二维复值Ginzburg-Landau方程的一个高阶紧致ADI差分格式

对二维复值金兹堡朗道(Ginzburg-Landau,GL)方程提出一个基于时间分裂的高阶紧致交替方向隐式有限差分格式。本文通过时间分裂法将GL方程分裂成一个非线性子问题及两个线性子问题,对非线性子问题以及其中一个线性子问题均通过精确积分进行计算,并对另一线性子问题构造紧致交替方向隐式差分格式进行数值计算。实际计算中,在每一时间步,利用追赶法求解一族常系数三对角线性代数方程组,从而使得算法既具有较高精度又拥有较快的计算速度。数值实验表明该算法在时间和空间方向分别具有二阶和四阶精度,并模拟了方程的一些动力学行为。

High-Order Compact Alternating Direction Implicit Scheme for Complex Ginzburg-Landau Equations in Two Dimensions

In the paper, we propose a time-splitting high-order compact alternating direction implicit (ADI) finite difference scheme for two-dimensional complex Ginzburg-Landau (GL) equation. The GL equation is split into a nonlinear sub-problem and two linear sub-problems. The nonlinear sub-problem and one of the linear subproblems are solved exactly. Then a compact alternating direction implicit difference scheme is constructed for another linear subproblem. In practical computation, a family of constant coefficient tri-diagonal linear algebraic equations by using the catch-up method at each time step is solved to make the algorithm get high accuracy and efficiency. Numerical experiments show that the algorithm has second-order and fourth-order accuracy in time and space direction, respectively. And some dynamics behaviors of the equation are simulated.