a new class of well-balanced finite volume …folk.uio.no/siddharm/mafelap2.pdfoutline the problem...

OutlineThe Problem

Numerical DifficultiesExisting Well-Balanced Schemes

New Well-Balanced SchemesNumerical Experiments

Summary and Future Work

A New Class of Well-Balanced Finite Volumeschemes for Conservation laws with source terms

Siddhartha Mishra

Centre of Mathematics for Applications (CMA),University of Oslo, Norway

Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms

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Joint Work with:

I Kenneth Hvistendahl Karlsen (CMA, Oslo).

I Nils Henrik Risebro (CMA, Oslo).

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The Problem

Numerical Difficulties

Existing Well-Balanced Schemes

New Well-Balanced Schemes

Numerical Experiments

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Basic Equations

Ut + (f (U))x + (g(U))y + (h(U))z = S(x ,U)

I System of Conservation laws in multi-D.

I Together with source terms.

I Source can be spatially dependent (maybe singular).

I Also termed Balance laws.

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Flow on a non-trivial topography

Non−Trivial Smooth Bottom Topography Discontinuous Bottom Topography

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An Example

• Shallow water equations with Non-trivial Bottom Topography.

ht + (hu)x + (hv)y = 0(hu)t + (hu2 + 1

2gh2)x + (huv)y = −ghbx

(hv)t + (huv)x + (hv2 + 12gh2)y = −ghby

I h is height of the free surface.

I (u, v) is the velocity vector.

I g - gravity constant.

I b Topography function (can be discontinuous).

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The Model Equation

• Single conservation law in 1-d.

ut + f (u)x = A(x , u)

I Unknown u, flux f and source A.

I Source can even be singular. (A can be a measure)

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Special Cases

• Autonomous source

ut + f (u)x = g(u)

• Scalar “Shallow Water” equations

ut + (f (u))x = z ′(x)b(u)

I z is the topography function (possibly discontinuous)

• Singular Sources

ut + (f (u))x = z ′(x)

• z Heaviside funtion ⇒ RHS is a measure.

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Weak Solutions

• Well Defined when A(x , u) ∈ L∞ .

• u ∈ L∞(R×R+) ∩ L1loc is a weak solution if for all test functions

ϕ,∫R+

uϕt + f (u)ϕx +A(x , u)ϕ dxdt +

u(x , 0)ϕ(x , 0) = 0 (1)

• Special attention when A /∈ L∞.• Make sense of the non-conservative product

z ′(x)b(u)

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Entropy Solutions

• Well Defined when A(x , u) ∈ L∞ .

• u ∈ L∞(R× R+) ∩ L1loc is a entropy solution if for all test

functions ϕ ≥ 0,∫R+

S(u)ϕt+Q(u)ϕx+S ′(u)A(x , u)ϕ dxdt+

u(x , 0)ϕ(x , 0) ≥ 0

• For any entropy-entropy flux pair (S ,Q).• Entropy solutions exist and are unique when A ∈ L∞.• No general theory in the singular case except whenA(x , u) = z ′(x)

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A Naive Numerical Scheme

ut + f (u)x = z ′(x)b(u)

I Explicit Euler in Time.

I Godunov type numerical fluxes for the flux.

I Central differences for the source.

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ut + f (u)x = z ′(x)b(u)

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ut + f (u)x = z ′(x)b(u)

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ut + f (u)x = z ′(x)b(u)

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A Numerical Experiment

ut + f (u)x = z ′(x)b(u)

• With

f (u) = 12u2 b(u) = u

−z(x) =

{cos(πx) if 4.5 < x < 5.50 Otherwise

u(t, 0) = 2 u(0, x) = 0

• Explicit steady state is given by

u(x) = 2 + z(x)

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At the steady state

0 2 4 6 8 10−1

−0.5

Exact:−−−−−−−−−−−−−CS :− − − − − − −BT: + + + + + +

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Key Numerical Issues

I Resolution of steady states.

I At a steady state ⇔ Flux-Source balance.

f (u)x ≈ A(x , u)

I Numerical schemes have to preserve Flux-Source balance.

I Centered Source/Operator splitting doesn’t respect it.

