High-resolution finite volume methods for

hyperbolic PDEs on manifolds

Randall J. LeVequeDepartment of Applied Mathematics

University of Washington

Supported in part by NSF, DOE

Overview

• High-resolution methods for first-order hyperbolic systems• Shock waves in nonlinear problems• Heterogeneous media with discontinuous properties• Godunov-type methods based on Riemann solvers• Second-order correction terms with limiters to minimize

dissipation and dispersion

Software

CLAWPACK (Conservation LAWs Package):

http://www.amath.washington.edu/˜claw

Includes a preliminary version of CLAWMAN for manifolds.

Developed by• Derek Bale (relativistic flow)• James Rossmanith (geophysical flow on the sphere)

Also includes AMRCLAW for adaptive mesh refinement onrectangular patches (with Marsha Berger).

AMR on a manifold

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Pressure at time t = 1

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Pressure at time t = 1.5

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Pressure at time t = 2

Outline

• Brief review of Godunov-type methods• wave propagation approach• f-wave approach for discontinous fluxes and source terms• curvilinear grids in flat space• manifolds:

• parallel transport data to cell edges• Express in local orthonormal frame• Solve locally-flat Riemann problem• Metric terms and geometric source term naturally

incorporated

References:

http://www.amath.washington.edu/˜rjl/publications and /students

J. Pons, Font, Ibanez, Marti, Miralles, General relativistic hydrodynamics with specialrelativistic Riemann solvers, Astron. Astrophys. 339 (1998), 638-642

Finite-difference Methods

• Pointwise values Qni ≈ q(xi, tn)• Approximate derivatives by finite differences• Assumes smoothness

Finite-volume Methods

• Approximate cell averages: Qni ≈1

∆x

∫ xi+1/2

xi−1/2

q(x, tn) dx

• Integral form of conservation law,

∂

∂t

∫ xi+1/2

xi−1/2

q(x, t) dx = f(q(xi−1/2, t)) − f(q(xi+1/2, t))

leads to conservation law qt + fx = 0 but also directly tonumerical method.

Godunov’s method

Qni defines a piecewise constant function

q̃n(x, tn) = Qni for xi−1/2 < x < xi+1/2

Discontinuities at cell interfaces =⇒ Riemann problems.

PSfrag replacementstn

tn+1

Qni

Qn+1i

q̃n(xi−1/2, t) ≡ q∨|

(Qi−1, Qi) for t > tn.

Fni−1/2 =1

k

∫ tn+1

tn

f(q∨|

(Qni−1, Qni )) dt = f(q

∨|

(Qni−1, Qni )).

Riemann solution for the Euler equations

PSfrag replacements

ρl

ul

pl

ρ∗l

u∗

p∗

ρ∗r

u∗

p∗

ρr

ur

pr

rarefaction wave shockcontact

The Roe solver uses the solution to a linear system

qt + Âi−1/2qx = 0.

All waves are simply discontinuities.

Typically a fine approximation if jumps are approximatelycorrect.

Wave-propagation viewpoint

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

waves Wp propagating at constant speed λp.

PSfrag replacements

λ2∆t

W1i−1/2W1i+1/2

W2i−1/2

W3i−1/2

Qi −Qi−1 =m∑

p=1

αpi−1/2

rp ≡m∑

p=1

Wpi−1/2

.

Qn+1i = Qni −

∆t

∆x

[λ2W2i−1/2 + λ3W3i−1/2 + λ1W1i+1/2

].

High-resolution method for systems

Qn+1i = Qni −

∆t

∆x

[A+∆Qi−1/2 + A−∆Qi+1/2

]−∆t

∆x(F̃i+1/2−F̃i−1/2)

A−∆Qi−1/2 =Mw∑

p=1

(spi−1/2

)−Wpi−1/2

,

A+∆Qi−1/2 =Mw∑

p=1

(spi−1/2

)+Wpi−1/2

,

Correction flux:

F̃i−1/2 =1

2

Mw∑

p=1

|spi−1/2

|(

1 − ∆t∆x

|spi−1/2

|)W̃p

i−1/2

where W̃pi−1/2

is a limited version of Wpi−1/2

.

CLAWPACK

http://www.amath.washington.edu/˜claw/

• Fortran codes with Matlab graphics routines.• Many examples and applications to run or modify.• 1d, 2d, and 3d.

