An implicit high accuracy variable mesh scheme for 1-D non-linear singular parabolic partial differential equations

• Published in: Applied mathematics and computation Vol. 186; no. 1; pp. 219 - 229
• Main Author: Mohanty, R.K.
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 01.03.2007; Elsevier
• Subjects: Burgers’ equation; Diffusion equation; Exact sciences and technology; Finite difference scheme; Functional analysis; Mathematical analysis; Mathematics; Non-linear parabolic equation; Normal derivative; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations; Partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Sciences and techniques of general use; Singular perturbation; Singularity; Variable mesh; Finite difference scheme; Singularity; Singular perturbation; Non-linear parabolic equation; Burgers’ equation; Diffusion equation; Variable mesh; Normal derivative; Difference scheme; Support; Singular equation; Grid; Boundary condition; Partial differential equation; Non linear equation; Function evaluation; Parabolic equation; Accuracy; Linear equation; Addition; Grid pattern; Initial condition; Initial value problem; Burgers equation; Numerical analysis; Boundary value problem; Coordinate; Applied mathematics; Derivative; Burgers' equation; Linear differential equations; Finite difference method
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2006.06.122

In this paper, we propose a new two-level implicit difference scheme of O ( k 2 h l - 1 + kh l + h l 3 ) for the solution of non-linear parabolic equation εu xx = ϕ( x, t, u, u x , u t ), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions prescribed, where k > 0, h l > 0 are mesh sizes in t- and x-coordinates respectively and ε > 0 is a small parameter. In addition, we also discuss a new explicit variable mesh difference scheme of O ( kh l + h l 3 ) for the estimates of (∂ u/∂ x). In all cases, we require only three spatial variable grid points. The proposed schemes require less algebra and three evaluations of function ϕ. A special technique is required to solve singular parabolic equations. The proposed variable mesh scheme when applied to a linear diffusion equation is shown to be stable for all h l > 0 and k > 0. Computational results are provided to support our derived schemes and analysis.