Adaptive multi scale scheme based on numerical density of entropy - - PowerPoint PPT Presentation
Adaptive multi scale scheme based on numerical density of entropy production for conservation laws. eric Golay 2 and Lyudmyla Yushchenko 3 Mehmet Ersoy 1 , Fr ed Workshop MTM2011-29306, 18-19 February 2013 1. Mehmet.Ersoy@univ-tln.fr 2.
Adaptive multi scale scheme based on numerical density of entropy production for conservation laws.
Mehmet Ersoy 1, Fr´ ed´ eric Golay 2 and Lyudmyla Yushchenko 3 Workshop MTM2011-29306, 18-19 February 2013
Outline of the talk
Outline of the talk 1 Introduction 2 Construction of the adaptive multi scale scheme
Numerical approximation and properties Mesh refinement algorithm The local time stepping algorithm
3 Numerical results
Sod’s shock tube problem Shu and Osher test case
4 Concluding remarks& perspectives
Entropy production MTM 2 / 35
Outline
Outline 1 Introduction 2 Construction of the adaptive multi scale scheme
Numerical approximation and properties Mesh refinement algorithm The local time stepping algorithm
3 Numerical results
Sod’s shock tube problem Shu and Osher test case
4 Concluding remarks& perspectives
Entropy production MTM 3 / 35
Framework
We focus on general non linear hyperbolic conservation laws ∂w ∂t + ∂f(w) ∂x = 0, (t, x) ∈ R+ × R w(0, x) = w0(x), x ∈ R. (1) where w ∈ Rd : vector state, f : flux governing the physical description of the flow. It is well-known, even if the initial data are smooth, that : at a finite time : solutions develop complex discontinuous structure uniqueness is lost
Serre D., Systems of conservation laws, (99) ; Eymard R., Gallou¨ et T., Herbin R., The finite volume method, (00) ;
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Framework
We focus on general non linear hyperbolic conservation laws ∂w ∂t + ∂f(w) ∂x = 0, (t, x) ∈ R+ × R w(0, x) = w0(x), x ∈ R. (1) where w ∈ Rd : vector state, f : flux governing the physical description of the flow. It is well-known, even if the initial data are smooth, that : at a finite time : solutions develop complex discontinuous structure uniqueness is lost and is recovered (weak physical solution) by completing the system (1) with an entropy inequality of the form : ∂s(w) ∂t + ∂ψ(w) ∂x where (s, ψ) stands for a convex entropy-entropy flux pair.
Serre D., Systems of conservation laws, (99) ; Eymard R., Gallou¨ et T., Herbin R., The finite volume method, (00) ;
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The role of the entropy inequality
The inequality ∂s(w) ∂t + ∂ψ(w) ∂x provides information on the regularity of the solution : ≡ 0 if solutions are smooth < 0 if solutions are discontinuous
Houston P., Mackenzie J.A., S¨ uli E., Warnecke G., Numer. Math., (99) ; Karni S., Kurganov A., Petrova G., J. Comp. Phys., (02) ; Puppo G., SIAM J. Sci. Comput. ICOSAHOM, (03) ; Puppo G., Semplice M., Commun. Comput. Phys., (03) ; Golay F., C.R. M´ ecanique, (09) ;
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The role of the entropy inequality
The inequality ∂s(w) ∂t + ∂ψ(w) ∂x provides information on the regularity of the solution : ≡ 0 if solutions are smooth < 0 if solutions are discontinuous Numerical approximation of this inequality, called numerical density of entropy production, measure the amount of violation of the entropy equation (as a measure of the local residual).
