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FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Numerical schemes for Hyperbolic CL overnon-uniform adaptively redistributed meshes
Nikolaos Sfakianakis
RICAM, Vienna
Paris / January 2011
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Scalar Conservation Laws
u(x , t)t + f (u(x , t))x = 0, x ∈ [a,b], t ∈ [0,T ]
f smooth, convex flux function.
• Loss of smoothness =⇒Weak solutions• Loss of uniqueness =⇒ Entropy conditions
x
t
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Scalar Conservation Laws
u(x , t)t + f (u(x , t))x = 0, x ∈ [a,b], t ∈ [0,T ]
f smooth, convex flux function.
• Loss of smoothness =⇒Weak solutions
• Loss of uniqueness =⇒ Entropy conditions
x
t
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Scalar Conservation Laws
u(x , t)t + f (u(x , t))x = 0, x ∈ [a,b], t ∈ [0,T ]
f smooth, convex flux function.
• Loss of smoothness =⇒Weak solutions• Loss of uniqueness =⇒ Entropy conditions
x
t
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Uniform mesh
{(xi , tn), xi+1 − xi = ∆x, tn+1 − tn = ∆t
}Derivative approximations
∂∂t u(xi , tn) ≈ u(xi ,tn+1)−u(xi ,tn)
∆t∂∂x u(xi , tn) ≈ u(xi+1,tn)−u(xi−1,tn)
2∆x
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
x
t
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Non-uniform mesh
{(xn
i , tn), xn
i+1 − xni = ∆xn
i+1/2, tn+1 − tn = ∆tn+1/2}
Derivative approximations
∂∂t u(xn
i , tn) ≈ u(xn
i ,tn+1)−u(xn
i ,tn)
∆tn+1/2
∂∂x u(xn
i , tn) ≈ u(xn
i+1,tn)−u(xn
i−1,tn)
∆xni−1/2+∆xn
i+1/2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
x
t
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Non-uniform mesh
• How do we create/update such grids?
• Do we use the same numerical schemes?• Do they have the same properties?• How does the mesh adaptation affect the numerical results?
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Non-uniform mesh
• How do we create/update such grids?• Do we use the same numerical schemes?
• Do they have the same properties?• How does the mesh adaptation affect the numerical results?
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Non-uniform mesh
• How do we create/update such grids?• Do we use the same numerical schemes?• Do they have the same properties?
• How does the mesh adaptation affect the numerical results?
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Non-uniform mesh
• How do we create/update such grids?• Do we use the same numerical schemes?• Do they have the same properties?• How does the mesh adaptation affect the numerical results?
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
Step 1 Mesh reconstruction Mn+1x = {xn+1
1 , . . . , xn+1N }
Step 2 Solution update Un = {un1 , . . . , u
nN}
Step 3 Time evolution Un+1 ={
un+11 , . . . , un+1
N
}
valuesnodes
valuesnodes
new nodes
valuesnodes
new nodesupd values
valuesnodes
new nodesupd valuesnext values
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
Step 1 Mesh reconstruction Mn+1x = {xn+1
1 , . . . , xn+1N }
Step 2 Solution update Un = {un1 , . . . , u
nN}
Step 3 Time evolution Un+1 ={
un+11 , . . . , un+1
N
}valuesnodes
valuesnodes
new nodes
valuesnodes
new nodesupd values
valuesnodes
new nodesupd valuesnext values
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
Step 1 Mesh reconstruction Mn+1x = {xn+1
1 , . . . , xn+1N }
Step 2 Solution update Un = {un1 , . . . , u
nN}
Step 3 Time evolution Un+1 ={
un+11 , . . . , un+1
N
}valuesnodes
valuesnodes
new nodes
valuesnodes
new nodesupd values
valuesnodes
new nodesupd valuesnext values
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
Step 1 Mesh reconstruction Mn+1x = {xn+1
1 , . . . , xn+1N }
Step 2 Solution update Un = {un1 , . . . , u
nN}
Step 3 Time evolution Un+1 ={
un+11 , . . . , un+1
N
}
valuesnodes
valuesnodes
new nodes
valuesnodes
new nodesupd values
