论文标题
对流扩散方程的显式欧拉法的最佳收敛速率II：高维情况
Optimal convergence rate of the explicit Euler method for convection-diffusion equations II: high dimensional cases
论文作者
论文摘要
这是对对流扩散方程的时间上显式Euler离散化最佳收敛率的第二部分[Appl。数学。 Lett。 \ textbf {131}（2022）108048]，重点介绍在迪里奇（Dirichlet）或诺伊曼（Neumann）边界条件下的高维线性/非线性案例。根据空间衍生物的时间导数和中心差异化离散化的显式欧拉离散化提出了几种新的校正差异方案。校正方案的先验估计值与恒定对流系数的应用详细说明，当每个方向沿每个方向等于$ 1/6 $的步骤比率时，都证明了最佳收敛速率四。校正后的差异方案基本上改进了{\ rm \ textbf {cfl}}条件和与经典差异方案相比的数值精度。涉及Dirichlet/Neumann边界条件下的二维线性/非线性问题的数值示例，例如Fisher方程，Chafee-Infante方程，汉堡方程和分类，以证明较差异差异方案所声称的良好属性。
This is the second part of study on the optimal convergence rate of the explicit Euler discretization in time for the convection-diffusion equations [Appl. Math. Lett. \textbf{131} (2022) 108048] which focuses on high-dimensional linear/nonlinear cases under Dirichlet or Neumann boundary conditions. Several new corrected difference schemes are proposed based on the explicit Euler discretization in temporal derivative and central difference discretization in spatial derivatives. The priori estimate of the corrected scheme with application to constant convection coefficients is provided at length by the maximum principle and the optimal convergence rate four is proved when the step ratios along each direction equal to $1/6$. The corrected difference schemes have essentially improved {\rm \textbf{CFL}} condition and the numerical accuracy comparing with the classical difference schemes. Numerical examples involving two-/three-dimensional linear/nonlinear problems under Dirichlet/Neumann boundary conditions such as the Fisher equation, the Chafee-Infante equation, the Burgers' equation and classification to name a few substantiate the good properties claimed for the corrected difference scheme.