Houston P., Mackenzie J.A., S¨ uli E., Warnecke G., Numer. Math., (99) ; Karni S., Kurganov A., Petrova G., J. Comp. Phys., (02) ; Puppo G., SIAM J. Sci. Comput. ICOSAHOM, (03) ; Puppo G., Semplice M., Commun. Comput. Phys., (03) ; Golay F., C.R. M´ ecanique, (09) ;
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The role of the entropy inequality
The inequality ∂s(w) ∂t + ∂ψ(w) ∂x provides information on the regularity of the solution : ≡ 0 if solutions are smooth < 0 if solutions are discontinuous Numerical approximation of this inequality, called numerical density of entropy production, measure the amount of violation of the entropy equation (as a measure of the local residual). As a consequence, the numerical density of entropy production provides information on the need : to coarsen the mesh if solutions are smooth locally refine the mesh if solutions are discontinuous
Houston P., Mackenzie J.A., S¨ uli E., Warnecke G., Numer. Math., (99) ; Karni S., Kurganov A., Petrova G., J. Comp. Phys., (02) ; Puppo G., SIAM J. Sci. Comput. ICOSAHOM, (03) ; Puppo G., Semplice M., Commun. Comput. Phys., (03) ; Golay F., C.R. M´ ecanique, (09) ;
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The role of the entropy inequality
The inequality ∂s(w) ∂t + ∂ψ(w) ∂x provides information on the regularity of the solution : ≡ 0 if solutions are smooth < 0 if solutions are discontinuous Numerical approximation of this inequality, called numerical density of entropy production, measure the amount of violation of the entropy equation (as a measure of the local residual). As a consequence, the numerical density of entropy production provides information on the need : to coarsen the mesh if solutions are smooth locally refine the mesh if solutions are discontinuous Conclusion : an intrinsic a posteriori error indicator = ⇒ automatic mesh refinement
Houston P., Mackenzie J.A., S¨ uli E., Warnecke G., Numer. Math., (99) ; Karni S., Kurganov A., Petrova G., J. Comp. Phys., (02) ; Puppo G., SIAM J. Sci. Comput. ICOSAHOM, (03) ; Puppo G., Semplice M., Commun. Comput. Phys., (03) ; Golay F., C.R. M´ ecanique, (09) ;
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An important time restriction
Explicit adaptive schemes are well-know to be time consuming due to a CFL stability condition since : the CFL imposes an upper bound on δt h where δt is the time step and h the finest mesh size.
M¨ uller S., Stiriba Y., SIAM J. Sci. Comput., (07) ; Tan Z., Zhang Z., Huang Y., Tang T., J. Comp. Phys., (04) ; Ersoy M., Golay F., Yushchenko L., CEJM, (13) ;
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An important time restriction, local time stepping approach
Explicit adaptive schemes are well-know to be time consuming due to a CFL stability condition since : the CFL imposes an upper bound on δt h where δt is the time step and h the finest mesh size. Nevertheless, the cpu-time can be significantly reduced using the local time stepping algorithm
M¨ uller S., Stiriba Y., SIAM J. Sci. Comput., (07) ; Tan Z., Zhang Z., Huang Y., Tang T., J. Comp. Phys., (04) ; Ersoy M., Golay F., Yushchenko L., CEJM, (13) ;
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An important time restriction, local time stepping approach & Aims
Explicit adaptive schemes are well-know to be time consuming due to a CFL stability condition since : the CFL imposes an upper bound on δt h where δt is the time step and h the finest mesh size. Nevertheless, the cpu-time can be significantly reduced using the local time stepping algorithm Aims : save the cpu-time by making use of the local time stepping algorithm
M¨ uller S., Stiriba Y., SIAM J. Sci. Comput., (07) ; Tan Z., Zhang Z., Huang Y., Tang T., J. Comp. Phys., (04) ; Ersoy M., Golay F., Yushchenko L., CEJM, (13) ;
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An important time restriction, local time stepping approach & Aims
Explicit adaptive schemes are well-know to be time consuming due to a CFL stability condition since : the CFL imposes an upper bound on δt h where δt is the time step and h the finest mesh size. Nevertheless, the cpu-time can be significantly reduced using the local time stepping algorithm Aims : save the cpu-time keeping the order of accuracy by making use of the automatic mesh refinement algorithm local time stepping algorithm
M¨ uller S., Stiriba Y., SIAM J. Sci. Comput., (07) ; Tan Z., Zhang Z., Huang Y., Tang T., J. Comp. Phys., (04) ; Ersoy M., Golay F., Yushchenko L., CEJM, (13) ;
Entropy production MTM 6 / 35
Outline
Outline 1 Introduction 2 Construction of the adaptive multi scale scheme
Numerical approximation and properties Mesh refinement algorithm The local time stepping algorithm
3 Numerical results
Sod’s shock tube problem Shu and Osher test case
4 Concluding remarks& perspectives
Entropy production MTM 7 / 35
Outline
Outline 1 Introduction 2 Construction of the adaptive multi scale scheme
Numerical approximation and properties Mesh refinement algorithm The local time stepping algorithm
3 Numerical results
Sod’s shock tube problem Shu and Osher test case
4 Concluding remarks& perspectives
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Finite volume formulation of the problem Figure: a cell Ck
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Finite volume formulation of the problem
Integrating ∂w ∂t + ∂f(w) ∂x = ∂s(w) ∂t + ∂ψ(w) ∂x
w (tn+1, x) dx −
w (tn, x) dx + tn+1
tn
f(w(t, xi+1/2)) − f(w(t, xi−1/2))dt = 0 S =
s(w(tn+1, x))dx −
s(w(tn, x))dx + tn+1
tn
ψ(w(t, xi+1/2)) − ψ(w(t, xi−1/2))dt where is the density of entropy production and should satisfy S 0.
