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Adaptive mesh refinement method : numerical density of entropy production and automatic thresholding. Ersoy, M. 1 a Pons, K. 1 , 2 b 1 EA 2134, IMATH, Universit e de Toulon 2 Principia S.A.S.,13705 La Ciotat cedex 2017, 17 February, LAMA

  1. Mesh refinement indicator : principle & illustration Compute w n k Compute S n k : S n k � = 0 = ⇒ the cell is refined or coarsened More precisely : for a given mesh refinement threshold α 1 � ◮ S n S n k � αS = ⇒ the cell is refined with, for instance, S = k | Ω | Ω ◮ S n ⇒ the cell is coarsened k < αS = M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 6 / 22

  2. Mesh refinement indicator : principle & illustration Compute w n k Compute S n k : S n k � = 0 = ⇒ the cell is refined or coarsened More precisely : for a given mesh refinement threshold α 1 � ◮ S n S n k � αS = ⇒ the cell is refined with, for instance, S = k | Ω | Ω ◮ S n ⇒ the cell is coarsened k < αS = M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 6 / 22

  3. Mesh refinement indicator : principle & illustration Compute w n k Compute S n k : S n k � = 0 = ⇒ the cell is refined or coarsened More precisely : for a given mesh refinement threshold α 1 � ◮ S n S n k � αS = ⇒ the cell is refined with, for instance, S = k | Ω | Ω ◮ S n ⇒ the cell is coarsened k < αS = ◮ A simple projection method but the scheme is locally non consistent [S088, TW05] Shu C. W., Osher S., Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys., 77(2) :439–471, 1988. Tang H., Warnecke G., A class of high resolution difference schemes for nonlinear Hamilton-Jacobi equations with varying time and space grids. SIAM J. Sci. Comput., 26(4) :1415–1431, 2005. M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 6 / 22

  4. Mesh refinement indicator : principle & illustration Compute w n k Compute S n k : S n k � = 0 = ⇒ the cell is refined or coarsened More precisely : for a given mesh refinement threshold α 1 � ◮ S n S n k � αS = ⇒ the cell is refined with, for instance, S = k | Ω | Ω ◮ S n ⇒ the cell is coarsened k < αS = ◮ A simple projection method but the scheme is locally non consistent [S088, TW05] ⋆ less time time consuming than other methods Shu C. W., Osher S., Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys., 77(2) :439–471, 1988. Tang H., Warnecke G., A class of high resolution difference schemes for nonlinear Hamilton-Jacobi equations with varying time and space grids. SIAM J. Sci. Comput., 26(4) :1415–1431, 2005. M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 6 / 22

  5. Mesh refinement indicator : principle & illustration Compute w n k Compute S n k : S n k � = 0 = ⇒ the cell is refined or coarsened More precisely : for a given mesh refinement threshold α 1 � ◮ S n S n k � αS = ⇒ the cell is refined with, for instance, S = k | Ω | Ω ◮ S n ⇒ the cell is coarsened k < αS = ◮ A simple projection method but the scheme is locally non consistent [S088, TW05] ⋆ less time time consuming than other methods ⋆ non consistency is almost negligible if the level of adjacent cells is limited by 2 Shu C. W., Osher S., Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys., 77(2) :439–471, 1988. Tang H., Warnecke G., A class of high resolution difference schemes for nonlinear Hamilton-Jacobi equations with varying time and space grids. SIAM J. Sci. Comput., 26(4) :1415–1431, 2005. M. Ersoy, F. Golay, L. Yushchenko. Adaptive multi-scale scheme based on numerical entropy production for conservation laws. CEJM, Central European Journal of Mathematics, 11(8), pp 1392–1415, 2013. F. Golay, M. Ersoy, L. Yuschenko, D. Sous. Block-based adaptive mesh refinement scheme using numerical density of entropy production for three-dimensional two-fluid flows. International Journal of Computational Fluid Dynamics, Taylor & Francis, 2015. M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 6 / 22

  6. Mesh refinement indicator : principle & illustration Compute w n k Compute S n k : S n k � = 0 = ⇒ the cell is refined or coarsened More precisely : for a given mesh refinement threshold α 1 � ◮ S n S n k � αS = ⇒ the cell is refined with, for instance, S = k | Ω | Ω ◮ S n ⇒ the cell is coarsened k < αS = ◮ A simple projection method but the scheme is locally non consistent [S088, TW05] ⋆ less time time consuming than other methods ⋆ non consistency is almost negligible if the level of adjacent cells is limited by 2 ◮ Improvement (cpu-time) : local time stepping [EGY13] See more details Shu C. W., Osher S., Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys., 77(2) :439–471, 1988. Tang H., Warnecke G., A class of high resolution difference schemes for nonlinear Hamilton-Jacobi equations with varying time and space grids. SIAM J. Sci. Comput., 26(4) :1415–1431, 2005. M. Ersoy, F. Golay, L. Yushchenko. Adaptive multi-scale scheme based on numerical entropy production for conservation laws. CEJM, Central European Journal of Mathematics, 11(8), pp 1392–1415, 2013. F. Golay, M. Ersoy, L. Yuschenko, D. Sous. Block-based adaptive mesh refinement scheme using numerical density of entropy production for three-dimensional two-fluid flows. International Journal of Computational Fluid Dynamics, Taylor & Francis, 2015. M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 6 / 22

