A new stable variable mesh method for 1-D non-linear parabolic partial differential equations

• Published in: Applied mathematics and computation Vol. 181; no. 2; pp. 1423 - 1430
• Main Authors: Arora, Urvashi, Karaa, Samir, Mohanty, R.K.
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 15.10.2006; Elsevier
• Subjects: Arithmetic average discretization; Burgers’ equation; Diffusion equation; Exact sciences and technology; Finite difference method; Implicit method; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; 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; Variable mesh; Burgers’ equation; Diffusion equation; Implicit method; Variable mesh; Arithmetic average discretization; Finite difference method; Singularity; Initial condition; Initial value problem; Boundary condition; Burgers equation; Partial differential equation; Convergence; Non linear equation; Function evaluation; Parabolic equation; Numerical analysis; Boundary value problem; Linear equation; Burgers' equation; Mesh method
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2006.02.032

We propose a new stable variable mesh implicit difference method for the solution of non-linear parabolic equation u xx = ϕ( x, t, u, u x , u t ), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions prescribed. We require only (3 + 3)-spatial grid points and two evaluations of the function ϕ. The proposed method is directly applicable to solve parabolic equation having a singularity at x = 0. The proposed method when applied to a linear diffusion equation is shown to be unconditionally stable. The numerical tests are performed to demonstrate the convergence of the proposed new method.