This paper reports on a novel explicit numerical method for the spatially discretized diffusion or heat equation. After discretizing the space variables as in conventional finite difference methods, this method does not use a finite difference approximation for the time derivatives, it instead combines constant-neighbor and linear-neighbor approximations, which decouple the ordinary differential equations, thus they can be solved analytically. In the obtained three-stage method, the time step size appears in exponential form with negative coefficients in the final expression. This property guarantees unconditional stability, as it is shown using von Neumann stability analysis. It is also proved that the convergence of the method is third order in the time step size. After verification, by solving Fisher's and Huxley's equations, it is demonstrated that it works for nonlinear equations as well. The new algorithm is tested against widely used numerical solvers for cases where the media is strongly inhomogeneous. According to the results, the new method is significantly more effective than the traditional explicit or implicit methods, especially for extremely large stiff systems. It is believed that this new method is unique in the sense that it is the first unconditionally stable explicit method with third-order convergence.

中文翻译：

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扩散型问题的一种新的稳定、显式、三阶方法

本文报告了空间离散扩散或热方程的一种新的显式数值方法。该方法在像传统有限差分法那样对空间变量进行离散化后，对时间导数不使用有限差分逼近，而是结合常邻逼近和线性邻邻逼近，将常微分方程解耦，从而求解分析地。在得到的三阶段方法中，时间步长以指数形式出现，最终表达式中的系数为负。该属性保证了无条件的稳定性，正如使用冯诺依曼稳定性分析所显示的那样。还证明了该方法在时间步长上的收敛性是三阶的。经过验证，通过求解Fisher's and Huxley's equations，证明它也适用于非线性方程。新算法针对广泛使用的数值求解器进行了测试，适用于介质非常不均匀的情况。结果表明，新方法明显比传统的显式或隐式方法更有效，特别是对于非常大的刚性系统。相信这种新方法是独一无二的，因为它是第一个具有三阶收敛性的无条件稳定显式方法。