基于快速正弦离散变换的有限差分方法求解半线性抛物型方程

Finite Difference Method for Solving Semi Linear Parabolic Equations Based on Fast Discrete Sine Transform

摘要: 针对带有Dirichlet边界条件的二维半线性抛物方程给出二阶中心差分格式，利用Kronecker积写出二维拉普拉斯算子的微分矩阵。进而应用Crank-Nicolson方法进行时间离散，采用Picard迭代求解离散得到的非线性代数方程组。具体实现过程中结合快速离散正弦变换，本文方法的优点是减少了存储量并大幅度降低了计算时间。数值算例验证本文的方法可以更好地捕捉解的爆破现象。

Abstract: In this paper, the second-order finite difference scheme is applied for the numerical solution of the two-dimensional semi-linear parabolic equations with Dirichlet boundary conditions. We construct the differentiation matrix of two-dimensional Laplacian operator by Kronecker product. The time discretization method is chosen as Crank-Nicolson method. In every time level we solve the nonlinear algebraic equations by Picard iteration method. The fast discrete Sine transform is applied in the process of implementation. The major feature of the proposed method is that the memory requirement and CPU time are reduced obviously. Numerical examples show that the proposed method can better capture the blow up phenomenon of the solution.