Abstract
The main contribution of this article is to propose a compact explicit scheme for solving time-dependent partial differential equations (PDEs). The proposed explicit scheme has an advantage over the corresponding implicit compact scheme to find solutions of nonlinear and linear convection–diffusion type equations because the implicit existing compact scheme fails to obtain the solution. In addition, the present scheme provides fourth-order accuracy in space and second-order accuracy in time, and is constructed on three grid points and three time levels. It is a compact multistep scheme and conditionally stable, while the existing multistep scheme developed on three time levels is unconditionally unstable for parabolic and considered a type of equations. The mathematical model of the heat transfer in a mixed convective radiative fluid flow over a flat plate is also given. The convergence conditions of dimensionless forms of these equations are given, and also the proposed scheme solves equations, and results are compared with two existing schemes. It is hoped that the results in the current report are a helpful source for future fluid-flow studies in an industrial environment in an enclosure area.
Highlights
The linear reaction convection diffusion equation has a broad range of application in astrophysics, aerospace sciences, biology, and environmental sciences
Energy movements [1,2], fluid clotting [3], and hemodynamics [4] are the areas with relevance to the convection–diffusion equation
The mathematical conversion of a linear reaction convection–diffusion equation into a nonstandard finite difference equation was revealed by [9]; [10] proposed a study that comprised the family of positivity-preserving finite-difference methods for the classical Fisher–Kolmogorov–Petrovsky–Piscounov equation
Summary
The linear reaction convection diffusion equation has a broad range of application in astrophysics, aerospace sciences, biology, and environmental sciences. The fourth-order compact finite difference scheme was studied by Sun and Zhang [11] using parameters such as space and time. The existing implicit compact scheme consisted of the mixed partial derivative term in space and time, and this term was responsible for establishing an implicit compact scheme rather than an explicit one In this contribution, this mixed derivative term is turned into a second-order partial derivative in time term, and this term is discretized using a second-order standard–classical central difference formula constructed on three time levels. The secondorder difference formula is used to obtain the second accurate scheme in time and the fourth order in space. Equation (10) is an explicit compact difference equation that is fourth order in space and second order in time, and it is constructed on three time levels It requires an additional scheme evaluated on the first time level
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