I Search for better schemes

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f (u)x ≈ A(x , u)

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f (u)x ≈ A(x , u)

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f (u)x ≈ A(x , u)

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f (u)x ≈ A(x , u)

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f (u)x ≈ A(x , u)

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I Greenberg, Leroux.

I Greenberg, Leroux, Baraille and Noussair.

I Gosse, Leroux.

I Botchorischvili, Perthame and Vasseur. (BPV)

I Bermudez, Vasquez

I Perthame, Bouchut, Bristeau, Klien, Audusse.

I Russo, Noelle, Kurganov, Levy and many others.

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WBS (condensed)

• Consider Scalar Shallow water equations,

ut + f (u)x = z ′(x)b(u)

• The steady state is formally,

f (u)x = z ′(x)b(u)⇒ f ′(u)ux = z ′(x)b(u)

⇒ f ′(u)b(u) ux = z ′(x)

⇒ D(u)x = z ′(x)

D(u) =∫ u f ′(s)

b(s) ds

• Steady State evaluated from

D − z = Constant

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WBS (Condensed)

I At each time step, the cell values are projected unto “local”steady states i.e

I At nth time step let vnj be the cell-averages and zj be averages

of the topography, then define “local” steady states solving

D(vnj −)− zj = D(vn

j−1)− zj−1

D(vnj +)− zj = D(vn

j+1)− zj+1

I Use the local steady states to define a Godonov type schemewith update

vn+1j = vn

j −∆t

∆x(F (vn

j , vnj +)− F (vn

j −, vnj ))

• with F being Standard (Godunov, Enquist-Osher) fluxcorresponding to f

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WBS (Condensed)

j−1)− zj−1

j+1)− zj+1

vn+1j = vn

j −∆t

∆x(F (vn

j , vnj +)− F (vn

j −, vnj ))

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WBS (Condensed)

j−1)− zj−1

j+1)− zj+1

vn+1j = vn

j −∆t

∆x(F (vn

j , vnj +)− F (vn

j −, vnj ))

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WBS (Condensed)

j−1)− zj−1

j+1)− zj+1

vn+1j = vn

j −∆t

∆x(F (vn

j , vnj +)− F (vn

j −, vnj ))

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WBS (Advantages)

I Discrete steady states are preserved exactly.

I Shown to Converge to entropy solutions (via Kineticformulation).

I Basis for WBS for systems.

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WBS (Advantages)

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WBS (Advantages)

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WBS (Advantages)

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WBS (Problems)

I Expensive: 2 Algebraic equations to be solved for each meshpoint.

I Complicated: Steady state equations may have nosolutions/multiple solutions.

I Specialized: Difficult to extend when source is not in productform.

I Non-entropic: In some cases with discontinuous z .

I Possible loss of accuracy away from steady states.

I Subtle deficiences (see sequel)

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WBS (Problems)

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WBS (Problems)

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WBS (Problems)

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WBS (Problems)

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WBS (Problems)

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WBS (Problems)

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Along the lines of

I Greenberg, Leroux, Baraille and Noussair. (Singular Sources)

I Noussair.

I LeVeque.

I Bale, LeVeque, Mitran, Rossmanith (Flux - Differencing)

I Adimurthi, Gowda, Mishra (Singular Sources)

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x x x xj j jx j− 3/2 − 1/2 + 1/2 + 3/2

tn + 1

tn + 2

−1 + 1

j , (u u j + 1)

1−D Finite Volume Grid

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The Scheme: Design

I At time level n, let unj be the cell averages,

• Step 1: Freeze the source at tn and define the piecewiseconstant

un(x) =∑

unj 1{Ij}(x)

with Ij being the jth cell. Formally (“local” in time) we havethe equation

ut + (f (u))x = A(x , un(x))

I Primitive Reconstruction: Define the function

B̃n(x) =

A(y , un(y))dy

• We obtain the following discontinuous flux problem,

OutlineThe Problem

The Scheme: Design

I At time level n, let unj be the cell averages,

• Step 1: Freeze the source at tn and define the piecewiseconstant

un(x) =∑

unj 1{Ij}(x)

with Ij being the jth cell. Formally (“local” in time) we havethe equation

ut + (f (u))x = A(x , un(x))

