Multiresolution schemes for conservation laws

The concept of fully adaptive multiresolution finite volume schemes has been developed and investigated during the past decade. By now it has been successfully employed in numerous applications arising in engineering. In the present work a review on the methodology is given that aims to summarize the underlying concepts and to give an outlook on future developments.

[1]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

[2]  W. Dahmen,et al.  Biorthogonal Spline Wavelets on the Interval—Stability and Moment Conditions , 1999 .

[3]  E. Süli,et al.  A dual graph-norm refinement indicator for finite volume approximations of the Euler equations , 1998 .

[4]  Wolfgang Dahmen,et al.  Local Decomposition of Refinable Spaces and Wavelets , 1996 .

[5]  Vipin Kumar,et al.  Multilevel Algorithms for Multi-Constraint Graph Partitioning , 1998, Proceedings of the IEEE/ACM SC98 Conference.

[6]  M. Berger,et al.  Adaptive mesh refinement for hyperbolic partial differential equations , 1982 .

[7]  Endre Süli,et al.  A posteriori error analysis for numerical approximations of Friedrichs systems , 1999, Numerische Mathematik.

[8]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[9]  Björn Sjögreen,et al.  Numerical experiments with the multiresolution scheme for the compressible Euler equations , 1995 .

[10]  J. Edwards,et al.  Low-Diffusion Flux-Splitting Methods for Flows at All Speeds , 1997 .

[11]  Ralf Hartmann.  Adaptive Fe Methods for Conservation Equations , 2001 .

[12]  M. Farge,et al.  Non-Gaussianity and coherent vortex simulation for two-dimensional turbulence using an adaptive orthogonal wavelet basis , 1999 .

[13]  Kai Schneider,et al.  Coherent Vortex Simulation (CVS), A Semi-Deterministic Turbulence Model Using Wavelets , 2001 .

[14]  W. Dahmen.  Stability of Multiscale Transformations. , 1995 .

[15]  Thomas Sonar,et al.  Dynamic adaptivity and residual control in unsteady compressible flow computation , 1994 .

[16]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[17]  Siegfried Müller,et al.  Adaptive Multiscale Schemes for Conservation Laws , 2002, Lecture Notes in Computational Science and Engineering.

[18]  W. Dahmen.  Wavelet and multiscale methods for operator equations , 1997, Acta Numerica.

[19]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[20]  Albert Cohen,et al.  Fully adaptive multiresolution finite volume schemes for conservation laws , 2003, Math. Comput..

[21]  Marsha Berger,et al.  Three-Dimensional Adaptive Mesh Refinement for Hyperbolic Conservation Laws , 1994, SIAM J. Sci. Comput..

[22]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems IV: nonlinear problems , 1995 .

[23]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[24]  A. Harten.  Multiresolution representation of data: a general framework , 1996 .

[25]  A. Harten.  Adaptive Multiresolution Schemes for Shock Computations , 1994 .

[26]  Sidi Mahmoud Kaber,et al.  Multiresolution Analysis on Triangles: Application to Gas Dynamics , 2001 .

[27]  Vipin Kumar,et al.  A Parallel Algorithm for Multilevel Graph Partitioning and Sparse Matrix Ordering , 1998, J. Parallel Distributed Comput..

[28]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[29]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[30]  P. Colella,et al.  Local adaptive mesh refinement for shock hydrodynamics , 1989 .

[31]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems. I.: a linear model problem , 1991 .

[32]  Mario Ohlberger,et al.  A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions , 2000, Math. Comput..

[33]  Wim Sweldens,et al.  The lifting scheme: a construction of second generation wavelets , 1998 .

[34]  Jinhee Jeong,et al.  On the identification of a vortex , 1995, Journal of Fluid Mechanics.

[35]  Barna L. Bihari,et al.  Multiresolution Schemes for the Numerical Solution of 2-D Conservation Laws I , 1997, SIAM J. Sci. Comput..

[36]  S. Osher,et al.  Numerical approximations to nonlinear conservation laws with locally varying time and space grids , 1983 .

[37]  Barna L. Bihari,et al.  Multiresolution Schemes for Conservation Laws with Viscosity , 1996 .

[38]  Rolf Rannacher,et al.  A Feed-Back Approach to Error Control in Finite Element Methods: Basic Analysis and Examples , 1996 .

[39]  Kai Schneider,et al.  Numerical simulation of a mixing layer in an adaptive wavelet basis , 2000 .

[40]  R. LeVeque,et al.  Adaptive Mesh Refinement Using Wave-Propagation Algorithms for Hyperbolic Systems , 1998 .

[41]  A. Harti.  Discrete multi-resolution analysis and generalized wavelets , 1993 .

[42]  I. Daubechies,et al.  Biorthogonal bases of compactly supported wavelets , 1992 .

[43]  J. J. Quirk,et al.  An adaptive grid algorithm for computational shock hydrodynamics , 1991 .

[44]  A. Brandt.  Multi-Level Adaptive Techniques (MLAT) for Partial Differential Equations: Ideas and Software , 1977 .