  7. Mesh refinement indicator : principle & illustration Compute w n k Compute S n k : S n k � = 0 = ⇒ the cell is refined or coarsened More precisely : for a given mesh refinement threshold α 1 � ◮ S n S n k � αS = ⇒ the cell is refined with, for instance, S = k | Ω | Ω ◮ S n ⇒ the cell is coarsened k < αS = ◮ A simple projection method but the scheme is locally non consistent [S088, TW05] ⋆ less time time consuming than other methods ⋆ non consistency is almost negligible if the level of adjacent cells is limited by 2 ◮ Improvement (cpu-time) : local time stepping [EGY13] See more details ◮ 2D-3D computer management : BB-AMR [GEYS] See more details Shu C. W., Osher S., Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys., 77(2) :439–471, 1988. Tang H., Warnecke G., A class of high resolution difference schemes for nonlinear Hamilton-Jacobi equations with varying time and space grids. SIAM J. Sci. Comput., 26(4) :1415–1431, 2005. M. Ersoy, F. Golay, L. Yushchenko. Adaptive multi-scale scheme based on numerical entropy production for conservation laws. CEJM, Central European Journal of Mathematics, 11(8), pp 1392–1415, 2013. F. Golay, M. Ersoy, L. Yuschenko, D. Sous. Block-based adaptive mesh refinement scheme using numerical density of entropy production for three-dimensional two-fluid flows. International Journal of Computational Fluid Dynamics, Taylor & Francis, 2015. M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 6 / 22

  8. Outline Outline 1 Adaptive Mesh Refinement method Generality An example 2 Automatic mesh refinement threshold 3 Numerical simulations M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 7 / 22

  9. An example : the one-dimensional gas dynamics equations for ideal gas ρ ( t, x ) : density ∂t + ∂ρu ∂ρ u ( t, x ) : velocity ∂x = 0 p ( t, x ) : pressure ρu 2 + p � � ∂t + ∂ ∂ρu γ := 1 . 4 : ratio of the specific heats = 0 where ∂x E ( ε, u ) : total energy ∂ρE + ∂ ( ρE + p ) u ε : internal specific energy = 0 ε + u 2 ∂t ∂x E = p = ( γ − 1) ρε 2 Intel(R) Core(TM) i5-2500 CPU @ 3.30GHz M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 8 / 22

  10. An example : the one-dimensional gas dynamics equations for ideal gas ρ ( t, x ) : density ∂t + ∂ρu ∂ρ u ( t, x ) : velocity ∂x = 0 p ( t, x ) : pressure ρu 2 + p � � ∂t + ∂ ∂ρu γ := 1 . 4 : ratio of the specific heats = 0 where ∂x E ( ε, u ) : total energy ∂ρE + ∂ ( ρE + p ) u ε : internal specific energy = 0 ε + u 2 ∂t ∂x E = p = ( γ − 1) ρε 2 Conservative variables w = ( ρ, ρu, ρE ) t convex continuous entropy � p � s ( w ) = − ρ ln of flux ψ ( w ) = u s ( w ) . ρ γ Intel(R) Core(TM) i5-2500 CPU @ 3.30GHz M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 8 / 22

  11. Sod’s shock tube problem Mesh coarsening parameter α : 0 . 001 , 1 Mesh refinement parameter ¯ � S n S : k b | Ω | k b CFL : 0 . 25 , Simulation time ( s ) : 0 . 4 , Initial number of cells : 200 , Maximum level of mesh refinement : L max . M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 9 / 22

  12. Accuracy 1.2 1.4 5 0.1 ρ on adaptive mesh with L max = 4 level n ρ on uniform fixed mesh N = 681 S k 1.2 4.5 Numerical density of entropy production Numerical density of entropy production ρ ex | ρ - ρ ex | 1 0.08 n S k 1 4 Mesh refinement level 0.8 0.8 3.5 0.06 Density 0.6 3 0.6 0.04 0.4 2.5 0.4 0.2 2 0.02 0 1.5 0.2 1-0.2 1 1 0 -1 -0.5 0 0.5 -1 -0.5 0 0.5 x x (a) Density and numerical density of en- (b) Mesh refinement level, numerical tropy production. 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 L max = 5 and the AB1 scheme on a uniform fixed grid of 681 cells. M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 10 / 22

  13. Properties of S : shock criterion type Theorem ([P04]) Consider a p th convergent scheme. Let S n k be the corresponding numerical density of entropy production and ∆ t = λh be a fixed time step where h stands for the meshsize. Then O (∆ t p )  if the solution is smooth, � 1  n →∞ S n � lim k = O if the solution is discontinuous. ∆ t  Puppo G., Numerical entropy production for central schemes. SIAM J. Sci. Comput., 25(4) :1382–1415, 2004. M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 11 / 22

  14. Properties of S : shock criterion type Theorem ([P04]) Consider a p th convergent scheme. Let S n k be the corresponding numerical density of entropy production and ∆ t = λh be a fixed time step where h stands for the meshsize. Then O (∆ t p )  if the solution is smooth, � 1  n →∞ S n � lim k = O if the solution is discontinuous. ∆ t  Properties Consider a monotone scheme. Then, for almost every k , every n , S n k � 0 . M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 11 / 22

  15. Properties of S : shock criterion type Theorem ([P04]) Consider a p th convergent scheme. Let S n k be the corresponding numerical density of entropy production and ∆ t = λh be a fixed time step where h stands for the meshsize. Then O (∆ t p )  if the solution is smooth, � 1  n →∞ S n � lim k = O if the solution is discontinuous. ∆ t  Properties Consider a monotone scheme. Then, for almost every k , every n , S n k � 0 . Thus, even if locally S n k can take positive value, one has S n k � C ∆ t q , q � p . See example M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 11 / 22

