An Exploration Of High Order Difference Schemes For Differential Equations And Eigenvalue Problems

In the thesis, we adopt a fast algorithm to define the coefficients of difference stencil, which is used to produce the high-order difference schemes for the numerical solution of PDEs and eigenvalue problems. For the 3D problems in the thesis, we get the high-order difference scheme in terms of approximating the Laplace operator by 1D long-stencil difference independently in every direction, and solve the Poisson equation, Poisson-Boltzmann equation and eigenvalue problems. A parallel MINRES algorithm with the local ILU(0) factorization as a preconditioner is used successfully to solve the Poisson-Boltzmann equation, and get the high parallel efficiency. high performance of high-order difference schemes is shown in numerical experiments, and the long-stencil difference schemes could attach the very high accuracy compared with compact difference schemes.