Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
A New Class of Well-Balanced Finite Volume
schemes 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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Flow on a non-trivial topography
h
h
b
Non−Trivial Smooth Bottom Topography
Siddhartha Mishra
b
Discontinuous Bottom Topography
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
An Example
• Shallow water equations with Non-trivial Bottom Topography.
ht + (hu)x + (hv )y
(hu)t + (hu 2 + 12 gh2 )x + (huv )y
(hv )t + (huv )x + (hv 2 + 12 gh2 )y
= 0
= −ghbx
= −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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Special Cases
• Autonomous source
ut + f (u)x = g (u)
• Scalar “Shallow Water” equations
ut + (f (u))x = z 0 (x)b(u)
I
z is the topography function (possibly discontinuous)
• Singular Sources
ut + (f (u))x = z 0 (x)
• z Heaviside funtion ⇒ RHS is a measure.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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,
Z
Z Z
uϕt + f (u)ϕx + A(x, u)ϕ dxdt + u(x, 0)ϕ(x, 0) = 0 (1)
R+
R
R
• Special attention when A ∈
/ L∞ .
• Make sense of the non-conservative product
z 0 (x)b(u)
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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,
Z Z
Z
0
S(u)ϕt +Q(u)ϕx +S (u)A(x, u)ϕ dxdt+ u(x, 0)ϕ(x, 0) ≥ 0
R+
R
R
• 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 when
A(x, u) = z 0 (x)
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
A Naive Numerical Scheme
ut + f (u)x = z 0 (x)b(u)
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
A Naive Numerical Scheme
ut + f (u)x = z 0 (x)b(u)
I
Explicit Euler in Time.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
A Naive Numerical Scheme
ut + f (u)x = z 0 (x)b(u)
I
Explicit Euler in Time.
I
Godunov type numerical fluxes for the flux.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
A Naive Numerical Scheme
ut + f (u)x = z 0 (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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
A Numerical Experiment
ut + f (u)x = z 0 (x)b(u)
• With
1 2
= 2u
cos(πx) if 4.5 < x < 5.5
−z(x) =
0
Otherwise
u(t, 0) = 2
f (u)
b(u)
= u
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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
At the steady state
3.5
3
2.5
2
1.5
1
Exact:−−−−−−−−−−−−−
CS :− − − − − − −
BT: + + + + + +
0.5
0
−0.5
−1
0
2
4
Siddhartha Mishra
6
8
10
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Key Numerical Issues
I
Resolution of steady states.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Key Numerical Issues
I
Resolution of steady states.
I
At a steady state ⇔ Flux-Source balance.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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)
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (condensed)
• Consider Scalar Shallow water equations,
ut + f (u)x = z 0 (x)b(u)
• The steady state is formally,
⇒
⇒
⇒
f (u)x
f 0 (u)ux
f 0 (u)
b(u) ux
D(u)x
D(u)
=
=
=
=
=
z 0 (x)b(u)
z 0 (x)b(u)
z 0 (x)
z 0 (x)
R u f 0 (s)
b(s) ds
• Steady State evaluated from
D − z = Constant
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (Condensed)
I
At each time step, the cell values are projected unto “local”
steady states i.e
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (Condensed)
I
I
At each time step, the cell values are projected unto “local”
steady states i.e
At nth time step let vjn be the cell-averages and zj be averages
of the topography, then define “local” steady states solving
D(vjn −) − zj
D(vjn +) − zj
Siddhartha Mishra
n )−z
= D(vj−1
j−1
n )−z
= D(vj+1
j+1
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (Condensed)
I
I
At each time step, the cell values are projected unto “local”
steady states i.e
At nth time step let vjn be the cell-averages and zj be averages
of the topography, then define “local” steady states solving
D(vjn −) − zj
D(vjn +) − zj
I
n )−z
= D(vj−1
j−1
n )−z
= D(vj+1
j+1
Use the local steady states to define a Godonov type scheme
with update
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (Condensed)
I
I
At each time step, the cell values are projected unto “local”
steady states i.e
At nth time step let vjn be the cell-averages and zj be averages
of the topography, then define “local” steady states solving
D(vjn −) − zj
D(vjn +) − zj
I
I
n )−z
= D(vj−1
j−1
n )−z
= D(vj+1
j+1
Use the local steady states to define a Godonov type scheme
with update
∆t
(F (vjn , vjn +) − F (vjn −, vjn ))
∆x
• with F being Standard (Godunov, Enquist-Osher) flux
corresponding to f
vjn+1 = vjn −
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (Advantages)
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (Advantages)
I
Discrete steady states are preserved exactly.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (Advantages)
I
Discrete steady states are preserved exactly.
