Numerical experiments with the multiresolution scheme for the compressible Euler equations
Journal of Computational Physics
Multiresolution Schemes for the Numerical Solution of 2-D Conservation Laws I
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Point Value Multiscale Algorithms for 2D Compressible Flows
SIAM Journal on Scientific Computing
Exponential attractor for a chemotaxis-growth system of equations
Nonlinear Analysis: Theory, Methods & Applications
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
Fully Adaptive Multiscale Schemes for Conservation Laws Employing Locally Varying Time Stepping
Journal of Scientific Computing
Adaptive multiresolution WENO schemes for multi-species kinematic flow models
Journal of Computational Physics
Solving regularly and singularly perturbed reaction-diffusion equations in three space dimensions
Journal of Computational Physics
An adaptive multiresolution scheme with local time stepping for evolutionary PDEs
Journal of Computational Physics
Adaptive Multiresolution Methods for the Simulation of Waves in Excitable Media
Journal of Scientific Computing
Applied Numerical Mathematics
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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Spatially 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.