Abstract
In this paper, a type of high-order compact (HOC) finite difference method is developed for solving two- and three-dimensional unsteady convection diffusion reaction (CDR) equations with variable coefficients. Firstly, an HOC difference scheme is derived to solve the two-dimensional (2D) unsteady CDR equation. Discretization in time is carried out by Taylor series expansion and correction of the truncation error remainder, while discretization in space is based on the fourth-order compact difference formulas. The scheme is second-order accuracy in time and fourth-order accuracy in space. The unconditional stability is obtained by the von Neumann analysis method. Then, this scheme is extended to solve the three-dimensional (3D) unsteady CDR equation. It needs only a five-point stencil for 2D problems and a seven-point stencil for 3D problems. Moreover, the present schemes can solve the nonlinear Burgers equation. Finally, numerical experiments are conducted to show the good performances of the new schemes.
Highlights
The convection diffusion reaction (CDR) equation is a kind of basic mathematical physics equation, which is usually used to describe many physical and chemical processes
Tong et al [22] proposed two fourth-order methods by using a second-order scheme followed by the Richardson extrapolation and a direct fourth-order finite difference (FD) scheme for a steady CDR equation with variable coefficients
An high-order compact (HOC) FD scheme [31] was devised for the 2D variable convection coefficients equation, which is fourth-order in space and not more than second-order in time according to weighted discretization
Summary
The CDR equation is a kind of basic mathematical physics equation, which is usually used to describe many physical and chemical processes. For 1D problem, another scheme with fourth-order accuracy in both temporal and spatial directions was proposed in [26], which is transformed into a reaction diffusion equation and is unconditionally stable. An HOC FD scheme [31] was devised for the 2D variable convection coefficients equation, which is fourth-order in space and not more than second-order in time according to weighted discretization. The derivation process is simple, and it does not require discrete convection terms as some previous works do In this way, compact difference schemes with temporally second-order and spatially fourth-order accuracy can be obtained by using the minimum grid points.
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