valuesnodes
new nodesupd valuesnext values
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 1 (Mesh reconstruction)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
Comput. cost O(N)
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 1 (Mesh reconstruction)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
• Estimator function KUn (·)
• Curvature, continuous version for smooth u
Ku =|u′′(x)|
(1+(u′(x))2)32
• Curvature, discrete version
K dscrUn,i =
2xi+1−xi−1
∣∣∣∣∣ uni −un
i−1xni −xn
i−1−
uni+1−un
ixni+1−xn
i
∣∣∣∣∣1+
(un
i −uni−1
xni −xn
i−1
)21+
(uni+1−un
ixni+1−xn
i
)21+
(uni+1−un
i−1xni+1−xn
i−1
)21/2
• Interpolate to KUn (x)
Comput. cost O(N)
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 1 (Mesh reconstruction)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
• Estimator function KUn (·)• Curvature, continuous version for smooth u
Ku =|u′′(x)|
(1+(u′(x))2)32
• Curvature, discrete version
K dscrUn,i =
2xi+1−xi−1
∣∣∣∣∣ uni −un
i−1xni −xn
i−1−
uni+1−un
ixni+1−xn
i
∣∣∣∣∣1+
(un
i −uni−1
xni −xn
i−1
)21+
(uni+1−un
ixni+1−xn
i
)21+
(uni+1−un
i−1xni+1−xn
i−1
)21/2
• Interpolate to KUn (x)
Comput. cost O(N)
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 1 (Mesh reconstruction)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
• Estimator function KUn (·)• Curvature, continuous version for smooth u
Ku =|u′′(x)|
(1+(u′(x))2)32
• Curvature, discrete version
K dscrUn,i =
2xi+1−xi−1
∣∣∣∣∣ uni −un
i−1xni −xn
i−1−
uni+1−un
ixni+1−xn
i
∣∣∣∣∣1+
(un
i −uni−1
xni −xn
i−1
)21+
(uni+1−un
ixni+1−xn
i
)21+
(uni+1−un
i−1xni+1−xn
i−1
)21/2
• Interpolate to KUn (x)
Comput. cost O(N)
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 1 (Mesh reconstruction)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
• Estimator function KUn (·)• Curvature, continuous version for smooth u
Ku =|u′′(x)|
(1+(u′(x))2)32
• Curvature, discrete version
K dscrUn,i =
2xi+1−xi−1
∣∣∣∣∣ uni −un
i−1xni −xn
i−1−
uni+1−un
ixni+1−xn
i
∣∣∣∣∣1+
(un
i −uni−1
xni −xn
i−1
)21+
(uni+1−un
ixni+1−xn
i
)21+
(uni+1−un
i−1xni+1−xn
i−1
)21/2
• Interpolate to KUn (x)
Comput. cost O(N)
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 1 (Mesh reconstruction)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
• Estimator function KUn (·)• Monitor function MUn (·)
• Discrete monitor MdscrUn,i =
∫ xni
a KUn (x)dx• Interpolate to MUn (x)
Comput. cost O(N)
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 1 (Mesh reconstruction)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
• Estimator function KUn (·)• Monitor function MUn (·)
• Discrete monitor MdscrUn,i =
∫ xni
a KUn (x)dx
• Interpolate to MUn (x)
Comput. cost O(N)
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 1 (Mesh reconstruction)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
• Estimator function KUn (·)• Monitor function MUn (·)
• Discrete monitor MdscrUn,i =
∫ xni
a KUn (x)dx• Interpolate to MUn (x)
Comput. cost O(N)
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 1 (Mesh reconstruction)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
• Estimator function KUn (·)• Monitor function MUn (·)
• New nodes
{xn+1
1 = xn1 ,
MUn (xn+1i+1 )−MUn (xn+1
i ) = 1N MUn (xn
N), i
Comput. cost O(N)
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 1 (Mesh reconstruction)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
• Estimator function KUn (·)• Monitor function MUn (·)
• New nodes
{xn+1
1 = xn1 ,
MUn (xn+1i+1 )−MUn (xn+1
i ) = 1N MUn (xn
N), iComput. cost O(N)
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 1 (Mesh reconstruction)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
• Estimator function KUn (·)• Monitor function MUn (·)
• New nodes
{xn+1
1 = xn1 ,
MUn (xn+1i+1 )−MUn (xn+1
i ) = 1N MUn (xn
N), iComput. cost O(N)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
