Abstract
This paper addresses exponential basis and compact formulation for solving three-dimensional convection-diffusion-reaction equations that exhibit an accuracy of order three or four depending on exponential expanding or uniformly spaced grid network. The compact formulation is derived with three grid points in each spatial direction and results in a block-block tri-diagonal Jacobian matrix, which makes it more suitable for efficient computing. In each direction, there are two tuning parameters; one associated with exponential basis, known as the frequency parameter, and the other one is the grid ratio parameter that appears in exponential expanding grid sequences. The interplay of these parameters provides more accurate solution values in short computing time with less memory space, and their estimates are determined according to the location of layer concentration. The Jacobian iteration matrix of the proposed scheme is proved to be monotone and irreducible. Computational experiments with convection dominated diffusion equation, Schrödinger equation, Helmholtz equation, nonlinear elliptic Allen–Cahn equation, and sine-Gordon equation support the theoretical convergence analysis.
Highlights
We shall describe a numerical method to solve the general form of three-space dimensions mildly nonlinear elliptic partial differential equations ∂2U ∂2U ∂2U ∂U ∂U ∂U ∂x2 + ∂y2 + ∂z2= ψ x, y, z, U, ∂x ∂y ∂z 0
The nineteen-point extended compact finite-difference replacement (3.14) for approximating elliptic partial differential equations (PDEs) (1.1) exhibits truncation error of order three on the exponential expanding grid network, and it is fourth-order accurate on uniformly distributed grid points
We summarize the above result in the following manner: Theorem 4.1 The exponential basis compact scheme (3.14) on the exponential√expa√nding grid network has third order of convergence, provided ∂ψ/∂U ≥ 0, p, q, r ∈ (
Summary
The application of second central-difference operator and composite of averaging and first central-difference operator results in a scheme having second-order truncation errors, and it is compact too since it uses minimum grid points required to discretize the presence of maximum order partial differentials in the given mathematical model. Approximations of second-order partial derivatives at the grid-point location xi can be estimated on the same line by replacing δxiUix,j,k to δx2i Uix,jx,k in the linear combination (2.3) and giving different names to the unknown coefficients. In this way, we can get an approximation of diffusion terms.
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