Explicit Stable Finite Difference Methods for Diffusion-Reaction Type Equations

Humam Kareem Jalghaf, Endre Kovács, János Majár, Ádám Nagy and Ali Habeeb Askar
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Humam Kareem Jalghaf: Department of Fluid and Heat Engineering, University of Miskolc, 3515 Miskolc, Hungary
Endre Kovács: Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary
János Majár: Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary
Ádám Nagy: Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary
Ali Habeeb Askar: Department of Fluid and Heat Engineering, University of Miskolc, 3515 Miskolc, Hungary

Mathematics, 2021, vol. 9, issue 24, 1-21

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.

Keywords: UPFD method; diffusion equation; heat transfer; explicit time-integration; stiff equations; unconditional stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations: View citations in EconPapers (3)

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