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Finite volume approximation
Choosing Fk+1/2(wn
k , wn k+1) as a suitable approximation of
1 δtn tn+1
tn
f(w(xk±1/2, s) ds, noting δtn = tn+1 − tn and wn
k ≃ 1
hk
w (x, tn) dx we obtain : wn+1
k
= wn
k − δtn
hk
k+1/2 − F n k−1/2
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Finite volume approximation
Choosing Fk+1/2(wn
k , wn k+1) as a suitable approximation of
1 δtn tn+1
tn
f(w(xk±1/2, s) ds, noting δtn = tn+1 − tn and wn
k ≃ 1
hk
w (x, tn) dx we obtain : wn+1
k
= wn
k − δtn
hk
k+1/2 − F n k−1/2
Using the same discretisation, we get : Sn
k = sn+1 k
− sn
k
δtn + ψn
k+1/2 − ψn k−1/2
hk , called the numerical density of entropy production.
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the numerical density of entropy production as a mesh refinement indicator
The numerical density of entropy production : if the solution is smooth, Sn
k fails to be 0.
if the solution is discontinuous, Sn
k should be negative.
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the numerical density of entropy production as a mesh refinement indicator
The numerical density of entropy production : if the solution is smooth, Sn
k fails to be 0.
if the solution is discontinuous, Sn
k should be negative.
Thus, in both cases, Sn
k provides local information
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the numerical density of entropy production as a mesh refinement indicator
The numerical density of entropy production : if the solution is smooth, Sn
k fails to be 0.
if the solution is discontinuous, Sn
k should be negative.
Thus, in both cases, Sn
k provides local information
and it will be used as a mesh refinement indicator :
0.2 0.4 0.6 0.8 1 1.2
0.5 1-0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Density Numerical density of entropy production x ρ on adaptive mesh with Lmax = 2 ρex Sk
n
(a) Density and numerical density of entropy production (−Sn
k ).
1 1.5 2 2.5 3 3.5 4 4.5 5
0.5 1 0 0.02 0.04 0.06 0.08 0.1 Mesh refinement level Numerical density of entropy production x level Sk
n
|ρ-ρex|
(b) Numerical density of entropy production and local error (−Sn
k ).
Figure: Numerical example
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The numerical density of entropy production and its properties
In particular, one has :
Theorem
Consider a pth convergent scheme for Equations (1) and discretise the entropy inequality as Equations (1). Let Sn
k be the corresponding numerical density of entropy production
and ∆ = λh be a fixed time step where h stands for the meshsize. Then lim
n→∞ Sn k =
O(∆tp) if the solution is smooth, O 1 ∆t
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The numerical density of entropy production and its properties
In particular, one has :
Theorem
Consider a pth convergent scheme for Equations (1) and discretise the entropy inequality as Equations (1). Let Sn
k be the corresponding numerical density of entropy production
and ∆ = λh be a fixed time step where h stands for the meshsize. Then lim
n→∞ Sn k =
O(∆tp) if the solution is smooth, O 1 ∆t
and the following property is satisfied :
Properties
Consider a monotone scheme. Then, for almost every k, every n, Sn
k 0.
