二维退化反应扩散系统的自适应多分辨率方案与局部时间步进
Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction-diffusion systems
AI总结：
针对二维退化反应扩散系统，提出一种基于有限体积离散和分级树结构的全自适应多分辨率方案，通过最优阈值压缩和局部时间步进加速计算并控制误差。
摘要：
我们提出了一种全自适应多分辨率方案，用于空间二维的、可能退化的反应扩散系统，重点关注燃烧模型以及数学生物学中的模式形成和趋化性模型。在这些应用中，这些方程的解表现出陡峭的梯度，在退化情况下，则表现为尖锐的前沿和不连续性。该多分辨率方案基于具有显式时间步进的有限体积离散化。解的多元分辨率表示存储在一个分级树中。通过阈值化过程，即消除小于阈值的小叶子，实现了显著的数据压缩和CPU时间减少。阈值的选择是最优的，即自适应方案的总误差与参考有限体积方案的误差斜率相同。由于化学反应涉及大范围的时间尺度，但在空间上局部化良好（特别是在燃烧模型中），因此采用了局部变化的自适应时间步进策略。结果表明，局部时间步进将自适应多分辨率方法的速度提高了两倍，同时误差保持可控。
英文摘要：
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. 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. 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. It turns out that local time stepping accelerates the adaptive multiresolution method by a factor of two, while the error remains controlled.
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