Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies

In this paper, we investigate the coupling between operator splitting techniques and a time parallelization scheme, the parareal algorithm, as a numerical strategy for the simulation of reaction- diffusion equations modelling multi-scale reaction waves. This type of problems induces peculiar dif- ficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reactive fronts, spatially very localized. In a series of previous studies, the numerical analysis of the operator splitting as well as the parareal algorithm has been conducted and such approaches have shown a great potential in the framework of reaction-diffusion and convection-diffusion-reaction systems. However, complementary studies are needed for a more complete characterization of such techniques for these stiff configurations. Therefore, we conduct in this work a precise numerical analysis that considers the combination of time operator splitting and the parareal algorithm in the context of stiff reaction fronts. The impact of the stiffness featured by these fronts on the convergence of the method is thus quantified, and allows to conclude on an optimal strategy for the resolution of such problems. We finally perform some numerical simulations in the field of nonlinear chemical dynamics that validate the theoretical estimates and examine the performance of such strategies in the context of academical one-dimensional test cases as well as multi-dimensional configurations simulated on parallel architecture.

[1]  Irving R. Epstein,et al.  An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos , 1998 .

[2]  Yvon Maday,et al.  A Parareal in Time Semi-implicit Approximation of the Navier-Stokes Equations , 2005 .

[3]  Marc Massot,et al.  Operator splitting for nonlinear reaction-diffusion systems with an entropic structure : singular perturbation and order reduction , 2004, Numerische Mathematik.

[4]  L. Shampine,et al.  RKC: an explicit solver for parabolic PDEs , 1998 .

[5]  Christian Lubich,et al.  On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations , 2008, Math. Comput..

[6]  J. Brandts.  [Review of: W. Hundsdorfer, J.G. Verwer (2003) Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations] , 2006 .

[7]  R. Temam.  Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II) , 1969 .

[8]  Dwight Barkley,et al.  Fast Simulations of Waves in Three-Dimensional Excitable Media , 1997 .

[9]  Yves D'Angelo,et al.  Comparison and analysis of some numerical schemes for stiff complex chemistry problems , 1995 .

[10]  Marc Massot,et al.  Simulation of human ischemic stroke in realistic 3D geometry , 2010, Commun. Nonlinear Sci. Numer. Simul..

[11]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[12]  Marc Massot,et al.  New Resolution Strategy for Multiscale Reaction Waves using Time Operator Splitting, Space Adaptive Multiresolution, and Dedicated High Order Implicit/Explicit Time Integrators , 2012, SIAM J. Sci. Comput..

[13]  Yvon Maday,et al.  The Parareal in Time Iterative Solver: a Further Direction to Parallel Implementation , 2005 .

[14]  P. Deuflhard.  A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting , 1974 .

[15]  N. N. I︠A︡nenko.  The Method of Fractional Steps: The Solution of Problems of Mathematical Physics in Several Variables , 1971 .

[16]  R. Temam.  Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (I) , 1969 .

[17]  Y. Maday,et al.  A parareal in time procedure for the control of partial differential equations , 2002 .

[18]  Magne S. Espedal,et al.  A Convergent Algorithm for Time Parallelization Applied to Reservoir Simulation , 2005 .

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

[20]  Gilbert Strang,et al.  Accurate partial difference methods , 1964 .

[21]  Stéphane Descombes,et al.  A numerical study of the blocking of migraine by Rolando sulcus. , 2008, Progress in biophysics and molecular biology.

[22]  Suleyman A. Gokoglu,et al.  Significance of vapor phase chemical reactions on CVD rates predicted by chemically frozen and local thermochemical equilibrium boundary layer theories , 1988 .

[23]  A. Volpert,et al.  Traveling Wave Solutions of Parabolic Systems: Translations of Mathematical Monographs , 1994 .

[24]  Habib N. Najm,et al.  Modeling Low Mach Number Reacting Flow with Detailed Chemistry and Transport , 2005, J. Sci. Comput..

[25]  Pu Sun.  A Pseudo-Non-Time-Splitting Method in Air Quality Modeling , 1996 .

[26]  Charalambos Makridakis,et al.  Implicit-explicit multistep finite element methods for nonlinear parabolic problems , 1998, Math. Comput..

[27]  M S Day,et al.  Numerical simulation of laminar reacting flows with complex chemistry , 2000 .

[28]  Vitaly Volpert,et al.  Traveling Wave Solutions of Parabolic Systems , 1994 .

[29]  S. SIAMJ.,et al.  FOURTH ORDER CHEBYSHEV METHODS WITH RECURRENCE RELATION∗ , 2002 .

[30]  Y. Maday,et al.  A “Parareal” Time Discretization for Non-Linear PDE’s with Application to the Pricing of an American Put , 2002 .

