Tridiagonal solver-based L-stable one-step schemes for nonlinear parabolic equations

An attractive feature of the widely used Crank-Nicolson (C-N) scheme for parabolic equations is that it is a tridiagonal solver-based (TSB) scheme. But, in case of inconsistencies in the initial and boundary conditions or when the ratio of temporal to spatial steplengths is large, it can produce unwanted oscillations or an unacceptable solution. As alternative to C-N, Chawla et al. [2, 3] introduced L-stable generalized trapezoidal formulas (GTF(α)) which can give a more acceptable solution by a judicious choice of the parameter α; however, GTF are not TSB schemes. It is natural to ask for L-stable TSB schemes. In the present paper, we first introduce a one-parameter family of generalized midpoint formulas (GMF(α)); again GMF are not TSB schemes. We then introduce a two-parameter family through a linear combination of the GMF and the classical trapezoidal formula, and show the existence of a one-parameter subfamily of L-stable TSB schemes; these schemes are unconditionally stable. The computational performance of the obtained schemes is compared with the C-N scheme by considering a nonlinear reaction-diffusion equation.

Keywords:

• Parabolic Equations
• Nonlinear
• Crank-Nicolson
• Generalized Midpoint Formulas
• Tridiagonal Solver-based Schemes
• Unconditional Stability
• Reaction-diffusion Equation