Grid adaptivity for systems of conservation laws M. Semplice 1 G. - PowerPoint PPT Presentation

Grid adaptivity for systems of conservation laws M. Semplice 1 G. Puppo 2 1 Dipartimento di Matematica Universit` a degli Studi di Torino 2 Dipartimento di Scienze Matematiche Politecnico di Torino HYP2012 14 th International Conference on Hyperbolic Problems: Theory, Numerics and Applications Padova, 25-29 June 2012 M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 1 / 34

A recipe for an adaptive scheme Ingredients (finite volume) grid numerical fluxes time marching method refine/coarsen strategy (and indicator) Recipe Mix all ingredients, stir until it compiles and runs smoothly, enjoy the movie show! Tools Object Oriented Programming and a library like DUNE a allows to code each part independently (e.g. the Runge-Kutta does not know the number of spatial dimensions) a Dune (Distributed and Unified Numerics Environment), M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 2 / 34 http://www.dune-project.org/index.html

Locally refined grids 1 Timestepping 2 Adaptivity 3 Local recomputation 4 Two space dimensions 5 M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 3 / 34

Locally refined grids 2D grids and quad-trees Data are attached only to coloured cells (leaves) The grid manager ( dune-grid for us) allows to iterate easily on all the leaves on a given level Note: similar structures holds cubes, triangles, tetrahedra, . . . M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 4 / 34

Timestepping Locally refined grids 1 Timestepping 2 Adaptivity 3 Local recomputation 4 Two space dimensions 5 M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 5 / 34

Timestepping Semidiscrete scheme A ∂ t u + ∂ x f ( u ) = 0 R-K: b i − 1 λ = ∆ t stages u ( i ) � F ( k ) j +1 / 2 − F ( k ) � � = u n j − λ a ik j j − 1 / 2 h k =1 � � fluxes F ( i ) u ( i ) , − j +1 / 2 , u ( i ) , + j +1 / 2 = F j +1 / 2 ν � � � F ( i ) j +1 / 2 − F ( i ) step u n +1 = u n j − λ b i j j − 1 / 2 i =0 where u ( i ) , ± j +1 / 2 are some (non-oscillatory) reconstructions. M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 6 / 34

Timestepping Imposing the CFL globally ⇒ ∀ k : ∆ t ≤ λ h k = λ h 0 2 − k = ⇒ ∆ t = λ h 0 2 − k max CFL = t t n +4 t n +3 t n +2 t n +1 t n � b i F ( i ) : : cell averages i M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 7 / 34

Timestepping Local timestepping Choosing ∆ t globally: easier to program wastes CPU resources for big cells, where the solution is well approximated (in theory) unneeded numerical diffusion if many cells have sub-optimal (local) mesh ratio M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 8 / 34

Timestepping Local timestepping Choosing ∆ t globally: easier to program wastes CPU resources for big cells, where the solution is well approximated (in theory) unneeded numerical diffusion if many cells have sub-optimal (local) mesh ratio Local timestepping Each cell advances with different stepsizes t synchro ∆ t 1 ∆ t k = λ h k = λ h 0 2 − k ∆ t 2 ∆ t 3 x M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 8 / 34

Timestepping Review of local timestepping Motivated by approximating (on uniform grids) conservation laws with strong spatial variations of the velocity: First order Osher-Sanders (1983) Second order, TVD property Dawson-Kirby (2001), Kirby (2003) multiresolution (wavelets) Muller-Stiriba(2007) More recent work has focused on AMR: Second order Runge-Kutta Constantinescu-Sandu (2007), Krivodonova (2010) M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 9 / 34

Timestepping First order scheme on nonuniform grid – local timestepping ∆ t k = λ h k ∆ t = ∆ t macro cell averages temporaries fluxes t t n +1 t n x u n u n u n 3 u n u n u n 1 2 4 5 6 M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 10 / 34

Timestepping First order scheme on nonuniform grid – local timestepping ∆ t k = λ h k ∆ t = ∆ t macro cell averages temporaries fluxes t t n +1 t n x u n u n u n 3 u n u n u n 1 2 4 5 6 everywhere M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 10 / 34

Timestepping First order scheme on nonuniform grid – local timestepping ∆ t k = λ h k ∆ t = ∆ t macro temporaries cell averages fluxes t t n +1 t n x u n u n u n 3 u n u n u n 1 2 4 5 6 small M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 10 / 34

