Using two off-step points and a central point, we discuss a new two-time-level implicit method of order three based on polynomial cubic spline approximations for the solution of the system of 1D nonlinear parabolic equations on a quasi-variable mesh. The proposed method is derived directly from the consistency condition. The stability analysis for a model problem is discussed. The proposed method is tested to solve the coupled Burgers’ equations and Burgers–Huxley equation to demonstrate the utility of the method. We show that the proposed method enables us to obtain the high-accurate numerical solution for high Reynolds number.

中文翻译：

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一种基于离步三次多项式近似解耦合 Burgers 方程和 Burgers-Huxley 方程的高精度新方法

使用两个离步点和一个中心点，我们讨论了一种新的基于多项式三次样条近似的三阶两时间级隐式方法，用于求解准变量网格上的一维非线性抛物线方程组。所提出的方法直接来自于一致性条件。讨论了模型问题的稳定性分析。测试所提出的方法来求解耦合的 Burgers 方程和 Burgers-Huxley 方程，以证明该方法的实用性。我们表明，所提出的方法使我们能够获得高雷诺数的高精度数值解。