Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction–diffusion systems

Abstract

MSC: 35K65 35L65 35R05 65M06 76T20 92C17 Keywords: Degenerate parabolic equation Adaptive multiresolution scheme Pattern formation Finite volume schemes Chemotaxis Keller-Segel systems Flame balls interaction Locally varying time steppingSpatially two-dimensional, possibly degenerate reaction-diffusion systems, with a focus on models of combustion, pattern formation and chemotaxis, are solved by a fully adaptive multiresolution scheme. Solutions of these equations exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. This calls for a concentration of computational effort on zones of strong variation. The multiresolution scheme is based on finite volume discretizations with explicit time stepping. The multiresolution representation of the solution is stored in a graded tree ("quadtree"), whose leaves are the non-uniform finite volumes on whose borders the numerical divergence is evaluated. By a thresholding procedure, namely the elimination of leaves with solution values that are smaller than a threshold value, substantial data compression and CPU time reduction is attained. The threshold value is chosen such that the total error of the adaptive scheme is of the same order as that of the reference finite volume scheme. Since chemical reactions involve a large range of temporal scales, but are spatially well localized (especially in the combustion model), a locally varying adaptive time stepping strategy is applied. For scalar equations, this strategy has the advantage that consistence with a CFL condition is always enforced. Numerical experiments with five different scenarios, in part with local time stepping, illustrate the effectiveness of the adaptive multiresolution method. It turns out that local time stepping accelerates the adaptive multiresolution method by a factor of two, while the error remains controlled.

A multiresolution space‐time adaptive scheme for the bidomain model in electrocardiology

“…In [7], the authors prove for scalar, one-dimensional nonlinear conservation laws, that the threshold error is stable in the sense that the constant C is uniformly bounded and, in particular, does not depend on the threshold value ε R , the number of refinement levels L and the number of time steps n. In our case, even when a rigorous proof is still missing for the system considered in the present work, from the previous deduction and our numerical experiments (see Fig. 2 in Section VI) we see a similar behaviour for C. As in previous works [8,[10][11][12], here the reference tolerance ε R remains fixed for all times. It is certainly possible to recompute ε R at each time step, but this will usually mean that one has to perform additional computations to determine the value of C, and it seems unlikely that this procedure makes the scheme more efficient.…”

“…Moreover, as in [10], we may deduce that the explicit version of the FV method used herein, (3.2)-(3.5), is stable under the CFL condition…”

“…For finite volume discretizations of the bidomain and monodomain models, however, an exact rate of convergence has not yet been derived. Nevertheless, in [10][11][12] we demonstrate that also for degenerate parabolic equations and reaction-diffusion systems, an equation of the type (4.12) may be employed to determine ε R . We apply the same methodology to determine a reference tolerance for the problem at hand.…”

“…In the context of fully adaptive MR methods [8], the mathematical analysis is complete only in the case of a scalar conservation law, but in practice, these techniques have been used by several groups (see e.g., [7,9,10,23,26]) to successfully solve a wide class of problems, including applications to multidimensional systems. These results illustrate that the MR method has turned out to be a useful device for a series of problems with a similar structure to cardiac problems.…”

“…This deduction motivated by the rigorous analysis performed in [8] for scalar conservation laws in one space dimension and in [9] for strictly parabolic PDEs. Experience with degenerate parabolic equations and reaction-diffusion systems [9][10][11][12] suggests that the MR method should provide an efficient tool for solving the bidomain equations. Consequently, we herein extend this method to the novel application of the bidomain and monodomain models.…”