  16. Outline Outline 1 Adaptive Mesh Refinement method Generality An example 2 Automatic mesh refinement threshold 3 Numerical simulations M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 11 / 22

  17. Mesh refinement threshold Mesh refinement criterion S n k and the mesh refinement threshold α . . . ⇒ the cell is refined with S = 1 � S n S n k � αS = k | Ω | Ω S n k < αS = ⇒ the cell is coarsened . . . M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 12 / 22

  18. Mesh refinement threshold Mesh refinement criterion S n k and the mesh refinement threshold α . . . ⇒ the cell is refined with S = 1 � S n S n k � αS = k | Ω | Ω S n k < αS = ⇒ the cell is coarsened . . . how to set α suitably ? M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 12 / 22

  19. Mesh refinement threshold Mesh refinement criterion S n k and the mesh refinement threshold α . . . ⇒ the cell is refined with S = 1 � S n S n k � αS = k | Ω | Ω S n k < αS = ⇒ the cell is coarsened . . . how to set α suitably ? ideally ◮ automatically ◮ accuracy and computational time are“well-balanced” ⋆ catch the smallest maxima but not the smallest M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 12 / 22

  20. Mesh refinement threshold Mesh refinement criterion S n k and the mesh refinement threshold α . . . ⇒ the cell is refined with S = 1 � S n S n k � αS = k | Ω | Ω S n k < αS = ⇒ the cell is coarsened . . . how to set α suitably ? ideally ◮ automatically ◮ accuracy and computational time are“well-balanced” ⋆ catch the smallest maxima but not the smallest ⋆ simple method (” low-cost” ) M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 12 / 22

  21. Review of classical methods Mean method or deviation S ( x, t ) > α 1 � S ( x, t ) dx | Ω | Ω where α is a tunable dimensionless threshold parameter or deviation method S ( x, t ) > α 1 � S ( x, t ) dx + βσ ( x, t ) | Ω | Ω where σ is the standard deviation with β is a tunable dimensionless threshold parameter Kallinderis, Y.G., Baron, J.R. : Adaptation methods for a new navier-stokes algorithm. AIAA journal 27(1), 37–43 (1989) Ersoy, M., Golay, F., Yushchenko, L. : Adaptive multiscale scheme based on numerical density of entropy production for conservation laws. Cent. Eur. J. Math. 11(8), 1392–1415 (2013). M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 13 / 22

  22. Review of classical methods Mean method or deviation → pb : set α or ( α, β ) Filtering : two-steps 3 0.4 S ( x ) min( ¯ S, S ( x )) α ¯ S with α = 1 2.5 α min( ¯ S, S ( x )) with α = 1 0.3 2 1.5 0.2 1 0.1 0.5 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x (a) Filtering (mean me- (b) Filtering (mean me- thod) : first step thod) : second step Aftosmis, M. : Upwind method for simulation of viscous flow on adaptively refined meshes. AIAA journal 32(2), 268–277 (1994) Warren, G.P., Anderson,W.K., Thomas, J.L., Krist, S.L. : Grid convergence for adaptive methods. AIAA paper 1592, 1991 (1991) M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 13 / 22

  23. Review of classical methods Mean method or deviation → pb : set α or ( α, β ) Filtering : two-steps → pb : two-step ? Is it enough ? still to set α Filtering : wavelets 20 150 15 scales S ( x ) 100 10 50 5 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x (c) S (d) Wavelet coefficients ( x = space, y = scale) Figure: Illustration of the wavelet transformation for a given mesh refinement criterion computed with the Daubechies wavelet with four vanishing moments (warm colors correspond to large coefficients and cold colors to small coefficients) Leonard, S., Terracol, M., Sagaut, P. : A wavelet-based adaptive mesh refinement criterion for large-eddy simulation. Journal of Turbulence 7(64) (2006) M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 13 / 22

  24. Review of classical methods Mean method or deviation → pb : set α or ( α, β ) Filtering : two-steps → pb : two-step ? Is it enough ? still to set α Filtering : wavelets → pb : efficient but the cost ! still to set α Local maxima → pb : efficient but the cost ! still to set α 0.001 1 Criterion Flagged zone 0.0008 0.8 Flagged zone(x) 0.0006 0.6 S(x) 0.0004 0.4 0.0002 0.2 0 0 0 5 10 15 20 25 0 5 10 15 20 25 x (m) x (m) (a) S (b) Region to be refined Figure: Illustration of the local maxima method for a mesh refinement criterion involving multiple scales M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 13 / 22

  25. Distribution function (L-meas { S ( x ) > α } ) Assumptions and notations ◮ S is smooth and has p local maxima. ◮ S (0) = S ( L ) = S ′ (0) = S ′ ( L ) = 0 ◮ 0 < S ∞ = max x ∈ (0 ,L ) S ( x ) < ∞ 20 S ( x ) 18 16 14 p = 3 12 S ( x ) 10 8 6 4 2 0 0 2 4 6 8 10 x L M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 14 / 22

  26. Distribution function (L-meas { S ( x ) > α } ) Assumptions and notations ◮ S is smooth and has p local maxima. ◮ S (0) = S ( L ) = S ′ (0) = S ′ ( L ) = 0 ◮ 0 < S ∞ = max x ∈ (0 ,L ) S ( x ) < ∞ Then { x ∈ (0 , L ); ϕ α ( x ) = S ( x ) − α = 0 and S ′ ( x ) � = 0 } � = ∅ Z α = ◮ = { x 0 ( α ) < x 1 ( α ) < · · · < x 2 p α − 2 ( α ) < x 2 p α − 1 ( α ) } ◮ # Z α = 2 p α 20 S ( x ) 18 α 16 14 p = 3 p α = 2 12 S ( x ) 10 8 x 0 ( α ) x 1 ( α ) x 2 ( α ) x 3 ( α ) 6 4 2 0 0 2 4 6 8 10 x L M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 14 / 22

