New Resolution Strategy for Multiscale Reaction Waves using Time Operator Splitting, Space Adaptive Multiresolution, and Dedicated High Order Implicit/Explicit Time Integrators

Abstract

Abstract. We tackle the numerical simulation of reaction-diffusion equations modeling multiscale reaction waves. This type of problems induces peculiar difficulties 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 steep spatial gradients in the reaction fronts, spatially very localized. In this paper, we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts. It considers a high order implicit time integration of the reaction and an explicit one for the diffusion term in order to build a time operator splitting scheme that exploits efficiently the special features of each problem. Based on recent theoretical studies of numerical analysis such a strategy leads to a splitting time step which is not restricted neither by the fastest scales in the source term nor by stability constraints of the diffusive steps, but only by the physics of the phenomenon. We aim thus at solving complete models including all time and space scales within a prescribed accuracy, considering large simulation domains with conventional computing resources. The efficiency is evaluated through the numerical simulation of configurations which were so far, out of reach of standard methods in the field of nonlinear chemical dynamics for 2D spiral waves and 3D scroll waves as an illustration. Future extensions of the proposed strategy to more complex configurations involving other physical phenomena as well as optimization capability on new computer architectures are finally discussed. Key words.Reaction-diffusion equations, multi-scale reaction waves, operator splitting, adaptive multiresolution AMS subject classifications. 33K57, 35A18, 65M50, 65M081. Introduction. Numerical simulations of multi-scale phenomena are commonly used for modeling purposes in many applications such as combustion, chemical vapor deposition, or air pollution modeling. In general, all these models raise several difficulties created by the high number of unknowns, the wide range of temporal scales due to large and detailed chemical kinetic mechanisms, as well as steep spatial gradients associated with very localized fronts of high chemical activity. Furthermore, a natural stumbling block to perform 3D simulations with all scales resolution is either the unreasonably small time step due to stability requirements or the unreasonable memory requirements for implicit methods. In this context, one can consider various numerical strategies in order to treat the induced stiffness for time dependent * This research was supported by a fundamental project grant from ANR (French National Research Agency -ANR Blancs) Séchelles (project leader S. Descombes -2009Descombes - -2013, by a CNRS PEPS Maths-ST2I project MIPAC (project leader V. Louvet -2009Louvet - -2010, and by a DIGITEO RTRA project MUSE (project leader M. Massot -2010Massot - -2014 problems. The most natural id...

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

“…A new operator splitting for reaction-diffusion systems was recently introduced [6], which considers a high fifth order, A-stable, L-stable method like Radau5 [8], based on implicit Runge-Kutta schemes for stiff ODEs, that solves with a local cell by cell approach the reaction term: a system of stiff ODEs without spatial coupling. On the other hand, a high fourth order method was chosen, like ROCK4 [1], based on explicit stabilized Runge-Kutta schemes which features extended stability domains along the negative real axis, very appropriate for diffusion problems because of the usual predominance of negative real eigenvalues.…”

“…Moreover, there is no need to represent these quasi-stationary regions with the same spatial discretization needed to describe the reaction front, so that convection and diffusion problems might also be solved over a smaller number of nodes. An adapted mesh obtained by a multiresolution process which discriminates the various space scales of the phenomenon, turns out to be a very convenient solution to overcome these difficulties [6,7].…”

“…Even though, the strategy was developed for the resolution of general multi-scale phenomena in various domains as biomedical applications [7] or nonlinear chemical dynamics [6], we will focus here on multidimensional combustion problems at large Reynolds numbers in order to assess the capability of the method. The paper is organized as follows: section 2 describes briefly the numerical strategy, based on spatial adaptive multiresolution and second order adaptive time integration.…”

Time–space adaptive numerical methods for the simulation of combustion fronts

“…To address these problems, we have first proposed in [50] a new approach in the design of splitting schemes for propagating waves modeled by stiff reaction-diffusion systems, in which the time integration errors were uniquely related to the splitting errors, even for large splitting time scales, based on mathematical analyses carried out mainly in [51,52]. The underlying idea is to decouple time integration errors by choosing one-step and high-order dedicated methods with time-stepping features for the split subproblems, to independently handle and solve the cor-3 responding physical-numerical time spectra, and such that the corresponding numerical errors remain negligible when compared with the splitting ones.…”

“…The global error is then controlled by the splitting time step, defined according to the physical decoupling capabilities of the phenomenon and hence independently of standard stability constraints associated to mesh size or stiff source time scales. Additionally, the splitting scheme was coupled in [50] with a dynamic mesh refinement technique based on multiresolution (MR) analysis [53,54,55], previously restricted to non-stiff applications in the literature. For a given semi-discretized problem, the MR mathematical background allows a better monitoring of numerical errors introduced by the compressed spatial representations with respect to the original uniform grid discretization.…”