A New Stable, Explicit, Third‐Order Method for Diffusion‐Type Problems

Abstract

This paper reports on a novel explicit numerical method for the spatially discretized diffusion or heat equation. After discretizing the space variables as in conventional finite difference methods, this method does not use a finite difference approximation for the time derivatives, it instead combines constant-neighbor and linear-neighbor approximations, which decouple the ordinary differential equations, thus they can be solved analytically. In the obtained three-stage method, the time step size appears in exponential form with negative coefficients in the final expression. This property guarantees unconditional stability, as it is shown using von Neumann stability analysis. It is also proved that the convergence of the method is third order in the time step size. After verification, by solving Fisher's and Huxley's equations, it is demonstrated that it works for nonlinear equations as well. The new algorithm is tested against widely used numerical solvers for cases where the media is strongly inhomogeneous. According to the results, the new method is significantly more effective than the traditional explicit or implicit methods, especially for extremely large stiff systems. It is believed that this new method is unique in the sense that it is the first unconditionally stable explicit method with third-order convergence.

Stabil, explicit numerikus algoritmusok diffúziós és hővezetési problémák megoldására

“…Ezek bár nem teljesen közismertek, számos kutatócsoport foglalkozik velük, lásd pl. (Appadu, 2017;Karahan, 2007 Kovács et al, 2022). Cikkünk magyar nyelven először mutat be nagyszámú ilyen módszert és azokat a teszteket, amelyek a teljesítményüket összehasonlítják.…”

New Insights into a Three-Sub-Step Composite Method and Its Performance on Multibody Systems

“…Generally, conventional time integration methods [6,7] are divided into two categories, explicit and implicit methods [8]. Explicit methods [8][9][10] are not suitable for solving DAEs directly because they cannot satisfy constraint equations at the position level. In contrast, implicit methods together with an iterative procedure (e.g., Newton-Raphson method) are applicable.…”

A Comparative Study of Explicit and Stable Time Integration Schemes for Heat Conduction in an Insulated Wall

“…All the methods used, except, of course, Heun's method, are conditionally stable for the linear heat conduction equation, i.e., the previously mentio CFL limit is not relevant for them. This, however, does not mean they are always accur Actually, the price of unconditional stability is conditional consistency, which means spatial mesh refinement with a constant time step size yields worsening accuracy (in c trast to worsening stability properties as in mainstream methods), which is examined alytically and numerically in our previous papers [44] and [35], respectively. In Sec In this paper, we use only the combination already proven to be the best (S4 in [33]), which means θ = 0 is used at the first, θ = 1 at the fifth, and θ = 1 2 in all other stages.…”

“…This, however, does not mean they are always accurate. Actually, the price of unconditional stability is conditional consistency, which means that spatial mesh refinement with a constant time step size yields worsening accuracy (in contrast to worsening stability properties as in mainstream methods), which is examined analytically and numerically in our previous papers [44] and [35], respectively. In Section 3.2 we will show examples when the conditionally stable Heun's method is significantly more accurate for small time step sizes than our methods.…”