  27. Distribution function (L-meas { S ( x ) > α } ) Assumptions and notations ◮ S is smooth and has p local maxima. ◮ S (0) = S ( L ) = S ′ (0) = S ′ ( L ) = 0 ◮ 0 < S ∞ = max x ∈ (0 ,L ) S ( x ) < ∞ Then { x ∈ (0 , L ); ϕ α ( x ) = S ( x ) − α = 0 and S ′ ( x ) � = 0 } � = ∅ Z α = ◮ = { x 0 ( α ) < x 1 ( α ) < · · · < x 2 p α − 2 ( α ) < x 2 p α − 1 ( α ) } ◮ # Z α = 2 p α ◮ there exist sequence ( x ∗ k ) 1 � k � p 20 x ∗ S ( x ) 3 18 α 16 14 p = 3 p α = 2 12 x ∗ S ( x ) 1 10 8 x 0 ( α ) x 1 ( α ) x 2 ( α ) x 3 ( α ) 6 x ∗ 2 4 2 0 0 2 4 6 8 10 x L M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 14 / 22

  28. Distribution function (L-meas { S ( x ) > α } ) Assumptions and notations ◮ S is smooth and has p local maxima. ◮ S (0) = S ( L ) = S ′ (0) = S ′ ( L ) = 0 ◮ 0 < S ∞ = max x ∈ (0 ,L ) S ( x ) < ∞ Then { x ∈ (0 , L ); ϕ α ( x ) = S ( x ) − α = 0 and S ′ ( x ) � = 0 } � = ∅ Z α = ◮ = { x 0 ( α ) < x 1 ( α ) < · · · < x 2 p α − 2 ( α ) < x 2 p α − 1 ( α ) } ◮ # Z α = 2 p α ◮ there exist sequences ( x ∗ k ) 1 � k � p and ( α ∗ k ) 1 � k � p such that ⋆ ∀ k = 1 , . . . , p ⋆ S ( x ∗ k ) = α ∗ k and S ′ ( x ∗ k ) = 0 , S ′′ ( x ∗ k ) < 0 ⋆ x ∗ ∈ Z α and α ∗ k / p = S ∞ 20 x ∗ S ( x ) 3 18 α ∗ 1 α ∗ 2 α ∗ 3 = S ∞ 16 14 p = 3 12 x ∗ S ( x ) 1 10 8 6 x ∗ 2 4 2 0 0 2 4 6 8 10 x L M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 14 / 22

  29. Distribution function (L-meas { S ( x ) > α } ) Assumptions and notations ◮ S is smooth and has p local maxima. ◮ S (0) = S ( L ) = S ′ (0) = S ′ ( L ) = 0 ◮ 0 < S ∞ = max x ∈ (0 ,L ) S ( x ) < ∞ With these settings, we define the distribution d  1 if α = 0 ,   p α  1  � x 2 k +1 ( α ) − x 2 k ( α ) if 0 < α < S ∞ , α ∈ [0 , S ∞ ] �→ d ( α ) := L  k =1   0 if α = S ∞ .  M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 14 / 22

  30. Properties of d and α P E threshold Properties � S ∞ � L d ( α ) dα = S ( x ) dx . 1 0 0 2 d ∈ C 0 ([0 , S ∞ ] , R + ) and d ′ satisfies ∀ α ∈ [0 , S ∞ ] , d ′ ( α ) < 0 1 d ′ ( α ) = −∞ with the convention α ∗ ∀ k ∈ � 0 , p � , lim 0 := 0 2 α → α ∗ k p − 1 on the set D ∗ := 3 d ∈ C l � D ∗ , R + � � ( α ∗ k , α ∗ k +1 ) . k =0 20 1 0 S ( x ) x ∗ 3 18 d ( α ) α ∗ 1 0.9 α ∗ 2 16 α ∗ 3 = S ∞ 0.8 -1 14 p = 3 0.7 -2 12 0.6 d ′ ( α ) x ∗ 1 S ( x ) d ( α ) 10 0.5 p = 3 -3 8 0.4 6 x ∗ 0.3 2 -4 0.2 4 0.1 2 -5 0 5 10 15 20 0 α 0 0 5 10 15 20 0 2 4 6 8 10 α ∗ α ∗ α ∗ α ∗ 3 = S ∞ α x L 0 1 2 M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 15 / 22

  31. Automatic thresholding : distribution function (L-meas { S ( x ) > α } ) Hard to define α using only d 25 1 20 0.8 15 0.6 S ( x ) d ( α ) 10 0.4 5 0.2 0 0 0 2 4 6 8 10 0 5 10 15 20 x α M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 16 / 22

  32. Automatic thresholding : distribution function (L-meas { S ( x ) > α } ) Hard to define α using only d However, one can set α = α D or ◮ either α = α D = min { α | d ′ ( α ) = − 1 } 25 1 20 0.8 15 0.6 S ( x ) d ( α ) 10 0.4 5 0.2 0 0 0 2 4 6 8 10 0 5 10 15 20 x α 0 -1 -2 d ′ ( α ) -3 -4 -5 0 5 10 15 20 α Dannenhoffer, J.F. : Grid adaptation for complex two-dimensional transonic flows. Ph.D. thesis, Massachusetts Institute of Technology (1987) M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 16 / 22