Timestepping First order scheme on nonuniform grid – local timestepping ∆ t k = λ h k ∆ t = ∆ t macro temporaries cell averages fluxes t t n +1 h 2 ( 1 u n 2 − ∆ t 4 F 3 / 2 − 1 4 F 1 / 2 ) t n x u n u n u n 3 u n u n u n 1 2 4 5 6 small M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 10 / 34

Timestepping First order scheme on nonuniform grid – local timestepping ∆ t k = λ h k ∆ t = ∆ t macro cell averages temporaries fluxes t t n +1 u n 2 − ∆ t � h 2 4 F 1 4 F 2 1 3 / 2 + 1 3 / 2 2 F 3 collect all − 1 � 1 / 2 fluxes t n x u n u n u n 3 u n u n u n 1 2 4 5 6 small, medium M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 10 / 34

Timestepping First order scheme on nonuniform grid – local timestepping ∆ t k = λ h k ∆ t = ∆ t macro cell averages temporaries fluxes t u n +1 u n +1 u n +1 u n +1 u n +1 u n +1 1 2 3 4 5 6 t n +1 t n x u n u n u n 3 u n u n u n 1 2 4 5 6 M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 10 / 34

Timestepping Second order scheme on nonuniform grid – global CFL cell averages and stage values F (1) F (2) fluxes j +1 / 2 and temporaries j +1 / 2 t t n +1 t n x M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 11 / 34

Timestepping Second order scheme on nonuniform grid – global CFL cell averages and stage values F (1) F (2) fluxes j +1 / 2 and temporaries j +1 / 2 t t n +1 t n x M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 11 / 34

Timestepping Second order scheme on nonuniform grid – global CFL cell averages and stage values F (1) F (2) fluxes j +1 / 2 and temporaries j +1 / 2 t t n +1 t n x M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 11 / 34

Timestepping Second order scheme on nonuniform grid – global CFL cell averages and stage values F (1) F (2) fluxes j +1 / 2 and temporaries j +1 / 2 t t n +1 t n x M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 11 / 34

Timestepping Second order scheme on nonuniform grid – global CFL cell averages and stage values F (1) F (2) fluxes j +1 / 2 and temporaries j +1 / 2 t t n +1 t n x M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 11 / 34

Timestepping Second order scheme on nonuniform grid – global CFL cell averages and stage values F (1) F (2) fluxes j +1 / 2 and temporaries j +1 / 2 t t n +1 t n x automatically conservative! large reconstruction stencil ⇒ many temporaries M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 11 / 34

Timestepping Let’s test the performance of nonuniform grids 1 build a nonuniform grid ratio 1:2 ratio 1:4 ratio 1:4 ratio 1:4 ratio 1:4 2 leave it fixed in time 3 make different kinds of waves travel along them, crossing grid discontinuities. We expect: poor behaviour on error (the grid is fixed!) check if grid discontinuities can deform waves M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 12 / 34

Timestepping Smooth solution (linear transport) u 0 ( x ) = [cos(2 π x )] 4 M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 13 / 34

Timestepping Shock (Burgers equation) M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 14 / 34

Timestepping Shock (Burgers equation) M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 14 / 34

Timestepping Errors on uniform/nonuniform grids grid ratio 1:2 M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 15 / 34

Timestepping Smooth solution (Euler equations) M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 16 / 34

Timestepping Smooth solution (Euler equations) M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 16 / 34

Adaptivity Locally refined grids 1 Timestepping 2 Adaptivity 3 Local recomputation 4 Two space dimensions 5 M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 17 / 34

Adaptivity Semidiscrete scheme with numerical entropy production A ∂ t u + ∂ x f ( u ) = 0 ∂ t η ( u ) + ∂ x ψ ( u ) ≤ 0 R-K: b i − 1 λ = ∆ t � � stages u ( i ) F ( k ) j +1 / 2 − F ( k ) � = u n j − λ a ik j j − 1 / 2 h k =1 � � � � fluxes F ( i ) u ( i ) , − j +1 / 2 , u ( i ) , + Ψ ( i ) u ( i ) , − j +1 / 2 , u ( i ) , + j +1 / 2 = F j +1 / 2 = Ψ j +1 / 2 j +1 / 2 ν � � F ( i ) j +1 / 2 − F ( i ) step u n +1 � = u n j − λ b i j j − 1 / 2 i =0 � ν �� j = 1 � � � � � Ψ ( i ) j +1 / 2 − Ψ ( i ) entropy S n u n +1 � u n η − η + λ b i j j j − 1 / 2 ∆ t i =1 Note: most of the seminar is independent from this choice of error indicator, although the numerical entropy production works well in general M. Semplice et al. (Univ. Insubria) Grid adaptivity HYP2012 18 / 34