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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
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Joint Work with:
I Kenneth Hvistendahl Karlsen (CMA, Oslo).
I Nils Henrik Risebro (CMA, Oslo).
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Flow on a non-trivial topography
b
h
b
h
Non−Trivial Smooth Bottom Topography Discontinuous Bottom Topography
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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).
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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)
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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+
∫R
uϕt + f (u)ϕx +A(x , u)ϕ dxdt +
∫R
u(x , 0)ϕ(x , 0) = 0 (1)
• Special attention when A /∈ L∞.• Make sense of the non-conservative product
z ′(x)b(u)
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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+
∫R
S(u)ϕt+Q(u)ϕx+S ′(u)A(x , u)ϕ dxdt+
∫R
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)
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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)
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
At the steady state
0 2 4 6 8 10−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
Exact:−−−−−−−−−−−−−CS :− − − − − − −BT: + + + + + +
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Key Numerical Issues
I Resolution of steady states.
I At a steady state ⇔ Flux-Source balance.
I
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
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Key Numerical Issues
I Resolution of steady states.
I At a steady state ⇔ Flux-Source balance.
I
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
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Key Numerical Issues
I Resolution of steady states.
I At a steady state ⇔ Flux-Source balance.
I
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
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Key Numerical Issues
I Resolution of steady states.
I At a steady state ⇔ Flux-Source balance.
I
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
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Key Numerical Issues
I Resolution of steady states.
I At a steady state ⇔ Flux-Source balance.
I
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
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Key Numerical Issues
I Resolution of steady states.
I At a steady state ⇔ Flux-Source balance.
I
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
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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
I
vn+1j = vn
j −∆t
∆x(F (vn
j , vnj +)− F (vn
j −, vnj ))
• with F being Standard (Godunov, Enquist-Osher) fluxcorresponding to f
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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
I
vn+1j = vn
j −∆t
∆x(F (vn
j , vnj +)− F (vn
j −, vnj ))
• with F being Standard (Godunov, Enquist-Osher) fluxcorresponding to f
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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
I
vn+1j = vn
j −∆t
∆x(F (vn
j , vnj +)− F (vn
j −, vnj ))
• with F being Standard (Godunov, Enquist-Osher) fluxcorresponding to f
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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
I
vn+1j = vn
j −∆t
∆x(F (vn
j , vnj +)− F (vn
j −, vnj ))
• with F being Standard (Godunov, Enquist-Osher) fluxcorresponding to f
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
WBS (Advantages)
I Discrete steady states are preserved exactly.
I Shown to Converge to entropy solutions (via Kineticformulation).
I Basis for WBS for systems.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
WBS (Advantages)
I Discrete steady states are preserved exactly.
I Shown to Converge to entropy solutions (via Kineticformulation).
I Basis for WBS for systems.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
WBS (Advantages)
I Discrete steady states are preserved exactly.
I Shown to Converge to entropy solutions (via Kineticformulation).
I Basis for WBS for systems.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
WBS (Advantages)
I Discrete steady states are preserved exactly.
I Shown to Converge to entropy solutions (via Kineticformulation).
