Variable step implementation of ETD methods for semilinear problems

• Published in: Applied mathematics and computation Vol. 196; no. 2; pp. 627 - 637
• Main Authors: Calvo, M.P., Portillo, A.M.
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 01.03.2008; Elsevier
• Subjects: Exact sciences and technology; Exponential integrators; Implementation; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; 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; Semilinear equations; Variable steps; Variable steps; Semilinear equations; Exponential integrators; Implementation; Semi linear equation; Error estimation; Differential equation; Initial value problem; Nonlinear problems; Runge Kutta method; Partial differential equation; Discretization method; Variable step method; Extrapolation; Numerical analysis; Boundary value problem; Multistep method; Applied mathematics
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2007.06.025

Spatial discretization of many evolutionary partial differential equations leads to stiff semilinear systems of ordinary differential equations. Exponential integrators solve exactly the linear part, which is assumed to be responsible of the stiffness of the problem, while the nonlinear terms are approximated numerically. In this paper, a variable step implementation of the exponential time differencing methods of multistep type is considered. A local error estimate of exponential type, whose computation is for free, is introduced and some implementation issues for diagonalizable matrices are addressed. Numerical results showing the behaviour of the proposed variable step implementation are provided too.