Stable rotational symmetric schemes for nonlinear reaction-diffusion equations
Stable rotational symmetric schemes for nonlinear reaction-diffusion equations
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非线性反应扩散方程的稳定旋转对称格式
DOI：
10.1016/j.camwa.2022.01.026
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发表时间：
2022
影响因子：
2.9
通讯作者：
Kim, Seongjai
中科院分区：
文献类型：
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作者：
Lee, Philku;Popescu, George V.;Kim, Seongjai
在生物图案形成的模拟中，观察到数值解对空间网格分辨率比时间网格分辨率更敏感。这种对空间分辨率的较高灵敏度主要源于扩散微分算子的不精确近似，这可能会破坏旋转对称性，在低空间分辨率下是严重错误的。此外，众所周知，当初始数据或源项不光滑且时间步长设置得相对较大时，二阶Crank-Nicolson时间步长过程可能会引入虚假振荡。本文研究了增强旋转对称性的扩散算子的9点有限差分格式，利用Variable-θ方法实现了无振荡的二阶时间推进过程，并采用了一个有效的松弛线性求解器来高效地求解代数方程组。证明了Variable-θ方法满足最大值原理，保证了时步过程的无条件稳定。当代数求解器采用具有最优松弛参数的逐次超松弛方法时，迭代在大多数时间步长内以2-4次迭代收敛。整个算法在精度上是二阶的，在效率上是可扩展的。给出了各种算例，说明了该算法对非线性反应扩散方程组数值解的精度和效率。
In the simulation of biological pattern forming, it has been observed that the numerical solution is more sensitive to the spatial mesh resolution than the temporal one. Such a higher sensitivity to the spatial resolution is mainly originated from an inaccurate approximation of diffusion differential operators, which might violate the rotational symmetry to be seriously erroneous in low spatial resolutions. Also, it has been known that the second-order Crank-Nicolson time-stepping procedure may introduce spurious oscillations when the initial data or the source term is nonsmooth and the temporal step size is set relatively large. This article studies 9-point finite difference schemes for the diffusion operator to enhance the rotational symmetry, employs the variable-θmethod to achieve a nonoscillatory second-order time-stepping procedure, and adopts an effective relaxation linear solver to solve the algebraic systems efficiently. The variable-θmethod is proved to satisfy the maximum principle, which guarantees that the time-stepping procedure is unconditionally stable. When the successive over-relaxation method with an optimal relaxation parameter is adopted for the algebraic solver, the iteration converges in 2-4 iterations in most time steps. The overall algorithm is second-order in accuracy and scalable in efficiency. Various examples are given to show the accuracy and efficiency of the proposed algorithm for the numerical solution of the system of nonlinear reaction-diffusion equations.
耦合 Burgers 方程的外推 ADI Crank-Nicolson 正交样条配置
影响因子：
1.1
作者：
Nick Fisher;B. Bialecki
通讯作者：
B. Bialecki
反应扩散方程的非振荡二阶时间步进过程
影响因子：
2.3
作者：
Lee, Philku;Popescu, George V.;Kim, Seongjai
通讯作者：
Kim, Seongjai