I Primitive Reconstruction: Define the function

B̃n(x) =

A(y , un(y))dy

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Scheme: Design

ut + (f (u))x = (B̃n(x))x

I Local Discontinuous flux problems: By sampling define

Bn(x) =∑

B̃n(xj)1{Ij}(x)

ut + (f (u)− Bn(x))x = 0, u(x , tn) = un(x)

I Local Riemann problems at each interface

ut + (f (u)− Bnj )x = 0 u(x , 0) = un

j x < xj+1/2

ut + (f (u)− Bnj+1)x = 0 u(x , 0) = un

j+1 x > xj+1/2

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Scheme: Design

ut + (f (u))x = (B̃n(x))x

Bn(x) =∑

B̃n(xj)1{Ij}(x)

ut + (f (u)− Bnj )x = 0 u(x , 0) = un

j x < xj+1/2

ut + (f (u)− Bnj+1)x = 0 u(x , 0) = un

j+1 x > xj+1/2

OutlineThe Problem

Scheme: Design

ut + (f (u))x = (B̃n(x))x

Bn(x) =∑

B̃n(xj)1{Ij}(x)

ut + (f (u)− Bnj )x = 0 u(x , 0) = un

j x < xj+1/2

ut + (f (u)− Bnj+1)x = 0 u(x , 0) = un

j+1 x > xj+1/2

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Shape of Adjacent fluxes

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Scheme: Design

I Use a Exact Riemann Solver to solve the discontinuous fluxproblem

I RPs are simple to solve as the flux is additive.I The update formula is

un+1j = un

j −∆t

∆x(F n

j+1/2 − F nj − 1/2)

• with Fj+1/2 being the corresponding Godunov flux.I Explicit formulas are available in most cases e.g (f convex)

Fj+1/2 = max(f (max(uj , θ)− Bnj , f (min(uj+1, θ))− Bn

OutlineThe Problem

Scheme: Design

I RPs are simple to solve as the flux is additive.

I The update formula is

un+1j = un

j −∆t

∆x(F n

j+1/2 − F nj − 1/2)

OutlineThe Problem

Scheme: Design

un+1j = un

j −∆t

∆x(F n

j+1/2 − F nj − 1/2)

• with Fj+1/2 being the corresponding Godunov flux.

I Explicit formulas are available in most cases e.g (f convex)then

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Scheme: Design

un+1j = un

j −∆t

∆x(F n

j+1/2 − F nj − 1/2)

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Scheme: Properties

I Discrete steady states of the scheme

f (unj+1)− f (un

j ) = Bnj+1 − Bn

• Reflects Flux-Source balance.

I Rankine-Hugoniot Conditions + Jump entropy conditions ⇒Entropic Discrete steady states are preserved .

I Flexibility in the averaging steps to obtain equivalent discretesteady states.

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Scheme: Properties

f (unj+1)− f (un

j ) = Bnj+1 − Bn

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Scheme: Properties

f (unj+1)− f (un

j ) = Bnj+1 − Bn

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Scheme: Properties

I Growth assumptions of flux and source + Boundedness of A⇒ L∞ bounds.

I Entropy inequalities + Properties of the Riemann solution ⇒Rate of Blow-up of BV -norm.

I Blow-up estimates on BV -norm ⇒ Compactness ofApproximations (Compensated Compactness).

I Jump entropy conditions + Structure of the scheme ⇒Convergence to entropy solutions if A ∈ L∞

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Scheme: Properties

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Scheme: Properties

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Scheme: Properties

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Experiment 1 (Continuous Bottom)

ut + f (u)x = z ′(x)b(u)

• With

f (u) = 12u2 b(u) = u

−z(x) =

{cos(πx) if 4.5 < x < 5.50 if Otherwise

u(t, 0) = 2 u(0, x) = 0

u(x) = 2 + z(x)

• Comparision of Central Sources (CS), Existing Well-BalancedScheme (BPV) and New Well-Balanced Scheme (AWBS)

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At Steady State: ∆x = 0.1

0 2 4 6 8 10−1

−0.5

AWBS:−−−−−−−−−−−−−−−BPV:................................CS :− − − − − − −BT: o o o o o o o o o o o