I
Shown to Converge to entropy solutions (via Kinetic
formulation).
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (Advantages)
I
Discrete steady states are preserved exactly.
I
Shown to Converge to entropy solutions (via Kinetic
formulation).
I
Basis for WBS for systems.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (Problems)
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (Problems)
I
Expensive: 2 Algebraic equations to be solved for each mesh
point.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (Problems)
I
Expensive: 2 Algebraic equations to be solved for each mesh
point.
I
Complicated: Steady state equations may have no
solutions/multiple solutions.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (Problems)
I
Expensive: 2 Algebraic equations to be solved for each mesh
point.
I
Complicated: Steady state equations may have no
solutions/multiple solutions.
I
Specialized: Difficult to extend when source is not in product
form.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (Problems)
I
Expensive: 2 Algebraic equations to be solved for each mesh
point.
I
Complicated: Steady state equations may have no
solutions/multiple solutions.
I
Specialized: Difficult to extend when source is not in product
form.
I
Non-entropic: In some cases with discontinuous z.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (Problems)
I
Expensive: 2 Algebraic equations to be solved for each mesh
point.
I
Complicated: Steady state equations may have no
solutions/multiple solutions.
I
Specialized: Difficult to extend when source is not in product
form.
I
Non-entropic: In some cases with discontinuous z.
I
Possible loss of accuracy away from steady states.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
WBS (Problems)
I
Expensive: 2 Algebraic equations to be solved for each mesh
point.
I
Complicated: Steady state equations may have no
solutions/multiple solutions.
I
Specialized: Difficult to extend when source is not in product
form.
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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
1−D Finite Volume Grid
tn + 2
n+ 1
U j
n
Uj
n
U
j −1
xj
− 3/2
xj
− 1/2
Siddhartha Mishra
t
n
Uj +1
F (u j , u j + 1)
x j + 1/2
n+1
tn
x j + 3/2
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
The Scheme: Design
I
At time level n, let ujn be the cell averages,
• Step 1: Freeze the source at t n and define the piecewise
constant
X
u n (x) =
ujn 1{Ij } (x)
j
with Ij being the jth cell. Formally (“local” in time) we have
the equation
ut + (f (u))x = A(x, u n (x))
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
The Scheme: Design
I
At time level n, let ujn be the cell averages,
• Step 1: Freeze the source at t n and define the piecewise
constant
X
u n (x) =
ujn 1{Ij } (x)
j
with Ij being the jth cell. Formally (“local” in time) we have
the equation
ut + (f (u))x = A(x, u n (x))
I
Primitive Reconstruction: Define the function
Z x
n
B̃ (x) =
A(y , u n (y ))dy
• We obtain the following discontinuous flux problem,
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Scheme: Design
ut + (f (u))x = (B̃ n (x))x
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Scheme: Design
ut + (f (u))x = (B̃ n (x))x
I
Local Discontinuous flux problems: By sampling define
X
B n (x) =
B̃ n (xj )1{Ij } (x)
j
• We obtain the following discontinuous flux problem,
ut + (f (u) − B n (x))x = 0, u(x, t n ) = u n (x)
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Scheme: Design
ut + (f (u))x = (B̃ n (x))x
I
Local Discontinuous flux problems: By sampling define
X
B n (x) =
B̃ n (xj )1{Ij } (x)
j
• We obtain the following discontinuous flux problem,
ut + (f (u) − B n (x))x = 0, u(x, t n ) = u n (x)
I
Local Riemann problems at each interface
ut + (f (u) − Bjn )x
n )
ut + (f (u) − Bj+1
x
= 0
= 0
Siddhartha Mishra
u(x, 0) = ujn
n
u(x, 0) = uj+1
x < xj+1/2
x > xj+1/2
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Shape of Adjacent fluxes
f+
f −
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Scheme: Design
I
Use a Exact Riemann Solver to solve the discontinuous flux
problem
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Scheme: Design
I
I
Use a Exact Riemann Solver to solve the discontinuous flux
problem
RPs are simple to solve as the flux is additive.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Scheme: Design
I
I
I
Use a Exact Riemann Solver to solve the discontinuous flux
problem
RPs are simple to solve as the flux is additive.