  33. Automatic thresholding : distribution function (L-meas { S ( x ) > α } ) Hard to define α using only d However, one can set α = α D or α = α P ◮ either α = α P = min { α | d ′′ ( α ) = 0 } 25 1 20 0.8 15 0.6 S ( x ) d ( α ) 10 0.4 5 0.2 0 0 0 2 4 6 8 10 0 5 10 15 20 x α 0 20 15 -1 10 -2 d ′ ( α ) d ′′ ( α ) 5 -3 0 -4 -5 -5 -10 0 5 10 15 20 0 5 10 15 20 α α Powell, K.G., Murman, E.M. : An embedded mesh procedure for leading-edge vortex flows. In : NASA, Langley Research Center, Transonic Symposium : Theory, Application, and Experiment, vol. 1 (1989) M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 16 / 22

  34. Automatic thresholding : distribution function (L-meas { S ( x ) > α } ) Hard to define α using only d However, one can set α = α D or α = α P M − 1 � In practice, d is defined by d ( α ) = d j 1 ( α j ,α j +1 ) ( α ) with j =0 � j � � β � d j = # { k ; S n k > α j } and ( α j ) 0 � j � M = S m β � 1 M 0 � j � M 1 0 d ( α ) uniform grid d ( α ) non uniform grid 0.8 -0.5 0.6 d ′ ( α ) d ( α ) -1 0.4 -1.5 d ′ ( α ) uniform grid 0.2 fi ltered d ′ ( α ) uniform grid fi ltered d ′ ( α ) non uniform grid -2 0 0 5 10 15 20 0 5 10 15 20 α α Dannenhoffer, J.F. : Grid adaptation for complex two-dimensional transonic flows. Ph.D. thesis, Massachusetts Institute of Technology (1987) M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 16 / 22

  35. Automatic thresholding : distribution function (L-meas { S ( x ) > α } ) Hard to define α using only d However, one can set α = α D or α = α P α is usually too small (too sensitive to β ) ! ! ! → detects small fluctuations M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 16 / 22

  36. Automatic thresholding : distribution function (L-meas { S ( x ) > α } ) Hard to define α using only d However, one can set α = α D or α = α P α is usually too small (too sensitive to β ) ! ! ! → detects small fluctuations We propose to set α = α P E = min { α ∈ [0 , ¯ S ] | max { αd ( α ) }} 1 0.5 0.8 0.4 0.6 0.3 d ( α ) f ( α ) 0.4 0.2 0.2 0.1 0 0 0 5 10 15 20 0 5 10 15 20 α α Pons, K., Ersoy, M. Adaptive mesh refinement method. Part 1 : Automatic thresholding based on a distribution function Pons, K., Ersoy, M. Adaptive mesh refinement method. Part 2 : Application to tsunamis propagation M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 16 / 22

  37. Why αd ( α ) ? Properties Assume that S is twice differentiable and has p local maxima. Then, there exists ∀ k = 0 . . . p − 1 , α ∗∗ k +1 ∈ ( α ∗ k , α ∗ k +1 ) such that d ′′ ( α ∗∗ k +1 ) = 0 and the function f has p local maxima α 1 , . . . , α p such that ∀ k = 1 . . . p, α k ∈ ( α ∗∗ k , α ∗ k ) . As a consequence α P E > α P > α D . 20 0 S ( x ) x ∗ 0.5 p = 3 3 18 α ∗ f ( α ) 1 α ∗ 0.45 2 16 α ∗ 3 = S ∞ -1 0.4 14 p = 3 0.35 -2 12 d ′ ( α ) 0.3 x ∗ 1 S ( x ) f ( α ) 10 0.25 -3 8 0.2 6 x ∗ 0.15 2 -4 4 0.1 0.05 2 -5 0 5 10 15 20 0 0 α 0 5 10 15 20 0 2 4 6 8 10 α α ∗ α ∗∗ α ∗ α ∗∗ α ∗ α ∗∗ α ∗ 3 = S ∞ x L 0 1 1 2 2 3 M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 17 / 22

  38. Illustration : α P E threshold for “discontinuous” flows 10 0.12 S ( x ) α PE = 0.83974 0.1 8 S m = 2.4353 0.08 6 f ( α ) S ( x ) 0.06 4 0.04 2 0.02 0 0 0 0.5 1 1.5 2 0 1 2 3 4 5 x α (a) f ( α ) (b) S ( x ) Figure: The function f for the mesh refinement criterion S ( x ) = 200 exp( − 1000( x − 1 . 25) 2 ) + 1 . 25 exp( − 5( x − 3 . 75) 2 ) representing a shock type solution M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 18 / 22

  39. Illustration : α P E threshold for “smooth” flows 10 0.14 S ( x ) α PE = 0.41316 0.12 8 S m = 0.42152 0.1 6 f ( α ) S ( x ) 0.08 0.06 4 0.04 2 0.02 0 0 0 0.1 0.2 0.3 0.4 0 1 2 3 4 5 x α (a) f ( α ) (b) S ( x ) Figure: The function f for the mesh refinement criterion S ( x ) = 2 exp( − 10( x − 1 . 25) 2 ) + 1 . 25 exp( − 5( x − 3 . 75) 2 ) representing a smooth flow M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 18 / 22

  40. Outline Outline 1 Adaptive Mesh Refinement method Generality An example 2 Automatic mesh refinement threshold 3 Numerical simulations M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 18 / 22

