文章基本信息

标题：Stable, Explicit, Leapfrog-Hopscotch Algorithms for the Diffusion Equation
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作者：Ádám Nagy ; Issa Omle ; Humam Kareem 等
期刊名称：Computation
电子版ISSN：2079-3197
出版年度：2021
卷号：9
期号：8
页码：92
DOI：10.3390/computation9080092
语种：English
出版社：MDPI Publishing
摘要：In this paper, we construct novel numerical algorithms to solve the heat or diffusion equation. We start with 10<sup>5</sup> different leapfrog-hopscotch algorithm combinations and narrow this selection down to five during subsequent tests. We demonstrate the performance of these top five methods in the case of large systems with random parameters and discontinuous initial conditions, by comparing them with other methods. We verify the methods by reproducing an analytical solution using a non-equidistant mesh. Then, we construct a new nontrivial analytical solution containing the Kummer functions for the heat equation with time-dependent coefficients, and also reproduce this solution. The new methods are then applied to the nonlinear Fisher equation. Finally, we analytically prove that the order of accuracy of the methods is two, and present evidence that they are unconditionally stable.