In this paper, a new analytical method of symplectic system, Hamiltonian system, is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain. In the system, the fundamental problem...In this paper, a new analytical method of symplectic system, Hamiltonian system, is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain. In the system, the fundamental problem is reduced to an eigenvalue and eigensolution problem. The solution and boundary conditions can be expanded by eigensolutions using adjoint relationships of the symplectic ortho-normalization between the eigensolutions. A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space. The results show that fundamental flows can be described by zero eigenvalue eigensolutions, and local effects by nonzero eigenvalue eigensolutions. Numerical examples give various flows in a rectangular domain and show effectiveness of the method for solving a variety of problems. Meanwhile, the method can be used in solving other problems.展开更多
It is well known since 1960s that by exploring the tensor product structure of the discrete Laplacian on Cartesian meshes,one can develop a simple direct Poisson solver with an O(Nd+1/d)complexity in d-dimension,where...It is well known since 1960s that by exploring the tensor product structure of the discrete Laplacian on Cartesian meshes,one can develop a simple direct Poisson solver with an O(Nd+1/d)complexity in d-dimension,where N is the number of the total unknowns.The GPU acceleration of numerically solving PDEs has been explored successfully around fifteen years ago and become more and more popular in the past decade,driven by significant advancement in both hardware and software technologies,especially in the recent few years.We present in this paper a simple but extremely fast MATLAB implementation on a modern GPU,which can be easily reproduced,for solving 3D Poisson type equations using a spectral-element method.In particular,it costs less than one second on a Nvidia A100 for solving a Poisson equation with one billion degree of freedoms.We also present applications of this fast solver to solve a linear(time-independent)Schr¨odinger equation and a nonlinear(time-dependent)Cahn-Hilliard equation.展开更多
摘要In this paper, a new analytical method of symplectic system, Hamiltonian system, is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain. In the system, the fundamental problem is reduced to an eigenvalue and eigensolution problem. The solution and boundary conditions can be expanded by eigensolutions using adjoint relationships of the symplectic ortho-normalization between the eigensolutions. A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space. The results show that fundamental flows can be described by zero eigenvalue eigensolutions, and local effects by nonzero eigenvalue eigensolutions. Numerical examples give various flows in a rectangular domain and show effectiveness of the method for solving a variety of problems. Meanwhile, the method can be used in solving other problems.
基金supported in part by NSFC 12371409supported by NSF DMS-220815。
摘要It is well known since 1960s that by exploring the tensor product structure of the discrete Laplacian on Cartesian meshes,one can develop a simple direct Poisson solver with an O(Nd+1/d)complexity in d-dimension,where N is the number of the total unknowns.The GPU acceleration of numerically solving PDEs has been explored successfully around fifteen years ago and become more and more popular in the past decade,driven by significant advancement in both hardware and software technologies,especially in the recent few years.We present in this paper a simple but extremely fast MATLAB implementation on a modern GPU,which can be easily reproduced,for solving 3D Poisson type equations using a spectral-element method.In particular,it costs less than one second on a Nvidia A100 for solving a Poisson equation with one billion degree of freedoms.We also present applications of this fast solver to solve a linear(time-independent)Schr¨odinger equation and a nonlinear(time-dependent)Cahn-Hilliard equation.