非线性泊松问题的拟线性化边界积分方法研究

A STUDY OF BOUNDARY INTEGRAL METHOD FOR NONLINEAR POISSON PROBLEM BASED ON GENERALIZED QUASILINEARLIZATION THEORY

非线性泊松问题在热传导和多孔催化粒子的扩散反应等问题中是非常常见的,为此,利用广义拟线性化迭代理论,提出了一种非线性泊松问题的新的数值迭代方法.该方法将非线性方程转化成一序列线性方程的迭代,其优点是初始值的选取具有一定的理论基础,并且在一定的初始值条件下,迭代结果将单调地收敛于非线性问题的解.将此迭代方法与边界元和双互易杂交边界点方法结合,并用于非线性泊松问题的求解,比较了两种方法的结果精度,收敛速度及不同初始值下的稳定性.结果显示,基于拟线性化的双互易杂交边界点法具有较高的稳定性和计算效率,并且收敛速度为平方阶.

The nonlinear Poisson problems are very common in heat conduction and diffusion with simultaneous reaction in a porous catalyst particle,so the generalized quasilinearization theory is exploited and a new numerical iterative method is proposed for this type nonlinear Poisson problem.In this method,the non-linear equation is replaced by a set of iterative linear equation.An advantage of this method is that a theory background is substantial for the choice of the initial value of the iteration,and with a wide range of initial value the result of this iteration is monotonously converged to the exact value.This new iterative method is combined with boundary element method and dual reciprocity hybrid boundary node method for solving nonlinear Poisson problems,and the accuracy,the convergence rate and stability with different initial values of these two methods are compared with each other.It is shown that,the method based on dual reciprocity hybrid boundary node method and generalized quasilinearization theory,has the high stability and efficiency, and the iterative rate is quadratic.