  41. Numerical validation[PE] A Dam-break problem for the Saint-Venant equations : � ∂ t h + ∂ x ( hu ) = 0 , hu 2 + gh 2 / 2 = − ghZ ′ ( x ) � � ∂ t ( hu ) + ∂ x h ( t, x ) : density u ( t, x ) : velocity of the water column where g : gravity strength Z : topography K´ evin Pons, Mehmet Ersoy. Adaptive mesh refinement method. Part 1 : Automatic thresholding based on a distribution function M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 19 / 22

  42. Numerical validation[PE] h ex (t,x) h ex (t,x) h(t,x) on adaptive mesh with l max = 2 h(t,x) on adaptive mesh with l max = 2 n with α = 0.01265 and N m = 714 n with α = 0.05296 and N m = 667 S k S k Refined zones Refined zones 6 6 3 400 Mesh refinement criterion 350 Mesh refinement criterion 5 5 2.5 300 Water height Water height 4 2 4 250 3 3 1.5 200 150 2 2 1 100 1 1 0.5 50 0 0 0 0 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 x (m) x (m) (a) A priori error (b) Numerical density of entropy production h ex (t,x) h ex (t,x) h(t,x) on adaptive mesh with l max = 2 h(t,x) on adaptive mesh with l max = 2 n with α = 1.26730 and N m = 693 n with α = 0.05669 and N m = 691 S k S k Refined zones Refined zones 6 160 6 8 Mesh refinement criterion Mesh refinement criterion 140 7 5 5 120 6 Water height 4 Water height 4 100 5 3 80 3 4 60 3 2 2 40 2 1 1 20 1 0 0 0 0 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 x (m) x (m) (c) A posteriori error (d) Gradient of h Figure: Numerical results for the water height at time t = 2 s M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 19 / 22

  43. Numerical validation[PE] α α S m S m 0.016 7 0.014 6 0.012 5 0.01 4 α (t) α (t) 0.008 3 0.006 2 0.004 1 0.002 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t t (a) A priori error (b) Numerical density of entropy production α α S m S m 1.8 0.065 1.6 0.06 1.4 0.055 α (t) α (t) 1.2 0.05 1 0.045 0.8 0.04 0.6 0.035 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t t (c) A posteriori error (d) Gradient of h Figure: Time evolution of the threshold parameter and the mean value S m M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 19 / 22

  44. Test case I : Solitary wave propagation over a two dimensional reef Figure: Experimental settings and wave gauges locations K´ evin Pons, Mehmet Ersoy, Fr´ ed´ eric Golay, Richard Marcer. Adaptive mesh refinement method. Part 2 : Application to tsunamis propagation Roeber, V., Cheung, K.F. : Boussinesq-type model for energetic breaking waves in fringing reef environments. Coastal Engineering 70, 1–20 (2012) Roeber, V., Cheung, K.F., Kobayashi, M.H. : Shock-capturing boussinesq-type model for nearshore wave processes. Coastal Engineering 57(4), M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 20 / 22 407–423 (2010)

  45. Test case I : Solitary wave propagation over a two dimensional reef Adaptive mesh simula- Uniform mesh simula- tion tion Simulation time 240 240 Number of cells 200-560 1000 Re-meshing time step 0.05 s not applicable Time order integration 1 1 Space order integration 1 1 CFL 0.99 0.99 Table: Numerical parameters M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 20 / 22

  46. Test case I : Solitary wave propagation over a two dimensional reef Exp. h (uniform) h (adaptive) l Z 0.2 0 time ˜ t -0.2 -0.4 -0.6 -0.8 -1 0 20 40 60 80 x [ m ] (a) ˜ t = 55 . 03 Figure: Surface profiles of solitary wave propagation over an exposed reef crest. Confrontation of experimental data (blue circles) to numerical data computed on a uniform grid (solid green line) and on an adaptive grid (solid red lines). The solid cyan line represents the mesh level and the black one the bathymetry M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 20 / 22

  47. Test case I : Solitary wave propagation over a two dimensional reef Exp. h (uniform) h (adaptive) l Z 0.2 0 time ˜ t -0.2 -0.4 -0.6 -0.8 -1 0 20 40 60 80 x [ m ] (a) ˜ t = 66 . 53 Figure: Surface profiles of solitary wave propagation over an exposed reef crest. Confrontation of experimental data (blue circles) to numerical data computed on a uniform grid (solid green line) and on an adaptive grid (solid red lines). The solid cyan line represents the mesh level and the black one the bathymetry M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 20 / 22

  48. Test case I : Solitary wave propagation over a two dimensional reef Exp. h (uniform) h (adaptive) l Z 0.2 0 time ˜ t -0.2 -0.4 -0.6 -0.8 -1 0 20 40 60 80 x [ m ] (a) ˜ t = 69 . 13 Figure: Surface profiles of solitary wave propagation over an exposed reef crest. Confrontation of experimental data (blue circles) to numerical data computed on a uniform grid (solid green line) and on an adaptive grid (solid red lines). The solid cyan line represents the mesh level and the black one the bathymetry M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 20 / 22

  49. Test case I : Solitary wave propagation over a two dimensional reef Exp. h (uniform) h (adaptive) l Z 0.2 0 time ˜ t -0.2 -0.4 -0.6 -0.8 -1 0 20 40 60 80 x [ m ] (a) ˜ t = 70 . 68 Figure: Surface profiles of solitary wave propagation over an exposed reef crest. Confrontation of experimental data (blue circles) to numerical data computed on a uniform grid (solid green line) and on an adaptive grid (solid red lines). The solid cyan line represents the mesh level and the black one the bathymetry M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 20 / 22