[31]  M. Smooke,et al.  Error estimate for the modified Newton method with applications to the solution of nonlinear, two-point boundary-value problems , 1983 .

[32]  G. Marchuk.  Splitting and alternating direction methods , 1990 .

[33]  Izaskun Garrido Hernandez,et al.  Convergent iterative schemes for time parallelization , 2006, Math. Comput..

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

[35]  E. Hairer,et al.  Solving Ordinary Differential Equations I , 1987 .

[36]  J. Verwer,et al.  Numerical solution of time-dependent advection-diffusion-reaction equations , 2003 .

[37]  Stéphane Descombes,et al.  Strang's formula for holomorphic semi-groups , 2002 .

[38]  M. Markus,et al.  On-off intermittency and intermingledlike basins in a granular medium. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[40]  Bernard Philippe,et al.  A parallel shooting technique for solving dissipative ODE's , 1993, Computing.

[41]  L. Reichl,et al.  An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos By Irving R. Epstein (Brandeis University) and John A. Pojman (University of S. Mississippi). Oxford University Press: New York. 1998. 408 pp. $75.00. ISBN 0-19-509670-3. , 2000 .

[42]  Willem Hundsdorfer,et al.  RKC time-stepping for advection-diffusion-reaction problems , 2004 .

[43]  Bruno Sportisse,et al.  Reduction of chemical kinectics in air pollution modeling , 2000 .

[44]  Jan G. Verwer,et al.  An Implicit-Explicit Runge-Kutta-Chebyshev Scheme for Diffusion-Reaction Equations , 2004, SIAM J. Sci. Comput..

[45]  Lawrence F. Shampine,et al.  IRKC: an IMEX solver for stiff diffusion-reaction PDEs , 2005 .

[46]  I. Epstein,et al.  An Introduction to Nonlinear Chemical Dynamics , 1998 .

[47]  Stéphane Descombes,et al.  On the local and global errors of splitting approximations of reaction–diffusion equations with high spatial gradients , 2007, Int. J. Comput. Math..

[48]  Stéphane Descombes,et al.  Convergence of a splitting method of high order for reaction-diffusion systems , 2001, Math. Comput..

[49]  P. S. Wyckoff,et al.  A Semi-implicit Numerical Scheme for Reacting Flow , 1998 .

[50]  Bruno Sportisse.  Contribution a la modelisation des ecoulements reactifs : reduction des modeles de cinetique chimique et simulation de la pollution atmospherique , 1999 .

[51]  J. Verwer,et al.  A note on operator splitting in a stiff linear case , 1998 .

[52]  Seog Yeon Cho,et al.  Computation accuracy and efficiency of the time-splitting method in solving atmospheric transport/chemistry equations , 1997 .

[53]  Thierry Dumont,et al.  Numerical simulation of a stroke: computational problems and methodology. , 2008, Progress in biophysics and molecular biology.

[54]  N. N. Yanenko,et al.  The Method of Fractional Steps , 1971 .

[55]  B. Sportisse.  An Analysis of Operator Splitting Techniques in the Stiff Case , 2000 .

[56]  Charbel Farhat,et al.  Time‐decomposed parallel time‐integrators: theory and feasibility studies for fluid, structure, and fluid–structure applications , 2003 .

[57]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[58]  Marc Massot,et al.  Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure , 2002 .

[59]  Tarek Echekki,et al.  Topical review: Multiscale methods in turbulent combustion: strategies and computational challenges , 2009 .

[60]  Einar M. Rønquist,et al.  Stability of the Parareal Algorithm , 2005 .

[61]  Yves d'Angelo.  Analyse et simulation numérique de phénomènes liés à la combustion supersonique , 1994 .

[62]  Y Maday,et al.  Parallel-in-time molecular-dynamics simulations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[63]  William R. Goodin,et al.  Numerical solution of the atmospheric diffusion equation for chemically reacting flows , 1982 .

[64]  W. Skaggs,et al.  Chemical vortex dynamics in the Belousov-Zhabotinskii reaction and in the two-variable oregonator model , 1989 .

[65]  Michelle Schatzman,et al.  Toward Non Commutative Numerical Analysis: High Order Integration in Time , 2002, J. Sci. Comput..

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

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

[68]  D. Barkley.  A model for fast computer simulation of waves in excitable media , 1991 .

[69]  Peter Deuflhard,et al.  Newton Methods for Nonlinear Problems , 2004 .

[70]  V. Giovangigli.  Multicomponent flow modeling , 1999 .

[71]  G. Strang.  Accurate partial difference methods I: Linear cauchy problems , 1963 .

[72]  P. Deuflhard.  Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms , 2011 .

[73]  Habib N. Najm,et al.  Regular Article: A Semi-implicit Numerical Scheme for Reacting Flow , 1999 .

[74]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.