi−1e−∆x/u)

This is the quantity we split into waves.Equals zero when solution is in equilibrium. Demos...

Equilibrium path

Example: qt + uqx = −qσx(x).

For u > 0 the Riemann solution for Wi−1,Wi has two waves:speed 0: jump from (Qi−1, σi−1) to (Q̂i, σi),speed u: jump from (Q̂i, σi) to (Qi, σi).

where Q̂i = Qi−1e−∆x/u.

So ∫ 1

0f ′(q(s))q′(s)− ψ(q(s))σ′(s) ds = f(Qi)− f(Q̂i)

= u(Qi −Qi−1e−∆x/u)

This is the quantity we split into waves.Equals zero when solution is in equilibrium. Demos...

A few references

• Clawpack: www.clawpack.org

• R. J. LeVeque. A Well-Balanced Path-Integral f-wave Method forHyperbolic Problems with Source Terms, preprint, 2010:www.clawpack.org/links/wbfwave10

• D. Bale, R. J. LeVeque, S. Mitran, and J. A. Rossmanith. Awave-propagation method for conservation laws and balance laws withspatially varying flux functions. SIAM J. Sci. Comput., 24:955–978,2002.

• B. Khouider and A. J. Majda, A non-oscillatory balanced scheme foran idealized tropical climate model, Theor. Comput. Fluid Dyn. 19(2005), 331-354.

• G. Dal Maso, P. G. LeFloch, F. Murat, Definition and weak stabilityof nonconservative products, J. Math. Pures Appl. 74 (1995),483–548.

• M. J. Castro, P. G. LeFloch, M. L. Munoz-Riuz, C. Parés, Why manytheories of shock waves are necessary: Convergence error informally path-consistent schemes, J. Comput. Phys. 227 (2008),8107–8129.

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