ABSTRACT

In this article, we discuss a fourth-order accurate scheme based on non-polynomial splines in tension approximations for solving quasi-linear parabolic partial differential equations (PDEs). The proposed numerical method requires only two half-step points and a central point on a uniform mesh in spatial direction. This method is derived directly from the continuity condition for the first-order derivative of the non-polynomial tension spline function. The stability of the scheme is discussed using a model linear PDE. The method is applicable for solving singular parabolic problems in polar systems. The proposed method is tested on the generalized Burgers–Huxley equation, generalized Burgers–Fisher equation, and Burgers’ equations in polar coordinates.

中文翻译：

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一维拟线性抛物方程基于半步非多项式样条近似的四阶数值格式

摘要

在本文中，我们讨论一种基于非多项式样条的四阶精确格式，用于求解近似线性抛物型偏微分方程（PDE）。所提出的数值方法仅需要在空间方向上均匀网格上的两个半步点和一个中心点。该方法直接从非多项式张力样条函数的一阶导数的连续性条件导出。使用模型线性PDE讨论了该方案的稳定性。该方法适用于求解极性系统中的奇异抛物线问题。所提出的方法在极坐标下的广义Burgers-Huxley方程，广义Burgers-Fisher方程和Burgers方程中进行了测试。