论文标题
关于对流扩散方程的显式有限差异方法的稳定性
On the Stability of Explicit Finite Difference Methods for Advection-Diffusion Equations
论文作者
论文摘要
在本文中，我们研究了线性对流扩散方程（ADE）的明确有限差异化离散稳定性，并在线条方法的背景下具有任意准确性的稳定性。该分析首先着重于通过在空间中离散ADE然后扩展到完全离散的方法来获得的普通微分方程（ODE）系统的稳定性，在该方法中使用了显式runge-kutta方法用于集成ODE系统。特别是，事实证明，如果时间积分器至少是一阶准确的，那么ADE的所有稳定半差异都会产生有条件稳定的完全离散的方法，而如果临时阶太低，则对流方程的高阶空间离散化无法产生稳定的方法。在本文的后半部分，我们将分析扩展到部分耗散波系统，并获得半消化和完全消失的方法的稳定性结果。最后，主要的理论预测得到数值验证。
In this paper we study the stability of explicit finite difference discretizations of linear advection-diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability of the system of ordinary differential equations (ODE) that is obtained by discretizing the ADE in space and then extends to fully discretized methods where explicit Runge-Kutta methods are used for integrating the ODE system. In particular, it is proved that all stable semi-discretization of the ADE gives rise to a conditionally stable fully discretized method if the time-integrator is at least first-order accurate, whereas high-order spatial discretization of the advection equation cannot yield a stable method if the temporal order is too low. In the second half of this paper, we extend the analysis to a partially dissipative wave system and obtain the stability results for both semi-discretized and fully-discretized methods. Finally, the major theoretical predictions are verified numerically.