adamon
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
adamon
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 1 (Mesh reconstruction)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}
• Estimator function KUn (·)• Monitor function MUn (·)
• New nodes
{xn+1
1 = xn1 ,
MUn (xn+1i+1 )−MUn (xn+1
i ) = 1N MUn (xn
N), iComput. cost O(N)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
adamon
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
adamon
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 2 (Solution update)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}, Mn+1
x ={
xn+11 , . . . , xn+1
N
}
• Grids• Vertex centered Cn
i = (xni−1/2, x
ni+1/2), xn
i−1/2 =xn
i−1+xni
2• Finite element cells Cn
i = (xni−1, x
ni )
• Approximation function Un(x)• Piecewise constant• Piecewise linear
• Update to Un = {un1 , . . . , u
nN}
• Conservative reconstruction• Interpolation
graph
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 2 (Solution update)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}, Mn+1
x ={
xn+11 , . . . , xn+1
N
}• Grids
• Vertex centered Cni = (xn
i−1/2, xni+1/2), xn
i−1/2 =xn
i−1+xni
2• Finite element cells Cn
i = (xni−1, x
ni )
• Approximation function Un(x)• Piecewise constant• Piecewise linear
• Update to Un = {un1 , . . . , u
nN}
• Conservative reconstruction• Interpolation
graph
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 2 (Solution update)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}, Mn+1
x ={
xn+11 , . . . , xn+1
N
}• Grids
• Vertex centered Cni = (xn
i−1/2, xni+1/2), xn
i−1/2 =xn
i−1+xni
2• Finite element cells Cn
i = (xni−1, x
ni )
• Approximation function Un(x)• Piecewise constant• Piecewise linear
• Update to Un = {un1 , . . . , u
nN}
• Conservative reconstruction• Interpolation
graph
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Main Adaptive Scheme (MAS)Step 2 (Solution update)
Mnx = {xn
1 , . . . , xnN}, Un = {un
1 , . . . ,unN}, Mn+1
x ={
xn+11 , . . . , xn+1
N
}• Grids
• Vertex centered Cni = (xn
i−1/2, xni+1/2), xn
i−1/2 =xn
i−1+xni
2• Finite element cells Cn
i = (xni−1, x
ni )
• Approximation function Un(x)• Piecewise constant• Piecewise linear
• More numerical viscosity⇒ “More” entropy stability• Implicit time discretisation⇒ Entropy dissipation• Explicit time discretisation⇒ Entropy production
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Entropy conservative schemes (Ch. Arvanitis, Ch. Makridakis, N. Sfakianakis)
PDE: ut + f (u)x = 0Entropy Pair: (U,F ), U ′′ > 0 and F ′(u) = U ′(u)f ′(u)
Symmetrisation “on the right” - MockEntropy variables: v = U ′(u)
• More numerical viscosity⇒ “More” entropy stability• Implicit time discretisation⇒ Entropy dissipation• Explicit time discretisation⇒ Entropy production
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Entropy conservative schemes (Ch. Arvanitis, Ch. Makridakis, N. Sfakianakis)
PDE: ut + f (u)x = 0Entropy Pair: (U,F ), U ′′ > 0 and F ′(u) = U ′(u)f ′(u)
Symmetrisation “on the right” - MockEntropy variables: v = U ′(u)
• More numerical viscosity⇒ “More” entropy stability• Implicit time discretisation⇒ Entropy dissipation• Explicit time discretisation⇒ Entropy production
FD schemes overnon-uniform grids
for HCL
NikolaosSfakianakis
Introduction
Uniform mesh
Non-uniform mesh
Main AdaptiveScheme
Steps
Some numericalschemes
Remarks
Consistency
EntropyConservativeSchemes
More remarks
TVB
Extremes
TV - Approach 1
TV - Approach 2
Summary
Entropy conservative schemes (Ch. Arvanitis, Ch. Makridakis, N. Sfakianakis)
PDE: ut + f (u)x = 0Entropy Pair: (U,F ), U ′′ > 0 and F ′(u) = U ′(u)f ′(u)
Symmetrisation “on the right” - MockEntropy variables: v = U ′(u)
• More numerical viscosity⇒ “More” entropy stability• Implicit time discretisation⇒ Entropy dissipation• Explicit time discretisation⇒ Entropy production