Block-structured Adaptive Mesh Refinement Methods for Conservation - PowerPoint PPT Presentation

Structure of the lectures Block-structured Adaptive Mesh Refinement Methods for Conservation Laws 1. Fundamentals ◮ Finite volume schemes for hyperbolic problems ◮ Discussion of mesh adaptation approaches Theory, Implementation and Application 2. Structured AMR for hyperbolic problems Multi-resolution Summer School ◮ Presentation of all algorithmic components Fr´ ejus, 06/14/2010 - 06/16/2010 ◮ Parallelization 3. Complex hyperbolic structured AMR applications ◮ Shock-induced combustion ◮ Fluid-structure interaction 4. Further topics ◮ Using the SAMR approach for multigrid methods Ralf Deiterding ◮ Practical implementation, discussion of SAMR systems Computer Science and Mathematics Division Oak Ridge National Laboratory P.O. Box 2008 MS6367, Oak Ridge, TN 37831, USA E-mail: deiterdingr@ornl.gov Block-structured Adaptive Mesh Refinement Methods for Conservation Laws Theory, Implementation and Application Block-structured Adaptive Mesh Refinement Methods for Conservation Laws Theory, Implementation and Application 1 2 Useful references I Useful references II Finite volume methods for hyperbolic problems Adaptive multigrid (finite difference and finite element based in textbooks) ◮ LeVeque, R. J. (2002). Finite volume methods for hyperbolic problems . ◮ Hackbusch, W. (1985). Multi-Grid Methods and Applications . Springer Verlag, Cambridge University Press, Cambridge, New York. Berlin, Heidelberg. ◮ Godlewski, E. and Raviart, P.-A. (1996). Numerical approximation of hyperbolic ◮ Briggs, W. L., Henson, V. E., and McCormick, S. F. (2001). A Multigrid systems of conservation laws . Springer Verlag, New York. Tutorial . Society for Industrial and Applied Mathematics, 2nd edition. ◮ Toro, E. F. (1999). Riemann solvers and numerical methods for fluid dynamics . ◮ Trottenberg, U., Oosterlee, C., and Sch¨ Springer-Verlag, Berlin, Heidelberg, 2nd edition. uller, A. (2001). Multigrid . Academic Press, San Antonio. ◮ Laney, C. B. (1998). Computational gasdynamics . Cambridge University Press, ◮ Martin, D. F. (1998). A cell-centered adaptive projection method for the Cambridge. incompressible Euler equations . PhD thesis, University of California at Berkeley. Structured Adaptive Mesh Refinement Implementation, parallelization ◮ Berger, M. and Colella, P. (1988). Local adaptive mesh refinement for shock ◮ Hornung, R. D., Wissink, A. M., and Kohn, S. H. (2006). Managing complex hydrodynamics. J. Comput. Phys. , 82:64–84. data and geometry in parallel structured AMR applications. Engineering with ◮ Bell, J., Berger, M., Saltzman, J., and Welcome, M. (1994). Three-dimensional Computers , 22:181–195. adaptive mesh refinement for hyperbolic conservation laws. SIAM J. Sci. Comp. , 15(1):127–138. ◮ Rendleman, C. A., Beckner, V. E., Lijewski, M., Crutchfield, W., and Bell, J. B. (2000). Parallelization of structured, hierarchical adaptive mesh refinement ◮ Berger, M. and LeVeque, R. (1998). Adaptive mesh refinement using algorithms. Computing and Visualization in Science , 3:147–157. wave-propagation algorithms for hyperbolic systems. SIAM J. Numer. Anal. , 35(6):2298–2316. Block-structured Adaptive Mesh Refinement Methods for Conservation Laws Theory, Implementation and Application Block-structured Adaptive Mesh Refinement Methods for Conservation Laws Theory, Implementation and Application 3 4

Conservation laws Finite volume methods Upwind schemes Meshes and adaptation References Useful references III ◮ Deiterding, R. (2005). Construction and application of an AMR algorithm for Lecture 1 distributed memory computers. In Plewa, T., Linde, T., and Weirs, V. G., editors, Adaptive Mesh Refinement - Theory and Applications , volume 41 of Fundamentals: Used schemes and mesh Lecture Notes in Computational Science and Engineering , pages 361–372. Springer. adaptation ◮ Deiterding, R. (2003). Parallel adaptive simulation of multi-dimensional detonation structures . PhD thesis, Brandenburgische Technische Universit¨ at Cottbus. Course Block-structured Adaptive Mesh Refinement Methods for Applications (from my own work) Conservation Laws ◮ Deiterding, R. (2009). A parallel adaptive method for simulating shock-induced Theory, Implementation and Application combustion with detailed chemical kinetics in complex domains. Computers & Structures , 87:769–783. ◮ Deiterding, R., Radovitzky, R., Mauch, S. P., Noels, L., Cummings, J. C., and Meiron, D. I. (2006). A virtual test facility for the efficient simulation of solid materials under high energy shock-wave loading. Engineering with Computers , 22(3-4):325–347. Ralf Deiterding Computer Science and Mathematics Division ◮ Pantano, C., Deiterding, R., Hill, D. J., and Pullin, D. I. (2007). A Oak Ridge National Laboratory low-numerical dissipation patch-based adaptive mesh refinement method for P.O. Box 2008 MS6367, Oak Ridge, TN 37831, USA large-eddy simulation of compressible flows. J. Comput. Phys. , 221(1):63–87. E-mail: deiterdingr@ornl.gov Block-structured Adaptive Mesh Refinement Methods for Conservation Laws Theory, Implementation and Application 5 Fundamentals: Used schemes and mesh adaptation 1 Conservation laws Finite volume methods Upwind schemes Meshes and adaptation References Conservation laws Finite volume methods Upwind schemes Meshes and adaptation References Outline Outline Conservation laws Conservation laws Mathematical background Mathematical background Examples Examples Finite volume methods Finite volume methods Basics of finite difference methods Basics of finite difference methods Splitting methods, second derivatives Splitting methods, second derivatives Upwind schemes Upwind schemes Flux-difference splitting Flux-difference splitting Flux-vector splitting Flux-vector splitting High-resolution methods High-resolution methods Meshes and adaptation Meshes and adaptation Elements of adaptive algorithms Elements of adaptive algorithms Adaptivity on unstructured meshes Adaptivity on unstructured meshes Structured mesh refinement techniques Structured mesh refinement techniques Fundamentals: Used schemes and mesh adaptation 2 Fundamentals: Used schemes and mesh adaptation 3