  50. Test case I : Solitary wave propagation over a two dimensional reef Exp. h (uniform) h (adaptive) l Z 0.2 0 time ˜ t -0.2 -0.4 -0.6 -0.8 -1 0 20 40 60 80 x [ m ] (a) ˜ t = 76 . 33 Figure: Surface profiles of solitary wave propagation over an exposed reef crest. Confrontation of experimental data (blue circles) to numerical data computed on a uniform grid (solid green line) and on an adaptive grid (solid red lines). The solid cyan line represents the mesh level and the black one the bathymetry M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 20 / 22

  51. Test case I : Solitary wave propagation over a two dimensional reef Exp. h (uniform) h (adaptive) l Z 0.2 0 time ˜ t -0.2 -0.4 -0.6 -0.8 -1 0 20 40 60 80 x [ m ] (a) ˜ t = 125 . 03 Figure: Surface profiles of solitary wave propagation over an exposed reef crest. Confrontation of experimental data (blue circles) to numerical data computed on a uniform grid (solid green line) and on an adaptive grid (solid red lines). The solid cyan line represents the mesh level and the black one the bathymetry M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 20 / 22

  52. Test case I : Solitary wave propagation over a two dimensional reef 0.4 0.4 Experiment Experiment Numerical (uniform) Numerical (uniform) 0.3 0.3 Numerical (adaptive) Numerical (adaptive) 0.2 0.2 h 0 h 0 η η 0.1 0.1 0 0 -0.1 -0.1 100 150 200 100 150 200 time ˜ time ˜ t t (a) x = 17 . 6 m (b) x = 50 . 4 m 0.4 0.4 Experiment Experiment Numerical (uniform) Numerical (uniform) 0.3 0.3 Numerical (adaptive) Numerical (adaptive) 0.2 0.2 h 0 h 0 η η 0.1 0.1 0 0 -0.1 -0.1 100 150 200 100 150 200 time ˜ time ˜ t t (c) x = 58 . 1 m (d) x = 65 . 2 m M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 20 / 22 Figure: Surface profiles of solitary wave propagation in time at wave gauges 1 to 6.

  53. Test case I : Solitary wave propagation over a two dimensional reef 0.07 1200 Threshold Adaptive mesh 0.06 S m Uniform mesh 1000 0.05 Number of cells 800 0.04 α 0.03 600 0.02 400 0.01 0 200 60 80 100 120 140 160 180 200 220 60 80 100 120 140 160 180 200 220 ~ ~ time t time t (a) Threshold (b) Number of cells Figure: Time evolution of the mesh refinement threshold and the number of cells : 95 s against 210 s M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 20 / 22

  54. Test case II : Solitary wave propagation over an irregular 3-d shallow shelf Figure: Experimental settings K´ evin Pons, Mehmet Ersoy, Fr´ ed´ eric Golay, Richard Marcer. Adaptive mesh refinement method. Part 2 : Application to tsunamis propagation. Lynett, P.J., Swigler, D., Son, S., Bryant, D., Socolofsky, S. : Experimental study of solitary wave evolution over a 3d shallow shelf. Coastal Engineering Proceedings 1(32), 1 (2011). M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 21 / 22

  55. Test case II : Solitary wave propagation over an irregular 3-d shallow shelf Adaptive mesh simula- Uniform mesh simula- tion tion Simulation time 30 s 30 s Number of blocks 128 128 Number of cells 7 500-25 000 33 000 Re-meshing time step 0.25 s not applicable Time order integration 2 2 Space order integration 2 2 CFL 0.5 0.5 Table: Numerical parameters M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 21 / 22

  56. Test case II : Solitary wave propagation over an irregular 3-d shallow shelf (a) t=0.5s Figure: Numerical water height (coloration is issue from the kinetic energy) M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 21 / 22

  57. Test case II : Solitary wave propagation over an irregular 3-d shallow shelf (a) t=2.5s Figure: Numerical water height (coloration is issue from the kinetic energy) M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 21 / 22

  58. Test case II : Solitary wave propagation over an irregular 3-d shallow shelf (a) t=5.75s Figure: Numerical water height (coloration is issue from the kinetic energy) M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 21 / 22

  59. Test case II : Solitary wave propagation over an irregular 3-d shallow shelf (a) t=23.75s Figure: Numerical water height (coloration is issue from the kinetic energy) M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 21 / 22

  60. Test case II : Solitary wave propagation over an irregular 3-d shallow shelf 0.4 0.45 Experiment Experiment 0.35 0.4 Numerical (uniform) Numerical (uniform) 0.3 0.35 Numerical (adaptive) Numerical (adaptive) Free surface [m] 0.25 Free surface [m] 0.3 0.2 0.25 0.15 0.2 0.1 0.15 0.05 0.1 0 0.05 -0.05 0 -0.1 -0.05 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 time [s] time [s] (a) WG2 results (b) WG4 results 0.2 0.25 Experiment Experiment 0.2 Numerical (uniform) Numerical (uniform) 0.15 Numerical (adaptive) Numerical (adaptive) 0.15 Free surface [m] Free surface [m] 0.1 0.1 0.05 0.05 0 0 -0.05 -0.05 -0.1 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 time [s] time [s] (c) WG6 results (d) WG7 results M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 21 / 22

  61. Test case II : Solitary wave propagation over an irregular 3-d shallow shelf 0.02 40000 Threshold Adaptive mesh 0.018 S m 35000 Uniform mesh 0.016 30000 0.014 Number of cells 0.012 25000 0.01 α 20000 0.008 0.006 15000 0.004 10000 0.002 0 5000 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 50 time [s] time [s] (a) Threshold (b) Number of cells Figure: Time evolution of the mesh refinement threshold and the number of cells : speed up the computation by 2.5 time M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 21 / 22

