陳宜良臺灣大學：數學研究所林煜鈞Lin, Yu-ChunYu-ChunLin2007-11-282018-06-282007-11-282018-06-282007http://ntur.lib.ntu.edu.tw//handle/246246/59431We present a first order accurate method for solving the partial differential equation, Poisson-Boltzmann Equation where the coefficients is assumed to be discontinuous across an interface and the source term is allowed to be a delta function. In one-dimension, we take a finite differential approach. Near the discontinuities, the unknown is approximated by a piecewise function. We extend it in two dimensions by taking a dimension-by-dimension in discretization. The underlying grid is regular. We also present two efficient iterative solvers; the algebraic multigrid method to solve the resulting linear system, and the Newton’s method to solve the corresponding nonlinear equations. The main point of this article is to propose an initialization based on geometric multigrid method to reduce number of Newton’s iterations. We show by numerical experiments that total CPU time is nearly proportional to the number of unknowns.Abstract . 1 I.Introduction………………………………………………2 I.1 About the Poisson-Boltzmann Equation……..…..2 I.2 Fast Algorithm for Solving Poisson-Boltzmann Equation…….….…................................4 II.Approach for solving PBE……………………….……5 II.1.Multigrid method …….……….……………………5 II.2 Algebraic Multigrid Method……………………….7 II.3 Newton method………….……………………………10 III.Multigrid Method for Solving the PBE with discontinuous coefficients……………………………12 III.1.Multigrid Method for Solving the Poisson Boltzmann Equation with Discontinuous Coefficients in One-Dimension…………............................….12 III.2.Multigrid Method for Solving the Poisson Boltzmann Equation with Discontinuous Coefficients in Two-Dimension……..………...........................17 III.3.Operator Review……………………….…………21 IV.Numerical Experiments………………………………22 IV.1 One-dimensional examples……….………..…..22 IV.2 Two-dimensional examples……………………….29 V.Conclusion……………………………………………..38 VI.References…………………………………………….391913716 bytesapplication/pdfen-US多重網格法波瓦松-波茲曼方程algebraic multigrid methodnewtonPoission-Boltzmann equationthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/59431/1/ntu-96-R93221035-1.pdf