On the Convergence and Stability of Finite Difference Method for Parabolic Partial Differential Equations

Abstract

In this paper, we verify the convergence and stability of implicit (modified) finite difference scheme. Knowing fully that consistency and stability are very important criteria for convergence, we have prove the stability of the modied implicit scheme using the von Newmann method and also verify the convergence by comparing the numerical solution with the exact solution. The results shows that the schemes converges even as the step size is rened.

Comparative Study of Crank-Nicolson and Modified Crank-Nicolson Numerical methods to solve linear Partial Differential Equations

“…& Sanghvi, R. Yang (15) , Omowo B. J. & Abhulimen, C. E. (16,17) , are notable comprehensive texts on numerical methods for partial differential equations.…”

A Comparison of Implicit and Modified Implicit Finite Difference Schemes for Solving Parabolic Equations

“…For the stability of modified Implicit scheme by von Newmann, see [5] but for the purpose of this paper we have the following after substituting the trigonometric identities (12),…”

“…[2] considered the practical methods for numerical solution to partial differential equations of heat conduction type. [3] Investigated the stability of Modified Crank-Nicolson scheme using Fourier method (von-Newmann method). They prove that the scheme is consistent, convergent and stable.…”

“…into equation(3) givesξ i ς zβjk = −r(ξ i+1 ς zβ(j−1)k + ξ i+1 ς zβ(j+1)k ) + (1 + 2r)ξ i+1 ς zβjk ξ i ς zβjk = −rξ i ς zβjk (ξς −zβk + ξς zβk ) + (1 + 2r)ξςzβjk dividing both sides by ξ i ς zβjk we get 1 = −r(ξς −zβk + ξς zβk ) + (1 + 2r)ξ which can be written as 1 = −r ς −zβk + ς zβk + (1 + 2r) ξ (11) using the following trigonometric Identities ς −zβk + ς zβk = 2 cos βk and 2…”