User supplies:• Riemann solver, splitting data into waves and fluctuations

(Need not be in conservation form)

• Boundary condition routine to extend data to ghost cellsStandard bc1.f routine includes many standard BC’s

• Initial conditions — qinit.f

Some applications• Gas dynamics, Euler equations• Waves in heterogeneous / random media• Acoustics, ultrasound, seismology• Elasticity, plasticity, soil liquifaction• Flow in porous media, groundwater contamination, oil recovery• Geophysical flow on the sphere• Shallow water equations, bottom topography, tsunami propagation• Chemotaxis and pattern formation• Traffic flow• Crystal growth• Multi-fluid, multi-phase flows, bubbly flow• Streamfunction–vorticity form of incompressible flow• Projection methods for incompressible flow• Combustion, detonation waves• Astrophysics: binary stars, planetary nebulae, jets• Magnetohydrodynamics, shallow water MHD• Relativistic flow, black hole accretion• Numerical relativity — Einstein equations, gravity waves, cosmology

Spatially-varying flux functions

In one dimension:qt + f(q, x)x = 0

Examples:• Nonlinear elasticity in heterogeneous materials• Traffic flow on roads with varying conditions• Flow through heterogeneous porous media• Solving conservation laws on curved manifolds

Riemann problem for spatially-varying flux

qt + f(q, x)x = 0

Cell-centered discretization: Flux fi(q) defined in ith cell.PSfrag replacements

qt + fi−1(q)x = 0 qt + fi(q)x = 0

Qi−1 Qi

Qli−1/2

Qri−1/2

W1i−1/2

W2i−1/2

W3i−1/2

Flux-based wave decomposition:

fi(Qi) − fi−1(Qi−1) =m∑

p=1

βpi−1/2

rpi−1/2

≡m∑

p=1

Zpi−1/2

Wave-propagation algorithm using f-waves

Qn+1i = Qni −

∆t

∆x[A+∆Qi−1/2 + A−∆Qi+1/2]

− ∆t∆x

[F̃i+1/2 − F̃i−1/2]

Standard version: Qi −Qi−1 =∑m

p=1 Wpi−1/2

A−∆Qi+1/2 =m∑

p=1

(spi+1/2)−Wpi+1/2

A+∆Qi−1/2 =m∑

p=1

(spi−1/2)+Wpi−1/2

F̃i−1/2 =1

2

m∑

p=1

|spi−1/2|(

1 − ∆t∆x

|spi−1/2|)W̃pi−1/2.

Wave-propagation algorithm using f-waves

Qn+1i = Qni −

∆t

∆x[A+∆Qi−1/2 + A−∆Qi+1/2]

− ∆t∆x

[F̃i+1/2 − F̃i−1/2]

Using f -waves: fi(Qi) − fi−1(Qi−1) =∑m

p=1 Zpi−1/2

A−∆Qi−1/2 =∑

p:spi−1/2

0

Zpi−1/2,

F̃i−1/2 =1

2

m∑

p=1

sgn(spi−1/2)

(1 − ∆t

∆x|spi−1/2|

)Z̃pi−1/2

Source termsqt + f(q)x = ψ(q)

Quasi-steady problems with near-cancellation:f(q)x and ψ both large but qt ≈ 0.

Examples:• Atmosphere or stellar dynamics with gravity balanced by

hydrostatic pressure

• Shallow water equations in a lake over bottom topography

Fractional step method: Alternate between

1. qt + f(x)x = 0,

2. qt = ψ(q)

Large motions induced in each step should cancel out, but won’tnumerically.

Source termsqt + f(q)x = ψ(q)

Quasi-steady problems with near-cancellation:f(q)x and ψ both large but qt ≈ 0.

Examples:• Atmosphere or stellar dynamics with gravity balanced by

hydrostatic pressure

• Shallow water equations in a lake over bottom topography

Fractional step method: Alternate between

1. qt + f(x)x = 0,

2. qt = ψ(q)

Large motions induced in each step should cancel out, but won’tnumerically.

Riemann problem with a delta-function source term

qt + f(q)x = ∆xΨi−1/2 δ(x− xi−1/2)PSfrag replacements

qt + fi−1(q)x = 0 qt + fi(q)x = 0

Qi−1 Qi

Qli−1/2

Qri−1/2

W1i−1/2

W2i−1/2

W3i−1/2

Flux is no longer continuous:f(Qri−1/2) − f(Qli−1/2) = ∆xΨi−1/2.

fi(Qi)−fi−1(Qi−1)−∆xΨi−1/2 =m∑

p=1

βpi−1/2

rpi−1/2

≡m∑

p=1

Zpi−1/2

Multidimensional Hyperbolic Problems

Integral form of conservation law:

d

dt

∫∫

Ω

q(x, y, t) dx dy = −∫

∂Ω~n · ~f(q) ds.

