Abstract
By the iteration of the theta-formula and treating the neighbors explicitly such as the unconditionally positive finite difference (UPFD) methods, we construct a new 2-stage explicit algorithm to solve partial differential equations containing a diffusion term and two reaction terms. One of the reaction terms is linear, which may describe heat convection, the other one is proportional to the fourth power of the variable, which can represent radiation. We analytically prove, for the linear case, that the order of accuracy of the method is two, and that it is unconditionally stable. We verify the method by reproducing an analytical solution with high accuracy. Then large systems with random parameters and discontinuous initial conditions are used to demonstrate that the new method is competitive against several other solvers, even if the nonlinear term is extremely large. Finally, we show that the new method can be adapted to the advection–diffusion-reaction term as well.
Highlights
The numerical error is the absolute difference of the numerical solutions unum produced by the examined method and the j reference solution uref j at final time tfin
We found that the unconditionally positive finite difference (UPFD) method is first order while the new pseudoimplicit methods are second order in the time step size
In connection with the UPFD methods, earlier papers referred to these phenomena [30,32], and we can see them in the case of the new pseudo-implicit methods as well
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Since the formerly fast increase of the CPU clock frequencies halted more than a decade and a half ago, and the trend towards increasing parallelism in high performance computing is reinforced [23,24], we believe that the parallelizable explicit and unconditionally stable methods for numerically solving these equations will have an increasing role in the future Albeit these methods are less known, some scholars work with them. Al-Bayati et al compared the ADE, the alternating direction implicit (ADI) and the Hopscotch method in the case of the Gray–Scott reaction-diffusion equation [29]. These methods perform quite well in an equidistant and regular mesh, but they heavily rely on these beneficial properties of the mesh.
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