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Adaptive Schemes for Reaction-Diffusion Systems
| Content Provider | Semantic Scholar |
|---|---|
| Author | Baier, Ricardo Ruiz |
| Copyright Year | 2008 |
| Abstract | We present a fully adaptive multiresolution scheme for spatially two-dimensional, possibly degenerate reaction-diffusion systems, focusing on combustion models and models of pattern formation and chemotaxis in mathematical biology. Solutions of these equations in these applications exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. This calls for a concentration of computational effort in 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 the borders of which the numerical divergence is evaluated. By a thresholding procedure, namely the elimination of leaves that are smaller than a threshold value, substantial data compression and CPU time reduction is attained. The threshold value is chosen optimally, in the sense that the total error of the adaptive scheme is of the same slope 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. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv.org/pdf/0807.1359v1.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Abnormal degeneration Approximation algorithm CPU (central processing unit of computer system) Cartesian closed category Central processing unit Chemotaxis Computation Converge Courant–Friedrichs–Lewy condition Cure for Lymphoma Foundation Data compression Discretization Excretory function Experiment Finite volume method Gradient Iteration Large Mathematics Multiresolution analysis Nonlinear system Numerical analysis Numerical method Numerical methods for ordinary differential equations Pattern formation Quadtree Small Solutions Steady state Stepping level Thresholding (image processing) Turing benefit |
| Content Type | Text |
| Resource Type | Article |
