一维空间强退化抛物方程的完全自适应多分辨率格式
Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension
AI总结：
针对一维强退化抛物方程，提出基于Engquist-Osher通量和显式时间步进的有限体积自适应多分辨率格式，通过L1收缩优化阈值，提高计算效率并保持收敛性。
摘要：
我们提出了一种完全自适应的多分辨率格式，用于空间一维拟线性强退化抛物方程，具有零通量和周期边界条件。该数值格式基于使用Engquist-Osher数值通量和显式时间步进的有限体积离散化。然后使用基于单元平均的自适应多分辨率格式来加速底层有限体积格式的CPU时间和内存需求，其一阶版本已知收敛到问题的熵解。该方法的一个特点是解的多次分辨率表示存储在一个分级树中，其叶子是最终评估数值散度的非均匀有限体积。此外，利用离散时间演化算子的$L^1$收缩性，我们推导出自适应多分辨率方法中阈值的最优选择。数值例子说明了计算效率以及收敛性质。
英文摘要：
We present a fully adaptive multiresolution scheme for spatially one-dimensional quasilinear strongly degenerate parabolic equations with zero-flux and periodic boundary conditions. The numerical scheme is based on a finite volume discretization using the Engquist-Osher numerical flux and explicit time stepping. An adaptive multiresolution scheme based on cell averages is then used to speed up the CPU time and the memory requirements of the underlying finite volume scheme, whose first-order version is known to converge to an entropy solution of the problem. A particular feature of the method is the storage of the multiresolution representation of the solution in a graded tree, whose leaves are the non-uniform finite volumes on which the numerical divergence is eventually evaluated. Moreover using the $L^1$ contraction of the discrete time evolution operator we derive the optimal choice of the threshold in the adaptive multiresolution method. Numerical examples illustrate the computational efficiency together with the convergence properties.
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