Abstract

 
 
 We tackle the numerical simulation of reaction-diffusion equations modeling multi-scale reac- tion 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 large spatial gradients in the reaction fronts, spatially very lo- calized. 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. 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 restric- tive diffusive step stability limits, but only by the physics of the phenomenon. We thus aim at solving accurately complete models including all time and space scales of the phenomenon, 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 are finally discussed.
 
 
Highlights
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
The present work proposes a simple and solid numerical strategy that couples adaptive multiresolution techniques with a new operator splitting strategy for multi-scale reactions waves modeled by stiff reaction-diffusion systems
The splitting time step is chosen on the sole basis of the structure of the continuous system and its decoupling capabilities, but not related to any stability requirement of the numerical methods involved in order to integrate each subsystem, even if strong stiffness is present
Summary
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. It is natural to envision a compromise, since the resolution of the fully coupled problem is most of the time out of reach and the appropriate definition of reduced models is normally difficult to establish In this context, time operator splitting methods have been used for a long time and there exists a large literature showing the efficiency of such methods for evolution problems. It is firstly necessary to decouple numerically the reaction part from the rest of the physical phenomena like convection, diffusion or both, for which there exist dedicated numerical methods These techniques allow a completely independent optimization of the resolution of each subsystem which usually yields lower requirements of computing resources.
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