If q is smooth then the divergence theorem gives

d

dt

∫∫

Ω

q(x, y, t) dx dy = −∫∫

Ω

~∇ · ~f(q) dx dy,

or ∫∫

Ω

[qt + ~∇ · ~f(q)

]dx dy = 0.

True for all Ω =⇒qt + f(q)x + g(q)y = 0,

Finite volume method in 2D

PSfrag replacementsxi−1/2

xi+1/2

yj−1/2

yj+1/2

Qij

xi−1/2

Qnij ≈1

∆x∆y

∫ yj+1/2

yj−1/2

∫ xi+1/2

xi−1/2

q(x, y, tn) dx dy.

d

dt

∫∫

Cij

q(x, y, t) dx dy =

∫ yj+1/2yj−1/2

f(q(xi+1/2, y, t) dy −

∫ yj+1/2yj−1/2

f(q(xi−1/2, y, t) dy

+

∫ xi+1/2xi−1/2

g(q(x, yj+1/2, t) dx −

∫ xi+1/2xi−1/2

g(q(x, yj−1/2, t) dx.

Suggests the unsplit method

Qn+1ij = Qnij −

∆t

∆x[Fni+1/2,j − Fni−1/2,j ] −

∆t

∆y[Gni,j+1/2 −Gni,j−1/2],

Unsplit Godunov in 2D

At each cell edge, the flux is determined by solving a Riemannproblem in the normal direction with data from the neighboringcells.

qt + f(q)x = 0

Fni+1/2,j = f(q∨|

(Qnij , Qni+1,j))

qt + g(q)y = 0

Gni,j−1/2 = g(q∨|

(Qni,j−1, Qnij))

Finite volume method on a curvilinear grid(Flat space)

Two possible approaches:

1. Transform equations to computational space.Discretize equations that include metric terms, source terms.

2. Update cell averages in physical space.Solve 1d Riemann problems for physical equations in directionnormal to cell edges to compute flux.

Example: Linear acoustics

Homogeneous medium with

density ρ ≡ 1, bulk modulus K ≡ 1, sound speed c ≡ 1,p = pressure, u = velocity, T = pI = stress tensor

∂

∂tp+

1√h

∂

∂xk

(√huk

)= 0

∂

∂tum +

1√h

∂

∂xk

(√hT km

)= −ΓmnkT kn .

One approach:Discretize these equations directly in computational coordinatesxk.

∂

∂tp+

1√h

∂

∂xk

(√huk

)= 0

∂

∂tum +

1√h

∂

∂xk

(√hT km

)= −ΓmnkT kn .

Note:• Spatially varying flux functions• Source term — conservative?

Better approach:

Update cell average of q over physical finite volume cell

• Store Cartesian velocity components u, v in each cell.• At each cell edge, use data on each side to

• compute normal velocities at edge,• solve 1d Riemann problem in normal direction,• scale resulting waves by length of side,• use to update cell average

Grid mapping:

PSfrag replacements

∆x1

∆x2

PSfrag replacements∆x1

∆x2

J =

[∂x/∂x1 ∂x/∂x2

∂y/∂x1 ∂y/∂x2

], H = JTJ =

[h11 h12h21 h22

], h = det(H).

Lengths of sides in physical space ≈ h11∆x1, h22∆x2,

Area of cell ≈√h∆x1∆x2.

Flux differencing around boundary approximates covariantdivergence of flux.

Shallow water flow into a cylinder

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5q(1) at time t = 1.5

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5q(1) at time t = 2.25

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5q(1) at time t = 1.5

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5q(1) at time t = 2.25

Acoustics on a manifold

∂

∂tp+

1√h

∂

∂xk

(√huk

)= 0

∂

∂tum +

1√h

∂

∂xk

(√hT km

)= −ΓmnkT kn .

Now we must represent velocities in “computational coordinates”

At each cell edge:

• Parallel transport cell-centered velocities to edge,• Change coordinates to a local orthonormal frame at cell edge

to obtain normal and tangential velocities,• Solve 1d Riemann problem normal to cell edge

(assuming locally flat)• Scale resulting waves by length of side,

transform back to cell-centered coordinates,• Update cell averages.

CLAWMAN software

Currently only 2d.

Requires metric tensor H

• 2 × 2 matrix as function of x1 and x2,• Used to compute scaling factors for edge lengths, cell areas,• Used for orthonormalization at cell edges.

Christoffel symbols are needed for parallel transport

• Computed by finite differencing H.

