Many reaction-diffusion systems in various applications exhibit traveling wave solutions that evolve on multiple spatio-temporal scales. These traveling wave solutions are crucial for understanding the underlying dynamics of the system. In this work, we present sixth-order weighted essentially non-oscillatory (WENO) methods within the finite difference framework to solve reaction-diffusion systems. The WENO method allows us to use fewer grid points and larger time steps compared to classical finite difference methods. Our focus is on solving the reaction-diffusion system for the traveling wave solution with the sharp front. Although the WENO method is popular for hyperbolic conservation laws, especially for problems with discontinuity, it can be adapted for the equations of parabolic type, such as reaction-diffusion systems, to effectively handle sharp wave fronts. Thus, we employed the WENO methods specifically developed for equations of parabolic type. We considered various reaction-diffusion equations, including Fisher's, Zeldovich, Newell-Whitehead-Segel, bistable equations, and the Lotka-Volterra competition-diffusion system, all of which yield traveling wave solutions with sharp wave fronts. Numerical examples in this work demonstrate that the central WENO method is highly more accurate and efficient than the commonly used finite difference method. We also provide an analysis related to the numerical speed of the sharp propagating front in the Newell-Whitehead-Segel equation. The overall results confirm that the central WENO method is highly efficient and is recommended for solving reaction-diffusion equations with sharp wave fronts.

翻译：众多应用中的反应-扩散系统常表现出具有多时空尺度演化特征的行波解。这些行波解对于理解系统底层动力学机制至关重要。本研究在有限差分框架内提出了六阶加权本质无振荡（WENO）方法用于求解反应-扩散系统。相较于经典有限差分方法，WENO方法能够以更少的网格点和更大的时间步长进行计算。我们重点研究具有尖锐波前的行波解在反应-扩散系统中的求解问题。尽管WENO方法在双曲型守恒律（特别是包含间断的问题）中应用广泛，但该方法经过适当改进可适用于抛物型方程（如反应-扩散系统），从而有效处理尖锐波前。为此，我们采用了专门针对抛物型方程开发的WENO方法。本研究考察了多种反应-扩散方程，包括Fisher方程、Zeldovich方程、Newell-Whitehead-Segel方程、双稳态方程以及Lotka-Volterra竞争扩散系统，这些方程均能产生具有尖锐波前的行波解。数值算例表明，中心WENO方法相较于常用有限差分方法具有显著更高的精度和计算效率。我们还针对Newell-Whitehead-Segel方程中尖锐传播波前的数值速度进行了相关分析。总体结果证实，中心WENO方法具有极高效率，是求解含尖锐波前的反应-扩散方程的推荐方法。