We present a class of new explicit and stable numerical algorithms to solve the spatially discretized linear heat or diffusion equation. After discretizing the space and the time variables like conventional finite difference methods, we do not approximate the time derivatives by finite differences, but use constant‐neighbor and linear‐neighbor approximations to decouple the ordinary differential equations and solve them analytically. During this process, the time step size appears not in polynomial, but in exponential form with negative exponents, which guarantees stability. We compare the performance of the new methods with analytical and numerical solutions. According to our results, the methods are first and second order in time and can be much faster than the commonly used explicit or implicit methods, especially in the case of extremely large stiff systems.

中文翻译：

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一类新的稳定，显式方法，用于求解非平稳热方程

我们提出了一类新的显式和稳定的数值算法，以解决空间离散的线性热或扩散方程。在像传统的有限差分法离散化空间和时间变量之后，我们不通过有限差分来近似时间导数，而是使用常数邻域近似和线性邻域近似解耦常微分方程并进行解析求解。在此过程中，时间步长不是以多项式形式出现，而是以带有负指数的指数形式出现，从而保证了稳定性。我们将新方法的性能与解析和数值解决方案进行比较。根据我们的结果，这些方法在时间上处于一阶和二阶，并且比常用的显式或隐式方法快得多，