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 Barkleyâeuro?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.