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The numerical density of entropy production and its properties
In particular, one has :
Theorem
Consider a pth convergent scheme for Equations (1) and discretise the entropy inequality as Equations (1). Let Sn
k be the corresponding numerical density of entropy production
and ∆ = λh be a fixed time step where h stands for the meshsize. Then lim
n→∞ Sn k =
O(∆tp) if the solution is smooth, O 1 ∆t
and the following property is satisfied :
Properties
Consider a monotone scheme. Then, for almost every k, every n, Sn
k 0.
Let us remark that : even if locally Sn
k can take positive value, from the previous results, one has
Sn
k C∆tq,
q p .
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The numerical density of entropy production : examples
Let us consider the transport equation : wt + wx = w(0, x) = w0(x) and the Godunov scheme : wn+1
k
= wn
k + δt
δx (wn
k − wn k−1)
Sn+1
k
= s(wn+1
k
) − s(wn
k )
δt + ψ(s(wn
k )) − ψ(s(wn k−1))
δx with s(w) = w2 and ψ(w) = w2.
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The numerical density of entropy production : examples
Let us consider the transport equation : wt + wx = w(0, x) = w0(x) and the Godunov scheme : wn+1
k
= wn
k + δt
δx (wn
k − wn k−1)
Sn+1
k
= s(wn+1
k
) − s(wn
k )
δt + ψ(s(wn
k )) − ψ(s(wn k−1))
δx with s(w) = w2 and ψ(w) = w2. Substituting wn+1
k
into Sn+1
k
, one find Sn+1
k
= −ε wn
k − wn k−1
δx 2 with ε = δx
δx
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The numerical density of entropy production : examples
Let us consider the transport equation : wt + wx = w(0, x) = w0(x) and the Godunov scheme : wn+1
k
= wn
k + δt
δx (wn
k − wn k−1)
Sn+1
k
= s(wn+1
k
) − s(wn
k )
δt + ψ(s(wn
k )) − ψ(s(wn k−1))
δx with s(w) = w2 and ψ(w) = w2. Substituting wn+1
k
into Sn+1
k
, one find Sn+1
k
= −ε wn
k − wn k−1
δx 2 0 with ε = δx
δx
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Outline
Outline 1 Introduction 2 Construction of the adaptive multi scale scheme
Numerical approximation and properties Mesh refinement algorithm The local time stepping algorithm
3 Numerical results
Sod’s shock tube problem Shu and Osher test case
4 Concluding remarks& perspectives
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Notations Figure: Example of hierarchical dyadic tree with three different cells
Ck0 : macro cell, b : binary number which contains the hierarchical information of a sub-cell, Ckb : sub-cell of Ck0, Lmax : maximum level of refinement Ck0, length(b) : level of refinement
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Mesh refinement process : refinement& unrefinement
define a mesh refinement parameter ¯ S, say, the mean value over the domain Ω : ¯ S = 1 |Ω|
Sn
kb
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Mesh refinement process : refinement& unrefinement
define a mesh refinement parameter ¯ S, say, the mean value over the domain Ω : ¯ S = 1 |Ω|
Sn
kb
define two coefficients 0 αmin 1 : ratio of mesh coarsening, 0 αmax 1 : ratio of mesh refinement,
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Mesh refinement process : refinement& unrefinement
define a mesh refinement parameter ¯ S, say, the mean value over the domain Ω : ¯ S = 1 |Ω|
Sn
kb
define two coefficients 0 αmin 1 : ratio of mesh coarsening, 0 αmax 1 : ratio of mesh refinement, Then, for each cell Ckb : if Sn
kb > ¯
Sαmax, the mesh is refined and split into two sub-cells Ckb0 and Ckb1 , if Sn
kb0 < ¯
Sαmin and Sn
kb1 < ¯
Sαmin, the mesh is coarsened into a cell Ckb.
