二维半线性扩散反应方程的高精度全隐格式及其多重网格方法

针对二维非定常半线性扩散反应方程,空间导数项采用四阶紧致差分公式离散,时间导数项采用四阶向后Euler公式进行离散,提出一种无条件稳定的高精度五层全隐格式.格式截断误差为O(τ4+τ2h2+h4),即时间和空间均具有四阶精度.对于第一、二、三时间层采用Crank-Nicolson方法进行离散,并采用Richardson外推公式将启动层时间精度外推到四阶.建立适用于该格式的多重网格方法,加快在每个时间层上迭代求解代数方程组的收敛速度,提高计算效率.最后通过数值实验验证格式的精确性和稳定性以及多重网格方法的高效性.

High-order Fully Implicit Scheme and Multigrid Method for Two-dimensional Semilinear Diffusion Reaction Equations

A finite difference method is used for high-order numerical solution of two-dimensional unsteady semilinear diffusion reaction equation. The spatial derivative term is discretized by a fourth-order compact difference formula, and the time derivative term is discretized by a fourth-order backward Euler formula. An unconditionally stable high-order five-level fully implicit scheme is proposed. Truncation error of the scheme is O(τ⁴+τ²h²+h⁴), that is, the time and space have fourth-order accuracy. In calculation of start-up steps, the first, second and third time levels are discretized by Crank-Nicolson method. Richardson extrapolation formula was used to extrapolate startup time accuracy to the fourth-order. A multigrid method based on the scheme is established, which accelerates convergence speed of the algebraic equations on each time level and improves computational efficiency. Finally, accuracy, stability and efficiency of the scheme and multigrid approach are verified with numerical experiments.