Adaptive sparse grid discontinuous Galerkin method: review and software implementation

TL;DR

This paper reviews the adaptive sparse grid discontinuous Galerkin method for high-dimensional PDEs, discusses its software implementation, and demonstrates its efficiency through benchmark tests on various equations.

Contribution

It provides a comprehensive review of the aSG-DG method and introduces the AdaM-DG software package for high-dimensional PDEs with detailed implementation insights.

Findings

The AdaM-DG package efficiently solves high-dimensional linear and nonlinear PDEs.

Numerical tests show accurate error control and reasonable CPU costs.

The software successfully handles complex equations like Hamilton-Jacobi and wave equations.

Abstract

This paper reviews the adaptive sparse grid discontinuous Galerkin (aSG-DG) method for computing high dimensional partial differential equations (PDEs) and its software implementation. The C\texttt{++} software package called AdaM-DG, implementing the aSG-DG method, is available on Github at \url{https://github.com/JuntaoHuang/adaptive-multiresolution-DG}. The package is capable of treating a large class of high dimensional linear and nonlinear PDEs. We review the essential components of the algorithm and the functionality of the software, including the multiwavelets used, assembling of bilinear operators, fast matrix-vector product for data with hierarchical structures. We further demonstrate the performance of the package by reporting numerical error and CPU cost for several benchmark test, including linear transport equations, wave equations and Hamilton-Jacobi equations.