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Mesh refinement process : refinement& unrefinement
define a mesh refinement parameter ¯ S, say, the mean value over the domain Ω : ¯ S = 1 |Ω|
Sn
kb
define two coefficients 0 αmin 1 : ratio of mesh coarsening, 0 αmax 1 : ratio of mesh refinement, Then, for each cell Ckb : if Sn
kb > ¯
Sαmax, the mesh is refined and split into two sub-cells Ckb0 and Ckb1 , if Sn
kb0 < ¯
Sαmin and Sn
kb1 < ¯
Sαmin, the mesh is coarsened into a cell Ckb.
1 1.5 2 2.5 3 3.5 4 4.5 5
0.5 1 0 0.0001 0.0002 0.0003 0.0004 0.0005 Mesh refinement level Numerical density of entropy production x level Sk
nFigure: Example of ¯ S (−Sn
k )
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Mesh refinement process : refinement& unrefinement
If a cell Ckb is split into two sub-cells Ckb0 and Ckb1, new subcells are initialized as follows :
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Mesh refinement process : refinement& unrefinement
If a cell Ckb is split into two sub-cells Ckb0 and Ckb1, new subcells are initialized as follows : if two sub-cells Ckb0 and Ckb1 is merged, the new cell Ckb is initialized as follows :
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Outline
Outline 1 Introduction 2 Construction of the adaptive multi scale scheme
Numerical approximation and properties Mesh refinement algorithm The local time stepping algorithm
3 Numerical results
Sod’s shock tube problem Shu and Osher test case
4 Concluding remarks& perspectives
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Notations&Principle& Illustration
Notations : Let ∆tn = 2Nδtn : be the macro time step δtn : be themicro time step Lk : be thelevel of refinement of the kth cell Ck defined by : 2N−Lkhmin hk < 2N+1−Lkhmin N = log2 hmax hmin
: be the maximum level of refinement
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Notations& Principle& Illustration
Notations : Let ∆tn = 2Nδtn : be the macro time step δtn : be themicro time step Lk : be thelevel of refinement of the kth cell Ck defined by : 2N−Lkhmin hk < 2N+1−Lkhmin N = log2 hmax hmin
: be the maximum level of refinement Principle : Sort cells in groups w.r.t. to their level Update the cells following the local time stepping algorithm.
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Notations& Principle& Illustration
Let us note δF n
k−1,k,k+1 :=
k+1/2(wk, wk+1) − F n k−1/2(wk−1, wk)
Figure: t = tn
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Notations& Principle& Illustration
Let us note δF n
k−1,k,k+1 :=
k+1/2(wk, wk+1) − F n k−1/2(wk−1, wk)
Figure: tn1 = tn + δtn
wn1
k000 = wn k000 − δtn
hk000 δF n
k00,k000,k001
wn1
k001 = wn k001 − δtn
hk001 δF n
k000,k001,k+1b
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Notations& Principle& Illustration
Let us note δF n
k−1,k,k+1 :=
k+1/2(wk, wk+1) − F n k−1/2(wk−1, wk)
Figure: tn2 = tn + 2δtn
wn2
k00 = wn1 k00 − δtn
hk00 δF n1
k−10,k00,k000
wn2
k000 = wn1 k000 − δtn
hk000 δF n1
k00,k000,k001
wn2
k001 = wn1 k001 − δtn
hk001 δF n1
k000,k001,k+1b
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Notations& Principle& Illustration
Let us note δF n
k−1,k,k+1 :=
k+1/2(wk, wk+1) − F n k−1/2(wk−1, wk)
Figure: tn3 = tn + 3δtn
wn3
k000 = wn2 k000 − δtn
hk000 δF n2
k00,k000,k001
wn3
k001 = wn2 k001 − δtn
hk001 δF n2
k000,k001,k+1b
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Notations& Principle& Illustration
Let us note δF n
k−1,k,k+1 :=
k+1/2(wk, wk+1) − F n k−1/2(wk−1, wk)
Figure: tn+1 = tn + 4δtn
wn+1
k−10 = wn3 k−10 −
δtn hk−10 δF n3
k−2b,k−10,k00
wn+1
k00 = wn3 k00 − δtn
hk00 δF n3
k−10,k00,k000
wn+1
