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++software package called AdaM-D...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++software package called AdaM-DG,implementing the aSG-DG method,is available on GitHub at 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 the numerical error and the CPU cost for several benchmark tests,including linear transport equations,wave equations,and Hamilton-Jacobi(HJ)equations.展开更多
This paper, using the theory of exponential dichotomy and fixed point theorem, discusses the existence of periodic solution for a class of neutral higher dimensional differential system with delay, the result extends ...This paper, using the theory of exponential dichotomy and fixed point theorem, discusses the existence of periodic solution for a class of neutral higher dimensional differential system with delay, the result extends and improves the result of paper [6].展开更多
基金supported by the NSF grant DMS-2111383Air Force Office of Scientific Research FA9550-18-1-0257the NSF grant DMS-2011838.
文摘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++software package called AdaM-DG,implementing the aSG-DG method,is available on GitHub at 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 the numerical error and the CPU cost for several benchmark tests,including linear transport equations,wave equations,and Hamilton-Jacobi(HJ)equations.
基金This project is supported by the National Natural Science Foundation of China under grant 10671012.
文摘This paper, using the theory of exponential dichotomy and fixed point theorem, discusses the existence of periodic solution for a class of neutral higher dimensional differential system with delay, the result extends and improves the result of paper [6].
