Parallel multi-core and multi-processor methods on point-value multiresolution algorithms for hyperbolic conservation laws

Abstract The underlying sequential behavior of the multiresolution (MR) method has been exploited for parallel computing by introducing a concept of multiresolution forest structures (MFS) along with two new load-balancing algorithms. Another easy-to-implement multithreading approach has also been introduced for the multicore architectures. Tests were conducted using an Euler solver based on a fifth-order shock capturing WENO scheme and a third-order Runge–Kutta algorithm. The methods have been rigorously analyzed in terms of speedup ratio and parallel efficiency to bring forth their benefits as well as limitations. The performance yielded through these methods indicates that the MFS is a new headway for the MR method in parallel computing that has a potential to harness better scalability.

[1]  Diego Rossinelli.  Multiresolution flow simulations on multi/many-core architectures , 2011 .

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

[3]  Diego Rossinelli,et al.  High order finite volume methods on wavelet-adapted grids with local time-stepping on multicore architectures for the simulation of shock-bubble interactions , 2010, J. Comput. Phys..

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

[5]  Nikolaus A. Adams,et al.  Adaptive multi-resolution method for compressible multi-phase flows with sharp interface model and pyramid data structure , 2014, J. Comput. Phys..

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

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

[8]  Abdellah Hadjadj,et al.  Numerical Study of Compressible Mixing Layers Using High-Order WENO Schemes , 2011, J. Sci. Comput..

[9]  Samuel Paolucci,et al.  WAMR: An adaptive wavelet method for the simulation of compressible reacting flow. Part II. The parallel algorithm , 2014, J. Comput. Phys..

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

[11]  Olivier Roussel,et al.  A conservative fully adaptive multiresolution algorithm for parabolic PDEs , 2003 .

[12]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

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

[14]  D. Andrews,et al.  A relaxation method for solving nonlinear stress equilibrium problems , 1973 .

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

[16]  Nira Dyn,et al.  Multiresolution Schemes on Triangles for Scalar Conservation Laws , 2000 .

[17]  O. Vasilyev,et al.  Wavelet Methods in Computational Fluid Dynamics , 2010 .

[18]  Kai Schneider,et al.  An adaptive multiresolution scheme with local time stepping for evolutionary PDEs , 2008, J. Comput. Phys..

[19]  Eleuterio F. Toro,et al.  Finite-volume WENO schemes for three-dimensional conservation laws , 2004 .

[20]  S. Mallat.  Multiresolution approximations and wavelet orthonormal bases of L^2(R) , 1989 .

[21]  Barna L. Bihari,et al.  Multiresolution Schemes for the Reactive Euler Equations , 1999 .

[22]  Siegfried Müller,et al.  Fully Adaptive Multiscale Schemes for Conservation Laws Employing Locally Varying Time Stepping , 2007, J. Sci. Comput..

[23]  Wolfgang Dahmen,et al.  Multiresolution schemes for conservation laws , 2001 .

[24]  W. Rheinboldt,et al.  Error Estimates for Adaptive Finite Element Computations , 1978 .

[25]  Brian van Straalen,et al.  A survey of high level frameworks in block-structured adaptive mesh refinement packages , 2014, J. Parallel Distributed Comput..

[26]  Martin G. Everett,et al.  Parallel Dynamic Graph Partitioning for Adaptive Unstructured Meshes , 1997, J. Parallel Distributed Comput..

[27]  George Karypis,et al.  METIS and ParMETIS , 2011, Encyclopedia of Parallel Computing.

[28]  Yuxin Ren,et al.  A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws , 2003 .

[29]  Peter MacNeice,et al.  Paramesh: A Parallel Adaptive Mesh Refinement Community Toolkit , 2013 .

[30]  Xu-Dong Liu,et al.  Solution of Two-Dimensional Riemann Problems of Gas Dynamics by Positive Schemes , 1998, SIAM J. Sci. Comput..

[31]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[32]  Rémi Abgrall,et al.  Multiresolution Representation in Unstructured Meshes , 1998 .

[33]  François Pellegrini,et al.  PT-Scotch: A tool for efficient parallel graph ordering , 2008, Parallel Comput..

[34]  A. Harten.  Multiresolution algorithms for the numerical solution of hyperbolic conservation laws , 2010 .

[35]  N.V. Kantartzis,et al.  An explicit weighted essentially nonoscillatory time-domain algorithm for 3-D EMC applications with arbitrary media interfaces , 2006, IEEE Transactions on Magnetics.

[36]  Scott B. Baden,et al.  Parallel Software Abstractions for Structured Adaptive Mesh Methods , 2001, J. Parallel Distributed Comput..

[37]  Rosa Donat,et al.  Point Value Multiscale Algorithms for 2D Compressible Flows , 2001, SIAM J. Sci. Comput..

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

[39]  Eric Brown-Dymkoski,et al.  Parallel adaptive wavelet collocation method for PDEs , 2011, J. Comput. Phys..

[40]  V. Soni,et al.  On the accuracy and efficiency of point-value multiresolution algorithms for solving scalar wave and Euler equations , 2017, J. Comput. Appl. Math..