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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
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KeyWord:Nonlinear parabolic partial differential equations (PDEs); Two-dimensions Burger's-Huxley equation; Boussinesq equation; Convection-diffusion equation; Compact-scheme; Stability; Errors and numerical order
Author NameAffiliation
Navnit Jha Department of Mathematics, South Asian University, Akbar Bhawan, 110021 New-Delhi, India 
Madhav Wagley Department of Mathematics, Florida A\&M University, Tallahassee, Florida, USA 
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      Existing numerical schemes, maybe high-order accurate, are obtained on uniformly spaced meshes and challenges to achieve high accuracy in the presence of singular perturbation parameter, and nonlinearity remains left on nonuniformly spaced meshes. A new scheme is proposed for nonlinear 2D parabolic partial differential equations (PDEs) that attain fourth-order accuracy in $xy$-space and second-order exact in the temporal direction for uniform and nonuniform mesh step-size. The method proclaims a compact character using nine-point single-cell finite-difference discretization on a nonuniformly spaced spatial mesh point. A description of splitting compact operator form to the convection-dominated equation is obtained for implementing alternating direction 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 coupled Burger's equation. The technique yields the tridiagonal matrix and computed by the Thomas algorithm. Numerical simulations with linear and nonlinear problems corroborate the theoretical outcome.