A Family of Variable Step-Size Meshes Fourth-Order Compact Numerical Scheme for (2+1)-Dimensions Burger’s-Huxley, Burger’s-Fisher and Convection-Diffusion Equations (Journal of Nonlinear Modeling and Analysis, 2024)
Existing numerical schemes, maybe high-order accurate, are obtained on uniformly spaced meshes and challenges to achieve high accuracy inthe presence of singular perturbation parameter, and nonlinearity remains lefton nonuniformly spaced meshes. A new scheme is proposed for nonlinear 2Dparabolic partial differential equations (PDEs) that attain fourth-order accuracy in $xy$-space and second-order exact in the temporal direction for uniformand nonuniform mesh step-size. The method proclaims a compact characterusing nine-point single-cell finite-difference discretization on a nonuniformly spaced spatial mesh point. A description of splitting compact operator form tothe convection-dominated equation is obtained for implementing alternatingdirection implicit scheme. The procedure is examined for consistency and stability. The scheme is applied to linear and nonlinear 2D parabolic equations:convection-diffusion equations, Burger’s-Huxley, Burger’s-Fisher and coupledBurger’s equation. The technique yields the tridiagonal matrix and computedby the Thomas algorithm. Numerical simulations with linear and nonlinearproblems corroborate the theoretical outcome.