Parallel Transport

∂

∂xkum+Γmnku

n = 0 or∂

∂xk

[u1

u2

]+

[Γ1

1k Γ12k

Γ21k Γ

22k

] [u1

u2

]= 0.

Approximate using Taylor series:

um(xk ± 1

2∆xk

)≈ um(xk) ∓ 1

2∆xkΓmnku

n(xk).

Parallel Transport for acoustics

For acoustics with q = (p, u1, u2)T , solve Riemann problem with

q`i−1/2,j =

(I −

[ 0 0 00 Γ1

1k Γ12k

0 Γ21k Γ

22k

])qi−1,j

and

qri−1/2,j =

(I +

[ 0 0 00 Γ1

1k Γ12k

0 Γ21k Γ

22k

])qi,j

f-wave formulation

Split jump in fluxes ∆F into waves.

∆F 1 =

I +

0 0 00 Γ1

1k Γ12k

0 Γ21k Γ

22k

ij

f1ij −

I −

0 0 00 Γ1

1k Γ12k

0 Γ21k Γ

22k

i−1,j

f1i−1,j

= f1ij − f1i−1,j − ∆x1ψ1i−1/2,j

where

ψ1i−1/2,j = −1

2

[

0Γ111T

11 + Γ112T21

Γ211T11 + Γ212T

21

]

ij

+

[0

Γ111T11 + Γ112T

21

Γ211T11 + Γ212T

21

]

i−1,j

.

This is n = 1 portion of the source term

ψ = −ΓmnkT kn.

Acoustics on a manifold

∂

∂tp+

1√h

∂

∂xk

(√huk

)= 0

∂

∂tum +

1√h

∂

∂xk

(√hT km

)= −ΓmnkT kn .

With finite volume formulation,• Source term is automatically incorporated by parallel

transport of fluxes,• Covariant divergence is handled by use of edge lengths and

cell volume,• Parallel transport and orthonormalization allows use of

standard flat-space Riemann solver at interface.

Cubed sphere grid

Six logically rectangular grids arepatched together.

Data is transferred between patchesusing ghost cells

Shallow water on the sphere

0 0.25 0.5 0.75 1

0

0.25

0.5

0.75

1 1D2D

PSfrag replacements

t = 0

Scatterplot of depth vs. lattitude(from north to south pole)

Solid line is “exact” solutionfrom 1D equation

0 0.25 0.5 0.75 1

0

0.25

0.5

0.75

1 1D2D

PSfrag replacements

t = 1.25

0 0.25 0.5 0.75 1

0

0.25

0.5

0.75

1 1D2D

PSfrag replacementst = 1.25

t = 2.5

Shallow water on the sphere

0 0.25 0.5 0.75 1

0

0.25

0.5

0.75

1 1D2D

PSfrag replacements

t = 0

Scatterplot of depth vs. lattitude(from north to south pole)

Solid line is “exact” solutionfrom 1D equation

0 0.25 0.5 0.75 1

0

0.25

0.5

0.75

1 1D2D

PSfrag replacements

t = 1.25

0 0.25 0.5 0.75 1

0

0.25

0.5

0.75

1 1D2D

PSfrag replacementst = 1.25

t = 2.5

Outline

• Brief review of Godunov-type methods• wave propagation approach• f-wave approach for discontinous fluxes and source terms• curvilinear grids in flat space• manifolds:

• parallel transport data to cell edges• Express in local orthonormal frame• Solve locally-flat Riemann problem• Metric terms and geometric source term naturally

incorporated

References:

http://www.amath.washington.edu/˜rjl/publications and /students

J. Pons, Font, Ibanez, Marti, Miralles, General relativistic hydrodynamics with specialrelativistic Riemann solvers, Astron. Astrophys. 339 (1998), 638-642

OverviewSoftwareAMR on a manifoldOutlineGodunov's methodRiemann solution for the Euler equationsWave-propagation viewpointHigh-resolution method for systemsCLAWPACKprnormal Some applicationsSpatially-varying flux functionsRiemann problem for spatially-varying fluxWave-propagation algorithm using f-wavesWave-propagation algorithm using f-waves

Source termsSource terms

Riemann problem with a delta-function source termMultidimensional Hyperbolic ProblemsFinite volume method in 2DUnsplit Godunov in 2DFinite volume method on a curvilinear gridExample: Linear acousticsShallow water flow into a cylinderAcoustics on a manifoldCLAWMAN softwareParallel TransportParallel Transport for acousticsf-wave formulationAcoustics on a manifoldCubed sphere gridShallow water on the sphereShallow water on the sphere

Outline