I Basis for WBS for systems.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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)
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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)
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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)
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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)
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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)
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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)
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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)
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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)
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
x x x xj j jx j− 3/2 − 1/2 + 1/2 + 3/2
t n
tn + 1
tn + 2
U jn
Ujn
Un
−1 + 1
Unjj
n + 1
+ 1
F (uj
j , (u u j + 1)
1−D Finite Volume Grid
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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) =∑
j
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) =
∫ x
A(y , un(y))dy
• We obtain the following discontinuous flux problem,
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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) =∑
j
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) =
∫ x
A(y , un(y))dy
• We obtain the following discontinuous flux problem,
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Scheme: Design
ut + (f (u))x = (B̃n(x))x
I Local Discontinuous flux problems: By sampling define
Bn(x) =∑
j
B̃n(xj)1{Ij}(x)
• We obtain the following discontinuous flux problem,
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
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Scheme: Design
ut + (f (u))x = (B̃n(x))x
I Local Discontinuous flux problems: By sampling define
Bn(x) =∑
j
B̃n(xj)1{Ij}(x)
• We obtain the following discontinuous flux problem,
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
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Scheme: Design
ut + (f (u))x = (B̃n(x))x
I Local Discontinuous flux problems: By sampling define
Bn(x) =∑
j
B̃n(xj)1{Ij}(x)
• We obtain the following discontinuous flux problem,
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
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Shape of Adjacent fluxes
f
f +
−
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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)
then
Fj+1/2 = max(f (max(uj , θ)− Bnj , f (min(uj+1, θ))− Bn
j+1))
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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)
then
Fj+1/2 = max(f (max(uj , θ)− Bnj , f (min(uj+1, θ))− Bn
j+1))
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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)then
Fj+1/2 = max(f (max(uj , θ)− Bnj , f (min(uj+1, θ))− Bn
j+1))
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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)
then
Fj+1/2 = max(f (max(uj , θ)− Bnj , f (min(uj+1, θ))− Bn
j+1))
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Scheme: Properties
I Discrete steady states of the scheme
f (unj+1)− f (un
j ) = Bnj+1 − Bn
j
• 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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Scheme: Properties
I Discrete steady states of the scheme
f (unj+1)− f (un
j ) = Bnj+1 − Bn
j
• 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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Scheme: Properties
I Discrete steady states of the scheme
f (unj+1)− f (un
j ) = Bnj+1 − Bn
j
• 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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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∞
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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∞
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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∞
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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∞
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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
• Explicit steady state is given by
u(x) = 2 + z(x)
• Comparision of Central Sources (CS), Existing Well-BalancedScheme (BPV) and New Well-Balanced Scheme (AWBS)
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
At Steady State: ∆x = 0.1
0 2 4 6 8 10−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
AWBS:−−−−−−−−−−−−−−−BPV:................................CS :− − − − − − −BT: o o o o o o o o o o o
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Transients: ∆x = 0.1
Figure: Left:AWBS, Right:BPV
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Transient snapshots
0 2 4 6 8 10−1
−0.5
0
0.5
1
1.5
2
2.5
3
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
0
0.5
1
1.5
2
2.5
3
3.5
AWBS:−−−−−−−−−−−BPV: − − − − − − −CS :o o o o o o
t = 5Delta x = 0.1
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
BPV at high resolution
0 2 4 6 8 10−0.5
0
0.5
1
1.5
2
2.5
3
BPV(Delta x =0.1):− − − − −
BPV(Delta x=0.01):−−−−−−−−−
t = 3
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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
• Explicit steady state is given by
u(x) = 2 + z(x)
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
At Steady State: ∆x = 0.1
0 2 4 6 8 10
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
AWBS:−−−−−−−−−−−−−−−
BPV: o o o o o o o o
CS :− − − − − − −
BT:−.. −. −. − . −. −. −.. −..
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
A Transient snapshot
0 2 4 6 8 10
−0.5
0
0.5
1
1.5
2
2.5
AWBS:−−−−−−−−−−BPV:− − − − −CS:o o o o o o
t = 2
0 2 4 6 8 10
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
AWBS:−−−−−−−−−−−−−
BPV:− − − − − − −
CS: o o o o o o o
t = 6
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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
• Explicit steady state is given by
u(x) = 2 + z(x)
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
At Steady State: ∆x = 0.1
0 2 4 6 8 10
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
AWBS:−−−−−−−−−−−−−−−
BPV:− − − − − − −
CS: o o o o o o o
BT:−.. −... −... −.. −..−.. −.
t = 10
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Transients
Figure: Left:AWBS, Right:BPV
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Numerical Results at : ∆x = 0.1
0 2 4 6 8 100.5
1
1.5
AWBS;−−−−−−−−−−
t = 3
0 2 4 6 8 10−1.5
−1
−0.5
0
0.5
1
1.5
2
AwBS:−−−−−−−−−
BT:− − − − −
t = 10
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Numerical results with : ∆x = 0.1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
AWBS:−−−−−−−−−−−−
RK4 :o o o o o o o
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Subtle problems with existing well-balanced schemes
I Incorrect Shock speeds and strengths due to non-lineartransformations.
I Problems at resonance u = 0
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
Subtle problems with existing well-balanced schemes
I Incorrect Shock speeds and strengths due to non-lineartransformations.
I Problems at resonance u = 0
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms
OutlineThe Problem
Numerical DifficultiesExisting Well-Balanced Schemes
New Well-Balanced SchemesNumerical Experiments
Summary and Future Work
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.
Siddhartha Mishra A New Class of Well-Balanced Finite Volume schemes for Conservation laws with source terms