The update formula is
∆t n
(F
− F n j − 1/2)
∆x j+1/2
being the corresponding Godunov flux.
ujn+1 = ujn −
• with Fj+1/2
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Scheme: Design
I
I
I
Use a Exact Riemann Solver to solve the discontinuous flux
problem
RPs are simple to solve as the flux is additive.
The update formula is
∆t n
(F
− F n j − 1/2)
∆x j+1/2
• with Fj+1/2 being the corresponding Godunov flux.
Explicit formulas are available in most cases e.g (f convex)
then
ujn+1 = ujn −
I
n
Fj+1/2 = max(f (max(uj , θ) − Bjn , f (min(uj+1 , θ)) − Bj+1
))
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Scheme: Properties
I
Discrete steady states of the scheme
n
n
f (uj+1
) − f (ujn ) = Bj+1
− Bjn
• Reflects Flux-Source balance.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Scheme: Properties
I
Discrete steady states of the scheme
n
n
f (uj+1
) − f (ujn ) = Bj+1
− Bjn
• Reflects Flux-Source balance.
I
Rankine-Hugoniot Conditions + Jump entropy conditions ⇒
Entropic Discrete steady states are preserved .
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Scheme: Properties
I
Discrete steady states of the scheme
n
n
f (uj+1
) − f (ujn ) = Bj+1
− Bjn
• 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 discrete
steady states.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Scheme: Properties
I
Growth assumptions of flux and source + Boundedness of A
⇒ L∞ bounds.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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 of
Approximations (Compensated Compactness).
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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 of
Approximations (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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Experiment 1 (Continuous Bottom)
ut + f (u)x = z 0 (x)b(u)
• With
1 2
= 2u
cos(πx) if 4.5 < x < 5.5
−z(x) =
0
if Otherwise
u(t, 0) = 2
f (u)
b(u)
= u
u(0, x) = 0
• Explicit steady state is given by
u(x) = 2 + z(x)
• Comparision of Central Sources (CS), Existing Well-Balanced
Scheme (BPV) and New Well-Balanced Scheme (AWBS)
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
At Steady State: ∆x = 0.1
3.5
3
2.5
2
1.5
1
AWBS:−−−−−−−−−−−−−−−
BPV:................................
CS :− − − − − − −
BT: o o o o o o o o o o o
0.5
0
−0.5
−1
0
2
4
Siddhartha Mishra
6
8
10
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Errors at Steady State
CS
AWBS
BPV
L∞
0.1652
4.37 × 10−14
8.45 × 10−14
Siddhartha Mishra
L1
0.4824
2.22 × 10−13
2.26 × 10−13
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Transients: ∆x = 0.1
Figure: Left:AWBS,
Siddhartha Mishra
Right:BPV
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Transient snapshots
3
3.5
3
t = 2
Delta x = 0.1
2.5
t=5
Delta x = 0.1
2.5
AWBS:−−−−−−−−−−−−−−−−−
CS :o o o o o o o o o
BPV:− − − − − − −
−
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−1
0
−0.5
2
4
6
8
10
Siddhartha Mishra
−1
0
AWBS:−−−−−−−−−−−
BPV: − − − − − − −
CS :o o o o o o
2
4
6
8
10
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
BPV at high resolution
3
2.5
t = 3
BPV(Delta x =0.1):− − − − −
BPV(Delta x=0.01):−−−−−−−−−
2
1.5
1
0.5
0
−0.5
0
2
4
Siddhartha Mishra
6
8
10
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Experiment 2 (Discontinuous Bottom)
ut + f (u)x = z 0 (x)b(u)
• With
1 2
= 2u
cos(πx) if 5 < x < 6
−z(x) =
0
if Otherwise
u(t, 0) = 2
f (u)
b(u)
= u
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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
At Steady State: ∆x = 0.1
3.5
3
2.5
2
1.5
AWBS:−−−−−−−−−−−−−−−
BPV: o o o o o o o o
1
0.5
CS :− − − −
− −
−
BT:−.. −. −. − . −. −. −.. −..
