Adaptive and parallel algorithms for solving partialdifferential equations with variable coefficients on sparse grids

求解稀疏网格上变系数偏微分方程的自适应并行算法

• 批准号: 418669609
• 负责人: Professor Dr. Christoph Pflaum
• 金额: --
• 依托单位: Lehrstuhl für Informatik 10: Systemsimulation
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/418669609?language=en
• 关键词: Adaptive; parallel; algorithms; solving; partialdifferential

Sparse grids are an innovative technique for reducing the computational amount for the numerical solution of partial differential equations. Applications are differential equations on complex domains with reentrant edges and corners or high dimensional problems like the time independent Schrödinger equation. In both cases, accurate numerical solutions are difficult to obtain. In order to apply sparse grids to such differential equations it is important to apply a Ritz-Galerkin discretization. However, such a discretization leads to several algorithmic difficulties in case of variable coefficients. These difficulties do not appear for a new discretization method on sparse grids, which was recently developed. This discretization applies prewavelets and a discretization with semi-orthogonality. By this concept, PDE’s with variable coefficients can efficiently be solved by suitable algorithms. The aim of the project is to continue the development of algorithms for solving PDE’s on sparse grids. In particular algorithms on adaptive sparse grids for variable coefficients and efficient algorithms for the calculation of the stiffness matrix have to be developed. Furthermore, new parallelization concepts are needed, since conventional parallelization concepts cannot be applied to sparse grids. The new algorithms will be implemented and analyzed for suitable applications.

稀疏网格是一种减少偏微分方程数值求解计算量的创新技术，适用于具有重入边角的复杂域上的微分方程或高维问题（例如与时间无关的薛定谔方程）。为了将稀疏网格应用于此类微分方程，应用 Ritz-Galerkin 离散化非常重要。然而，这种离散化在变量的情况下会导致一些算法困难。对于最近开发的稀疏网格上的新离散化方法来说，这些困难不会出现。这种离散化应用预小波和半正交离散化，可以通过适当的算法有效地解决具有可变系数的偏微分方程。该项目的重点是继续开发稀疏网格上求解偏微分方程的算法，特别是用于可变系数的自适应稀疏网格的算法和用于计算的高效算法。此外，由于传统的并行化概念不能应用于稀疏网格，因此需要新的并行化概念来实现和分析合适的应用。