k000 = wn3 k000 − δtn
hk000 δF n3
k00,k000,k001
wn+1
k001 = wn3 k001 − δtn
hk001 δF n3
k000,k001,k+1b
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Time& space high order extensions
Time and space high order extensions can be easily implemented in multi-scale framework : Time integration using Adams-Bashforth integration technique. For example, the second order Adams-Bashforth method is : wk(tn+1) = wk(tn) − δtn hk δFk(tn) − δt2
n
2δtn−1 hk (δFk(tn) − δFk(tn−1)) . with Sn
k := s(wk(tn+1)) − s(wk(tn))
δtn + δψk(tn) hk + δtn 2δtn−1 hk (δψk(tn) − δψk(tn−1))
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Time& space high order extensions
Time and space high order extensions can be easily implemented in multi-scale framework : Time integration using Adams-Bashforth integration technique. For example, the second order Adams-Bashforth method is : wk(tn+1) = wk(tn) − δtn hk δFk(tn) − δt2
n
2δtn−1 hk (δFk(tn) − δFk(tn−1)) . with Sn
k := s(wk(tn+1)) − s(wk(tn))
δtn + δψk(tn) hk + δtn 2δtn−1 hk (δψk(tn) − δψk(tn−1)) For instance, a second order MUSCL (Monotone Upstream-centered Schemes) reconstruction : wn
kb0 = wn kb − hk
4 ∂wn
kb
∂x , wn
kb1 = wn kb + hk
4 ∂wn
kb
∂x .
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Outline
Outline 1 Introduction 2 Construction of the adaptive multi scale scheme
Numerical approximation and properties Mesh refinement algorithm The local time stepping algorithm
3 Numerical results
Sod’s shock tube problem Shu and Osher test case
4 Concluding remarks& perspectives
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Numerical solutions are computed in the case of the one-dimensional gas dynamics equations for ideal gas : ∂ρ ∂t + ∂ρu ∂x = 0 ∂ρu ∂t + ∂
= 0 ∂ρE ∂t + ∂ (ρE + p) u ∂x = 0 p = (γ − 1)ρε where ρ(t, x) : density u(t, x) : velocity p(t, x) : pressure γ := 1.4 : ratio of the specific heats E(ε, u) : total energy where E := ε + u2 2 (where ε is the internal specific energy). Using the conservative variables w = (ρ, ρu, ρE)t, we classically define the convex continuous entropy s(w) = −ρ ln p ργ
We have used the Godunov solver and displayed −S instead of S. All tests have been performed on Intel(R) Core(TM) i5-2500 CPU @ 3.30GHz
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Outline
Outline 1 Introduction 2 Construction of the adaptive multi scale scheme
Numerical approximation and properties Mesh refinement algorithm The local time stepping algorithm
3 Numerical results
Sod’s shock tube problem Shu and Osher test case
4 Concluding remarks& perspectives
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Parameters : Mesh refinement parameter αmax : 0.01 , Mesh coarsening parameter αmin : 0.001 , Mesh refinement parameter ¯ S : 1 |Ω|
Sn
kb
CFL : 0.25, Simulation time (s) : 0.4, Initial number of cells : 200, Maximum level of mesh refinement : Lmax . The initial conditions are :
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Accuracy&Convergence&CPU
0.2 0.4 0.6 0.8 1 1.2
0.5 1-0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 Density Numerical density of entropy production x ρ on adaptive mesh with Lmax = 4 ρ on uniform fixed mesh N = 681 ρex Sk
n
(a) Density and numerical density of entropy produc- tion.
1 1.5 2 2.5 3 3.5 4 4.5 5
0.5 1 0 0.02 0.04 0.06 0.08 0.1 Mesh refinement level Numerical density of entropy production x level Sk
n
|ρ-ρex|
(b) Mesh refinement level, numerical density of entropy production and local error.
Figure: Sod’s shock tube problem : solution at time t = 0.4 s using the AB1M scheme on a dynamic grid with Lmax = 5 and the AB1 scheme on a uniform fixed grid of 681 cells.
