A model for fast computer simulation of waves in excitable media
Selcted papers from a meeting on Waves and pattern in chemical and biological media
Global existence of weak solutions for interface equations coupled with diffusion equations
SIAM Journal on Mathematical Analysis
Multiresolution Schemes for the Numerical Solution of 2-D Conservation Laws I
SIAM Journal on Scientific Computing
Mathematical physiology
Bifurcation to spiral waves in reaction-diffusion systems
SIAM Journal on Mathematical Analysis
Point Value Multiscale Algorithms for 2D Compressible Flows
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
Partitioning methods for reaction-diffusion problems
Applied Numerical Mathematics
Fully Adaptive Multiscale Schemes for Conservation Laws Employing Locally Varying Time Stepping
Journal of Scientific Computing
An adaptive multiresolution scheme with local time stepping for evolutionary PDEs
Journal of Computational Physics
Pseudospectral method of solution of the Fitzhugh-Nagumo equation
Mathematics and Computers in Simulation
Applied Numerical Mathematics
Pointwise nonlinear scaling for reaction--diffusion equations
Applied Numerical Mathematics
Numerical simulation of stochastic PDEs for excitable media
Journal of Computational and Applied Mathematics
An adaptive multiresolution method on dyadic grids: Application to transport equations
Journal of Computational and Applied Mathematics
Finite volume element approximation of an inhomogeneous Brusselator model with cross-diffusion
Journal of Computational Physics
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We present fully adaptive multiresolution methods for a class of spatially two-dimensional reaction-diffusion systems which describe excitable media and often give rise to the formation of spiral waves. A novel model ingredient is a strongly degenerate diffusion term that controls the degree of spatial coherence and serves as a mechanism for obtaining sharper wave fronts. The multiresolution method is formulated on the basis of two alternative reference schemes, namely a classical finite volume method, and Barkley's approach (Barkley in Phys. D 49:61---70, 1991), which consists in separating the computation of the nonlinear reaction terms from that of the piecewise linear diffusion. The proposed methods are enhanced with local time stepping to attain local adaptivity both in space and time. The computational efficiency and the numerical precision of our methods are assessed. Results illustrate that the fully adaptive methods provide stable approximations and substantial savings in memory storage and CPU time while preserving the accuracy of the discretizations on the corresponding finest uniform grid.