Convergence analysis for parallel‐in‐time solution of hyperbolic systems

Parallel-in-time algorithms have been successfully employed for reducing time-to-solution of a variety of partial differential equations, especially for diffusive (parabolic-type) equations. A major failing of parallel-in-time approaches to date, however, is that most methods show instabilities or poor convergence for hyperbolic problems. This paper focuses on the analysis of the convergence behavior of multigrid methods for the parallel-in-time solution of hyperbolic problems. Three analysis tools are considered that differ, in particular, in the treatment of the time dimension: (1) space-time local Fourier analysis, using a Fourier ansatz in space and time, (2) semi-algebraic mode analysis, coupling standard local Fourier analysis approaches in space with algebraic computation in time, and (3) a two-level reduction analysis, considering error propagation only on the coarse time grid. In this paper, we show how insights from reduction analysis can be used to improve feasibility of the semi-algebraic mode analysis, resulting in a tool that offers the best features of both analysis techniques. Following validating numerical results, we investigate what insights the combined analysis framework can offer for two model hyperbolic problems, the linear advection equation in one space dimension and linear elasticity in two space dimensions.

[1]  Stefan Vandewalle,et al.  Local Fourier Analysis of Multigrid for the Curl-Curl Equation , 2008, SIAM J. Sci. Comput..

[2]  U. Trottenberg,et al.  A note on MGR methods , 1983 .

[3]  I. Yavneh,et al.  On Multigrid Solution of High-Reynolds Incompressible Entering Flows* , 1992 .

[4]  Achi Brandt,et al.  Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics, Revised Edition , 2011 .

[5]  James L. Thomas,et al.  Half-Space Analysis of the Defect-Correction Method for Fromm Discretization of Convection , 2000, SIAM J. Sci. Comput..

[6]  Stefan Vandewalle,et al.  Multigrid Waveform Relaxation on Spatial Finite Element Meshes: The Discrete-Time Case , 1996, SIAM J. Sci. Comput..

[7]  Stefan Vandewalle,et al.  Multigrid Waveform Relaxation for Anisotropic Partial Differential Equations , 2002, Numerical Algorithms.

[8]  Jürg Nievergelt,et al.  Parallel methods for integrating ordinary differential equations , 1964, CACM.

[9]  Gary R. Consolazio,et al.  Finite Elements , 2007, Handbook of Dynamic System Modeling.

[10]  Martin J. Gander,et al.  Multigrid interpretations of the parareal algorithm leading to an overlapping variant and MGRIT , 2018, Comput. Vis. Sci..

[11]  Daniel Ruprecht,et al.  Wave propagation characteristics of Parareal , 2017, Comput. Vis. Sci..

[12]  Benjamin W. Ong,et al.  Algorithm 965 , 2014, ACM Trans. Math. Softw..

[13]  Shlomo Ta'asan,et al.  On the Multigrid Waveform Relaxation Method , 1995, SIAM J. Sci. Comput..

[14]  Ben S. Southworth,et al.  Necessary Conditions and Tight Two-level Convergence Bounds for Parareal and Multigrid Reduction in Time , 2018, SIAM J. Matrix Anal. Appl..

[15]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[16]  Charbel Farhat,et al.  Time‐parallel implicit integrators for the near‐real‐time prediction of linear structural dynamic responses , 2006 .

[17]  Achi Brandt,et al.  Multigrid solvers for the non-aligned sonic flow: the constant coefficient case , 1999 .

[18]  Robert D. Falgout,et al.  Convergence of the multigrid reduction in time algorithm for the linear elasticity equations , 2018, Numer. Linear Algebra Appl..

[19]  Cornelis W. Oosterlee,et al.  Multigrid Line Smoothers for Higher Order Upwind Discretizations of Convection-Dominated Problems , 1998 .

[20]  Wolfgang Joppich,et al.  Practical Fourier Analysis for Multigrid Methods , 2004 .

[21]  Stefan Vandewalle,et al.  Multigrid waveform relaxation on spatial finite element meshes , 1996 .

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

[23]  Hans De Sterck,et al.  Optimizing MGRIT and Parareal coarse-grid operators for linear advection , 2019, ArXiv.

[24]  Shlomo Ta'asan,et al.  Fourier-Laplace analysis of the multigrid waveform relaxation method for hyperbolic equations , 1996 .

[25]  Scott P. MacLachlan,et al.  A generalized predictive analysis tool for multigrid methods , 2015, Numer. Linear Algebra Appl..

[26]  Guillaume Bal,et al.  On the Convergence and the Stability of the Parareal Algorithm to Solve Partial Differential Equations , 2005 .

[27]  Martin J. Gander,et al.  Nonlinear Convergence Analysis for the Parareal Algorithm , 2008 .

[28]  C. Rodrigo,et al.  A multigrid waveform relaxation method for solving the poroelasticity equations , 2018 .

[29]  Graham Horton,et al.  Fourier mode analysis of the multigrid waveform relaxation and time-parallel multigrid methods , 2005, Computing.

[30]  Matthias Bolten,et al.  A multigrid perspective on the parallel full approximation scheme in space and time , 2016, Numer. Linear Algebra Appl..

[31]  K. Stüben,et al.  Multigrid methods: Fundamental algorithms, model problem analysis and applications , 1982 .

[32]  Martin J. Gander,et al.  Analysis of the Parareal Time-Parallel Time-Integration Method , 2007, SIAM J. Sci. Comput..

[33]  Scott P. MacLachlan,et al.  Local Fourier Analysis of Space-Time Relaxation and Multigrid Schemes , 2013, SIAM J. Sci. Comput..

[34]  Ludmil T. Zikatanov,et al.  On the Validity of the Local Fourier Analysis , 2017, Journal of Computational Mathematics.

[35]  Cornelis W. Oosterlee,et al.  Local Fourier analysis for multigrid with overlapping smoothers applied to systems of PDEs , 2011, Numer. Linear Algebra Appl..

[36]  Scott P. MacLachlan,et al.  Local Fourier analysis for mixed finite-element methods for the Stokes equations , 2019, J. Comput. Appl. Math..

[37]  N. Anders Petersson,et al.  Two-Level Convergence Theory for Multigrid Reduction in Time (MGRIT) , 2017, SIAM J. Sci. Comput..

[38]  Robert D. Falgout,et al.  Multilevel Convergence Analysis of Multigrid-Reduction-in-Time , 2018, SIAM J. Sci. Comput..

[39]  S. MacLachlan,et al.  TWO-LEVEL FOURIER ANALYSIS OF MULTIGRID FOR HIGHER-ORDER FINITE-ELEMENT METHODS∗ , 2018 .

[40]  Hans De Sterck,et al.  Parallel-In-Time Multigrid with Adaptive Spatial Coarsening for The Linear Advection and Inviscid Burgers Equations , 2019, SIAM J. Sci. Comput..

[41]  Martin J. Gander,et al.  Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems , 2014, SIAM J. Sci. Comput..

[42]  Jan S. Hesthaven,et al.  Communication-aware adaptive Parareal with application to a nonlinear hyperbolic system of partial differential equations , 2018, J. Comput. Phys..

[43]  Francisco José Gaspar,et al.  Multigrid Waveform Relaxation for the Time-Fractional Heat Equation , 2016, SIAM J. Sci. Comput..

[44]  Robert D. Falgout,et al.  Parallel time integration with multigrid , 2014 .

[45]  J. Lions,et al.  Résolution d'EDP par un schéma en temps « pararéel » , 2001 .

[46]  Martin J. Gander,et al.  50 Years of Time Parallel Time Integration , 2015 .