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Accuracy& Convergence&CPU time
log10(error) log10(average number of cells) AB1 uniform mesh: order = 0.69705 AB1 adaptive mesh: order = 1.861 AB1M adaptive mesh: order = 2.1539
(a) ρex − ρl1
t l1 x with respect to the averaged
number of cells for the schemes of order 1.
Figure: Sod’s shock tube problem : convergence rate
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Accuracy&Convergence& CPU time
log10(error) log10(average number of cells) AB1 uniform mesh: order = 0.69705 AB1 adaptive mesh: order = 1.861 AB1M adaptive mesh: order = 2.1539
(a) ρex − ρl1
t l1 x with respect to the averaged
number of cells for the schemes of order 1.
log10(error) log10(cpu-time) AB1 uniform mesh: order = 0.41231 AB1 adaptive mesh: order = 0.64897 AB1M adaptive mesh: order = 0.77387
(b) cpu-time–ρex − ρl1
t l1 x with respect to the
cpu-time for the schemes of order 1.
Figure: Sod’s shock tube problem : convergence rate and cpu-time
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Outline
Outline 1 Introduction 2 Construction of the adaptive multi scale scheme
Numerical approximation and properties Mesh refinement algorithm The local time stepping algorithm
3 Numerical results
Sod’s shock tube problem Shu and Osher test case
4 Concluding remarks& perspectives
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Parameters : CFL : 0.219, Simulation time (s) : 0.18, Initial number of cells : 500, Maximum level of mesh refinement : Lmax = 4. The initial conditions are :
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Reference solution&Numerical results
Reference solution computed on a fine grid with 100 000 cells
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 Density Numerical density of entropy production x
Reference solution Sk
n
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Reference solution&Numerical results
1 2 3 4 5 6 7 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 Density Numerical density of entropy production x AB1 Sk
n for AB1
AB2 Sk
n for AB2
RK2 Sk
n for RK2
reference solution Sk
n for reference solution
(a) Density and numerical density of entropy produc- tion.
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.7 Density x AB1 AB2 RK2 Reference solution
(b) Zoom on oscillating region.
Figure: Shu Osher test case.
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Reference solution& Numerical results
P ρ − ρrefl1
x
cpu-time NLmax maximum number of cells AB1 0.288 4.74 10−2 181 1574 2308 AB1M 0.288 4.80 10−2 120 1572 2314 AB2 0.287 2.75 10−2 170 1391 2023 AB2M 0.286 2.74 10−2 108 1357 1994 RK2 0.285 2.08 10−2 299 1375 2005
Table: Comparison of numerical schemes of order 1 and 2
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Outline
Outline 1 Introduction 2 Construction of the adaptive multi scale scheme
Numerical approximation and properties Mesh refinement algorithm The local time stepping algorithm
3 Numerical results
Sod’s shock tube problem Shu and Osher test case
4 Concluding remarks& perspectives
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With local time stepping (to compute fast) the numerical density of entropy production (to pilot locally the mesh)
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With local time stepping (to compute fast) the numerical density of entropy production (to pilot locally the mesh) we have obtain an efficient adaptive algorithm : accurate reduction of the computational time implementation of first and second order in space and time
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With local time stepping (to compute fast) the numerical density of entropy production (to pilot locally the mesh) we have obtain an efficient adaptive algorithm : accurate reduction of the computational time implementation of first and second order in space and time We plan to improve the efficiency to capture accurately contact discontinuities (artificial entropy) to extend this work for 2D/3D numerical applications.
Figure: 2D dambreak problem : αmax = 0.1, αmax = 0.2, Lmax = 5 using ¯ S and 321
k , bottom left : level, bottom right :
1 |D|
Sn
k )
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local time stepping algorithm
foreach i ∈ {1, 2N} do Let j be the biggest integer such that 2j divides i foreach interface xk+1/2 such that Lk+1/2 N − j do
1 compute the integral of Fk+1/2(t) on the time interval 2N−Lk+1/2δtn, 2 distribute Fk+1/2(tn) to the two adjacent cells, 3 update only the cells of level greater than N − j.
end end
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