【作者】 刘璐璐；

【导师】 高卫国；

【作者基本信息】 复旦大学 ， 计算数学， 2010， 硕士

【摘要】 我们在本文中采用任意形式差分模板系数的快速算法生成高阶差分格式,探索求解偏微分方程和特征值问题。在本文中我们主要求解3D问题,通过在每个方向上独立做一维长格式差分的方案来近似Laplace算子,从而获得3D问题的高阶差分格式,并用于求解Poisson方程,Poisson-Boltzmann方程以及特征值问题。在求解Poisson-Boltzmann方程时,我们成功地采用局部作ILU(0)预条件的并行MINRES算法,获得了很高的并行效率。在数值实验中,我们可以看到长格式的高阶差分具有很好的数值效果,并且可以达到紧格式无法比拟的高精度。更多还原

【Abstract】 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.更多还原