Adaptive Time-Space Algorithms for the Simulation of Multi-scale Reaction Waves

We present a new resolution strategy for multi-scale reaction waves based on adaptive time operator splitting and space adaptive multiresolution, in the context of localized and stiff reaction fronts. The main goal is to perform computationally efficient simulations of the dynamics of multi-scale phenomena under study, considering large simulation domains with conventional computing resources. We aim at time-space accuracy control of the solution and splitting time steps purely dictated by the physics of the phenomenon and not by stability constraints associated with mesh size or source time scales. Numerical illustrations are provided for 2D and 3D combustion applications modeled by reaction-convection-diffusion equations.

[1]  Marc Massot,et al.  New Resolution Strategy for Multi-scale Reaction Waves using Time Operator Splitting and Space Adaptive Multiresolution: Application to Human Ischemic Stroke , 2011 .

[2]  Sébastien Candel,et al.  Computation of diffusion and premixed flames rolled up in vortex structures , 1989 .

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

[4]  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..

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

[6]  E. Hairer,et al.  Stiff and differential-algebraic problems , 1991 .

[7]  Marc Massot,et al.  A new numerical strategy with space-time adaptivity and error control for multi-scale streamer discharge simulations , 2011, J. Comput. Phys..

[8]  Marc Massot,et al.  Time–space adaptive numerical methods for the simulation of combustion fronts , 2013 .

[9]  Marc Massot,et al.  New Resolution Strategies for Multi-scale Reaction Waves: Optimal Time Operator Splitting and Space Adaptive Multiresolution , 2011, CLEI Electron. J..

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

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

[12]  Assyr Abdulle,et al.  Fourth Order Chebyshev Methods with Recurrence Relation , 2001, SIAM J. Sci. Comput..

[13]  Christian Tenaud,et al.  High order one-step monotonicity-preserving schemes for unsteady compressible flow calculations , 2004 .

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

[15]  Marc Massot,et al.  Adaptive time splitting method for multi-scale evolutionary partial differential equations , 2011, 1104.3697.

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

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

[18]  Marc Massot,et al.  Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies , 2011 .

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

[20]  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..