Explicit Runge-Kutta methods for parabolic partial differential equations
Applied Numerical Mathematics - Special issue celebrating the centenary of Runge-Kutta methods
Multiresolution Schemes for the Numerical Solution of 2-D Conservation Laws I
SIAM Journal on Scientific Computing
An analysis of operator splitting techniques in the stiff case
Journal of Computational Physics
Fourth Order Chebyshev Methods with Recurrence Relation
SIAM Journal on Scientific Computing
Fully adaptive multiresolution finite volume schemes for conservation laws
Mathematics of Computation
A conservative fully adaptive multiresolution algorithm for parabolic PDEs
Journal of Computational Physics
RKC time-stepping for advection-diffusion-reaction problems
Journal of Computational Physics
Modeling Low Mach Number Reacting Flow with Detailed Chemistry and Transport
Journal of Scientific Computing
IRKC: an IMEX solver for stiff diffusion-reaction PDEs
Journal of Computational and Applied Mathematics
International Journal of Computer Mathematics - Splitting Methods for Differential Equations
Journal of Computational Physics
Journal of Computational Physics
Dynamic adaptive chemistry with operator splitting schemes for reactive flow simulations
Journal of Computational Physics
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We tackle the numerical simulation of reaction-diffusion equations modeling multi-scale reaction waves. This type of problem 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. The paper 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 restricted by neither 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 two-dimensional spiral waves and three-dimensional 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 discussed.