论文标题
通过能量变异方法，用于多孔培养基方程的第二阶准确数值方案
A second order accurate numerical scheme for the porous medium equation by an energetic variational approach
论文作者
论文摘要
多孔培养基方程（PME）是典型的非线性退化抛物线方程。在最近的一项工作中，已经研究了一种能量变异方法[6]，其中获得了轨迹方程，并开发了一些一阶准确的数值方案和分析。在本文中，我们在时间和空间中构建和分析了二阶准确数值方案。基于凸性分析，建立了独特的可溶性，能量稳定性。此外，我们为提出的数值方案提供了详细的收敛分析。进行了仔细的高阶渐近扩展，并进行了两步误差估计。在更多详细信息中，需要进行粗略的估计来控制离散$ W^{1，\ infty} $ norm中的高度非线性术语，并应用了精制估算来得出最佳错误顺序。也提出了一些数值示例。
The porous medium equation (PME) is a typical nonlinear degenerate parabolic equation. An energetic variational approach has been studied in a recent work [6], in which the trajectory equation is obtained, and a few first order accurate numerical schemes have been developed and analyzed. In this paper, we construct and analyze a second order accurate numerical scheme in both time and space. The unique solvability, energy stability are established, based on the convexity analysis. In addition, we provide a detailed convergence analysis for the proposed numerical scheme. A careful higher order asymptotic expansion is performed and two step error estimates are undertaken. In more details, a rough estimate is needed to control the highly nonlinear term in a discrete $W^{1,\infty}$ norm, and a refined estimate is applied to derive the optimal error order. Some numerical examples are presented as well.
