A new linearly implicit trapezoidal formula for nonlinear parabolic equations

Authors:
M. M. Chawla
Affiliations:
Department of Mathematics & Computer Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
Venue:
Neural, Parallel & Scientific Computations
Year:
2003

Citing 5
Cited 0

Numerical solution of partial differential equations

Numerical solution of partial differential equations
Nonlinear partial differential equations: for scientists and engineers

Nonlinear partial differential equations: for scientists and engineers
Some user-oriented comparisons of adaptive grid methods for partial differential equations in one space dimension

Proceedings of international centre for mathematical sciences on Grid adaptation in computational PDES : theory and applications: theory and applications
On spatial adaptivity and interpolation when using the method of lines

Proceedings of international centre for mathematical sciences on Grid adaptation in computational PDES : theory and applications: theory and applications
Difference schemes of optimum degree of implicitness for a family of simple ODEs with blow-up solutions

Journal of Computational and Applied Mathematics - Special issue: nonlinear problems with blow-up solutions: applications and numerical analysis

Quantified Score

Hi-index	0.00

Visualization

Abstract

Richtmyer's linearization (Richtmyer & Morton, 1967) applied to the classical trapezoidal formula (TF) leads to a linearly implicit trapezoidal formula, the so-called Lintrap formula. Lintrap has found extensive applications in computational fluid dynamics, plasma physics, nonlinear parabolic equations and blow-up solutions for ODEs. But, Lintrap is only first order for non-autonomous problems, and its local truncation error does not match that of TF. We present a new linearly implicit trapezoidal formula (LITF). The presented LITF is second order for non-autonomous problems and its principal local truncation error matches that of TF. For nonlinear parabolic problems, in comparison with the implicit Crank-Nicolson method, the present LITF obviates the need to solve a nonlinear system at each time step of integration. Numerical experiments reported illustrate these properties of LITF over that of Lintrap. Our experiments include viscous Burgers' equation, nonlinear diffusion and nonlinear reaction-diffusion. In each case, an LITF scheme is much superior than the Lintrap scheme and it provides accuracy comparable with that of the implicit Crank-Nicolson scheme without the need to solve a nonlinear system at each time step of integration.