Conservation laws Finite volume methods Upwind schemes Meshes and adaptation References Conservation laws Finite volume methods Upwind schemes Meshes and adaptation References Mathematical background Mathematical background Hyperbolic Conservation Laws Weak solutions Integral form (Gauss’s theorem): d ∂ ∂ ∂ x n f n ( q ( x , t )) = s ( q ( x , t )) , D ⊂ { ( x , t ) ∈ R d × R + X 0 } ∂ t q ( x , t ) + Z Z q ( x , t + ∆ t ) d x − n =1 q ( x , t ) d x q = q ( x , t ) ∈ S ⊂ R M - vector of state, f n ( q ) ∈ C 1 ( S , R M ) - flux functions, Ω Ω s ( q ) ∈ C 1 ( S , R M ) - source term t +∆ t t +∆ t d Z Z Z Z X + f n ( q ( o , t )) σ n ( o ) d o dt = s ( q ( x , t )) d x Definition (Hyperbolicity) n =1 t t A ( q , ν ) = ν 1 A 1 ( q ) + · · · + ν d A d ( q ) with A n ( q ) = ∂ f n ( q ) /∂ q has M real ∂ Ω Ω eigenvalues λ 1 ( q , ν ) ≤ ... ≤ λ M ( q , ν ) and M linear independent right Theorem (Weak solution) eigenvectors r m ( q , ν ). q 0 ∈ L ∞ loc ( R d , S ) . q ∈ L ∞ loc ( D , S ) is weak solution if q satisfies If f n ( q ) is nonlinear, classical solutions q ( x , t ) ∈ C 1 ( D , S ) do not generally exist, not ∞ " # d Z Z ∂ϕ ∂ϕ Z even for q 0 ( x ) ∈ C 1 ( R d , S ) [Majda, 1984], X ∂ t · q + ∂ x n · f n ( q ) − ϕ · s ( q ) d x dt + ϕ ( x , 0) · q 0 ( x ) d x = 0 [Godlewski and Raviart, 1996], n =1 0 R d R d [Kr¨ oner, 1997] for any test function ϕ ∈ C 1 0 ( D , S ) Example: Euler equations Fundamentals: Used schemes and mesh adaptation 4 Fundamentals: Used schemes and mesh adaptation 5 Conservation laws Finite volume methods Upwind schemes Meshes and adaptation References Conservation laws Finite volume methods Upwind schemes Meshes and adaptation References Mathematical background Mathematical background Entropy solutions Entropy solutions II Definition (Entropy solution) Select physical weak solution as lim ε → 0 q ε = q almost everywhere in D of Weak solution q is called an entropy solution if q satisfies d d ∞ ∂ 2 q ε ∂ q ε ∂ f n ( q ε ) " d # T = s ( q ε ) , x ∈ R d , t > 0 Z Z ∂ϕ ∂ϕ ψ n ( q ) − ϕ ∂η ( q ) Z X X ∂ t + − ε X ∂ t η ( q ) + · s ( q ) d x dt + ϕ ( x , 0) η ( q 0 ( x )) d x ≥ 0 ∂ x 2 ∂ x n ∂ x n ∂ q n n =1 n =1 n =1 0 R d R d 0 ( D , R + for all entropy functions η ( q ) and all test functions ϕ ∈ C 1 0 ), ϕ ≥ 0 Theorem (Entropy condition) Assume existence of entropy η ∈ C 2 ( S , R ) and entropy fluxes ψ n ∈ C 1 ( S , R ) Theorem (Jump conditions) that satisfy An entropy solution q is a classical solution q ∈ C 1 (D,S) almost everywhere and T T satisfies the Rankine-Hugoniot (RH) jump condition ∂η ( q ) · ∂ f n ( q ) = ∂ψ n ( q ) , n = 1 , . . . , d ∂ q ∂ q ∂ q d ( q + − q − ) σ t + X f n ( q + ) − f n ( q − ) ` ´ σ n = 0 then lim ε → 0 q ε = q almost everywhere in D is weak solution and satisfies n =1 and the jump inequality d T ∂η ( q ) ∂ψ n ( q ) ≤ ∂η ( q ) X + · s ( q ) d ∂ t ∂ x n ∂ q ( η ( q + ) − η ( q − )) σ t + X ψ n ( q + ) − ψ n ( q − ) n =1 ` ´ σ n ≤ 0 n =1 in the sense of distributions. Proof: [Godlewski and Raviart, 1996] along discontinuities. Proof: [Godlewski and Raviart, 1996] Fundamentals: Used schemes and mesh adaptation 6 Fundamentals: Used schemes and mesh adaptation 7