求解Klein-Gordon-Zakharov方程的一个线性化差分格式(英文)

A Linearized Difference Scheme for the Klein-Gordon-Zakharov Equation

本文针对Klein-Gordon-Zakharov方程运用Crank-Nicolson格式和蛙跳格式的构造方法分别对线性项和非线性项进行离散,得到一个新的半显式有限差分格式.该格式在实际计算中是线性化解耦的,即在具体计算中的每一时间步,只需求解两个独立的三对角线性代数方程组,从而可以大幅提高计算效率.由于难以得到数值解的最大模先验估计,本文引入数学归纳方法并利标准的能量方法和不动点定理得到了数值解的存在唯一性,并证明了格式在时间和空间两个方向的二阶收敛性.数值结果验证了格式的精度和稳定性.

This paper presents a semi-explicit finite difference scheme for solving the the Klein-Gordon-Zakharov （KGZ） equation by the Crank-Nicolson/leap frog discretization on the linear/nonlinear terms. The new scheme is linearized and decoupled in the practical computation. Actually, at each time step, just only two independent tridiagonal systems of linear algebraic equations need to be solved. To obtain the a priori estimate of the numerical solution, we apply an induction argument, the energy method and the fixed point theorem to prove the unique existence of the nu-merical solution and second order convergence in both the time and space direction. Numerical results indicate the accuracy and the stability of the proposed scheme.