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Errors at Steady State

L∞ L1

CS 0.1652 0.4824AWBS 4.37× 10−14 2.22× 10−13

BPV 8.45× 10−14 2.26× 10−13

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Transients: ∆x = 0.1

Figure: Left:AWBS, Right:BPV

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Transient snapshots

0 2 4 6 8 10−1

−0.5

AWBS:−−−−−−−−−−−−−−−−−CS :o o o o o o o o oBPV:− − − − − − − −

t = 2 Delta x = 0.1

0 2 4 6 8 10−1

−0.5

AWBS:−−−−−−−−−−−BPV: − − − − − − −CS :o o o o o o

t = 5Delta x = 0.1

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BPV at high resolution

0 2 4 6 8 10−0.5

BPV(Delta x =0.1):− − − − −

BPV(Delta x=0.01):−−−−−−−−−

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Experiment 2 (Discontinuous Bottom)

ut + f (u)x = z ′(x)b(u)

• With

f (u) = 12u2 b(u) = u

−z(x) =

{cos(πx) if 5 < x < 60 if Otherwise

u(t, 0) = 2 u(0, x) = 0

u(x) = 2 + z(x)

OutlineThe Problem

0 2 4 6 8 10

−0.5

AWBS:−−−−−−−−−−−−−−−

BPV: o o o o o o o o

CS :− − − − − − −

BT:−.. −. −. − . −. −. −.. −..

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Errors at Steady State

L∞ L1

CS1 0.8027 1.6449AWBS 1.87× 10−12 8.12× 10−13

BPV 2.53× 10−9 6.34× 10−10

Table: Errors at the steady state for the three schemes with ∆x = 0.1 in

Experiment 2

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A Transient snapshot

0 2 4 6 8 10

−0.5

AWBS:−−−−−−−−−−BPV:− − − − −CS:o o o o o o

0 2 4 6 8 10

−0.5

AWBS:−−−−−−−−−−−−−

BPV:− − − − − − −

CS: o o o o o o o

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Experiment 3 (Another Discontinuous Bottom)

ut + f (u)x = z ′(x)b(u)

• With

f (u) = 12u2 b(u) = u

−z(x) =

{− cos(πx) if 5 < x < 60 if Otherwise

u(t, 0) = 2 u(0, x) = 0

u(x) = 2 + z(x)

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0 2 4 6 8 10

−0.5

AWBS:−−−−−−−−−−−−−−−

BPV:− − − − − − −

CS: o o o o o o o

BT:−.. −... −... −.. −..−.. −.

t = 10

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Transients

Figure: Left:AWBS, Right:BPV

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Experiment 4 (Non-Monotone D)

ut + f (u)x = z ′(x)b(u)

• With

f (u) = 13u3 b(u) = u

−z(x) =

{cos(πx) if 4.5 < x < 5.50 if Otherwise

u(t, 0) = 2 u(0, x) = 0

• Difficult to define BPV as D = u2 is not monotone. No problemswith AWBS

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Numerical Results at : ∆x = 0.1

0 2 4 6 8 100.5

AWBS;−−−−−−−−−−

0 2 4 6 8 10−1.5

−0.5

AwBS:−−−−−−−−−

BT:− − − − −

t = 10

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Experiment 5 (Source not in Product form)

ut + f (u)x = A(x , u)

• With

f (u) = 12u2 A(x , u) = sin(2πxu2)

u(t, 0) = 1 u(0, x) = 0

• Unclear how to define BPV in this case (other than using ODEsolvers at each mesh point) whereas AWBS is well-defined

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Numerical results with : ∆x = 0.1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

AWBS:−−−−−−−−−−−−

RK4 :o o o o o o o

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Subtle problems with existing well-balanced schemes

I Incorrect Shock speeds and strengths due to non-lineartransformations.

I Problems at resonance u = 0

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Subtle problems with existing well-balanced schemes

I Incorrect Shock speeds and strengths due to non-lineartransformations.

I Problems at resonance u = 0

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Summary

• New class of Well-Balanced Schemes are

I Very Simple to implement (Explicit formulas, No extraequations)

I Robust and proved to be convergent.

I Very General: Work with different type of fluxes and sources.

I Tailormade for discontinuous and singular sources.

I Numerically efficient at both transients and steady states.

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Summary

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Summary

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Summary

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Summary

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Summary

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Ongoing and Future Work

I Discontinuous and Singular A.

I Higher order schemes.

I Multi Dimensional problems.

I Systems: Shallow Water, Euler.

I Stiff Source terms.

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