  62. Test case III : Tsunami runup onto a complex three dimensional Monai-Walley beach (a) Top view (b) Side view Figure: Settings K´ evin Pons, Mehmet Ersoy, Fr´ ed´ eric Golay, Richard Marcer. Adaptive mesh refinement method. Part 2 : Application to tsunamis propagation M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  63. Test case III : Tsunami runup onto a complex three dimensional Monai-Walley beach Adaptive mesh simula- Uniform mesh simula- tion tion Simulation time 30 s 30 s Number of blocks 240 240 Number of cells 8 000-40 000 62 000 Re-meshing time step 0.25 s not applicable Time order integration 2 2 Space order integration 1 1 CFL 0.5 0.5 Table: Numerical parameters M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  64. Test case III : Tsunami runup onto a complex three dimensional Monai-Walley beach (a) t = 11 . 25 s Figure: Numerical water height (coloration is issue from the kinetic energy) M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  65. Test case III : Tsunami runup onto a complex three dimensional Monai-Walley beach (a) t = 13 . 25 s Figure: Numerical water height (coloration is issue from the kinetic energy) M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  66. Test case III : Tsunami runup onto a complex three dimensional Monai-Walley beach (a) t = 16 s Figure: Numerical water height (coloration is issue from the kinetic energy) M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  67. Test case III : Tsunami runup onto a complex three dimensional Monai-Walley beach (a) t = 17 . 5 s Figure: Numerical water height (coloration is issue from the kinetic energy) M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  68. Test case III : Tsunami runup onto a complex three dimensional Monai-Walley beach 7 7 Experiment Experiment 6 6 Numerical (uniform) Numerical (uniform) 5 Numerical (adaptive) 5 Numerical (adaptive) Free surface [cm] Free surface [cm] 4 4 3 3 2 2 1 1 0 0 -1 -1 0 5 10 15 20 25 30 0 5 10 15 20 25 30 time [s] time [s] (a) Gauge 1 (b) Gauge 2 7 Experiment 6 Numerical (uniform) 5 Numerical (adaptive) Free surface [cm] 4 3 2 1 0 -1 0 5 10 15 20 25 30 time [s] (c) Gauge 3 Figure: Free surface results at different positions : experimental data versus numerical simulation with and without mesh adaptivity M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  69. Test case III : Tsunami runup onto a complex three dimensional Monai-Walley beach 0.0002 Threshold Adaptive mesh 70000 S m Uniform mesh 60000 0.00015 Number of cells 50000 0.0001 40000 α 30000 5x10 -5 20000 10000 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 time [s] time [s] (a) Threshold (b) Number of cells Figure: Time evolution of the mesh refinement threshold and the number of cells : speed up the computation by 3 time M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  70. Thank you Thank you for your for your attention attention M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  71. Example Let us consider the transport equation : � w t + w x = 0 w (0 , x ) = w 0 ( x ) Back to slide M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  72. Example Let us consider the transport equation : � w t + w x = 0 w (0 , x ) = w 0 ( x ) and the Godunov scheme :  k − δt w n +1 w n w n k − w n = � �  k − 1  k δx       Back to slide M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  73. Example Let us consider the transport equation : � w t + w x = 0 w (0 , x ) = w 0 ( x ) and the Godunov scheme :  k − δt w n +1 w n w n k − w n = � �  k − 1  k δx   s ( w n +1 + ψ ( s ( w n k )) − ψ ( s ( w n ) − s ( w n k − 1 )) k )   S n +1 k =   k δt δx with s ( w ) = w 2 and ψ ( w ) = w 2 . Back to slide M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  74. Example Let us consider the transport equation : � w t + w x = 0 w (0 , x ) = w 0 ( x ) and the Godunov scheme :  k − δt w n +1 w n w n k − w n = � �  k − 1  k δx   s ( w n +1 + ψ ( s ( w n k )) − ψ ( s ( w n ) − s ( w n k − 1 )) k )   S n +1 k =   k δt δx with s ( w ) = w 2 and ψ ( w ) = w 2 . Substituting w n +1 into S n +1 , we get k k � 2 � w n k − w n � 1 − δt � S n +1 k − 1 = − ε � 0 with ε = δx > 0 . k δx δx Back to slide M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  75. Time restriction Explicit adaptive schemes : time consuming due to the restriction � w � δt h � 1 , h = min k h k M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  76. Time restriction, local time stepping approach Explicit adaptive schemes : time consuming due to the restriction � w � δt h � 1 , h = min k h k Local time stepping algorithm : save the cpu-time ◮ Sort cells in groups w.r.t. to their level Muller S., Stiriba Y., Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping. SIAM J. Sci. Comput., 30(3) :493 ?531, 2007. M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  77. Time restriction, local time stepping approach & Aims Explicit adaptive schemes : time consuming due to the restriction � w � δt h � 1 , h = min k h k Local time stepping algorithm : save the cpu-time ◮ Sort cells in groups w.r.t. to their level ◮ Update the cells following the local time stepping algorithm. Muller S., Stiriba Y., Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping. SIAM J. Sci. Comput., 30(3) :493 ?531, 2007. M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

  78. Illustration Figure: t = t n with � � δF n F n k +1 / 2 ( w k , w k +1 ) − F n k − 1 ,k,k +1 := k − 1 / 2 ( w k − 1 , w k ) M. Ersoy (IMATH) AMR 2017, 17 February, LAMA – Chamb´ ery 22 / 22

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