Stable, Explicit, Leapfrog-Hopscotch Algorithms for the Diffusion Equation

Computation

Gabriella Bognar¹  Issa Omle²  Ádám Nagy²  Humam Kareem²  Endre Kovács²  Imre Ferenc Barna³ 

[1] Institute of Machine and Product Design, University of Miskolc, 3515 Miskolc, Hungary;Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary;Wigner Research Center for Physics, 1051 Budapest, Hungary;

关键词: odd-even hopscotch method;    diffusion equation;    heat equation;    explicit time-integration;    stiff equations;    unconditional stability;   

DOI  :  10.3390/computation9080092

来源: DOAJ

【 摘 要 】

In this paper, we construct novel numerical algorithms to solve the heat or diffusion equation. We start with 10⁵ 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.

【 授权许可】

Unknown