0
−0.5
−1
0
2
4
Siddhartha Mishra
6
8
10
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Errors at Steady State
CS1
AWBS
BPV
L∞
0.8027
1.87 × 10−12
2.53 × 10−9
L1
1.6449
8.12 × 10−13
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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
A Transient snapshot
2.5
3.5
3
2
2.5
AWBS:−−−−−−−−−−
BPV:− − − − −
CS:o o o o o o
1.5
t =6
2
1
1.5
1
0.5
0.5
0
0
−0.5
0
AWBS:−−−−−−−−−−−−−
BPV:− − − − − − −
CS: o o o o o o o
t
−0.5
= 2
2
4
6
8
10
Siddhartha Mishra
0
2
4
6
8
10
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Experiment 3 (Another Discontinuous Bottom)
ut + f (u)x = z 0 (x)b(u)
• With
1 2
= 2u
− cos(πx) if 5 < x < 6
−z(x) =
0
if Otherwise
u(t, 0) = 2
f (u)
b(u)
= u
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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
At Steady State: ∆x = 0.1
3.5
3
2.5
2
1.5
1
0.5
AWBS:−−−−−−−−−−−−−−−
BPV:− − − − − − −
t = 10
CS: o o o o o o o
BT:−.. −... −... −.. −..−.. −.
0
−0.5
−1
0
2
4
Siddhartha Mishra
6
8
10
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Transients
Figure: Left:AWBS,
Siddhartha Mishra
Right:BPV
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Experiment 4 (Non-Monotone D)
ut + f (u)x = z 0 (x)b(u)
• With
1 3
= 3u
cos(πx) if 4.5 < x < 5.5
−z(x) =
0
if Otherwise
u(t, 0) = 2
f (u)
b(u)
= u
u(0, x) = 0
• Difficult to define BPV as D = u 2 is not monotone. No problems
with AWBS
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Numerical Results at : ∆x = 0.1
1.5
2
1.5
1
0.5
1
0
−0.5
AWBS;−−−−−−−−−−
t=3
0.5
0
AwBS:−−−−−−−−−
BT:− − − −
−1
2
4
6
8
10
Siddhartha Mishra
−1.5
0
−
t = 10
2
4
6
8
10
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Experiment 5 (Source not in Product form)
ut + f (u)x = A(x, u)
• With
f (u)
= 21 u 2
u(t, 0) = 1
A(x, u) = sin(2πxu 2 )
u(0, x) = 0
• Unclear how to define BPV in this case (other than using ODE
solvers at each mesh point) whereas AWBS is well-defined
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Numerical results with : ∆x = 0.1
1.3
1.2
1.1
AWBS:−−−−−−−−−−−−
RK4 :o o o o o o o
1
0.9
0.8
0.7
0.6
0.5
0.4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Subtle problems with existing well-balanced schemes
I
Incorrect Shock speeds and strengths due to non-linear
transformations.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Subtle problems with existing well-balanced schemes
I
Incorrect Shock speeds and strengths due to non-linear
transformations.
I
Problems at resonance u = 0
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Summary
• New class of Well-Balanced Schemes are
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Summary
• New class of Well-Balanced Schemes are
I
Very Simple to implement (Explicit formulas, No extra
equations)
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Summary
• New class of Well-Balanced Schemes are
I
Very Simple to implement (Explicit formulas, No extra
equations)
I
Robust and proved to be convergent.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Summary
• New class of Well-Balanced Schemes are
I
Very Simple to implement (Explicit formulas, No extra
equations)
I
Robust and proved to be convergent.
I
Very General: Work with different type of fluxes and sources.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Summary
• New class of Well-Balanced Schemes are
I
Very Simple to implement (Explicit formulas, No extra
equations)
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.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Summary
• New class of Well-Balanced Schemes are
I
Very Simple to implement (Explicit formulas, No extra
equations)
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 Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Ongoing and Future Work
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Ongoing and Future Work
I
Discontinuous and Singular A.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Ongoing and Future Work
I
Discontinuous and Singular A.
I
Higher order schemes.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical Experiments
Summary and Future Work
Ongoing and Future Work
I
Discontinuous and Singular A.
I
Higher order schemes.
I
Multi Dimensional problems.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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.
Siddhartha Mishra
A New Class of Well-Balanced Finite Volume schemes for Cons
Outline
The Problem
Numerical Difficulties
Existing Well-Balanced Schemes
New Well-Balanced Schemes
Numerical 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 Cons