Exponential time-differencing with embedded Runge–Kutta adaptive step control

• Published in: Journal of computational physics Vol. 280; pp. 579 - 601
• Main Authors: Whalen, P., Brio, M., Moloney, J.V.
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 01.01.2015
• Subjects: Burgers; Burgers equation; CHAOS THEORY; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Collapse; COMPARATIVE EVALUATIONS; Dynamical systems; Embedded Runge–Kutta; Embedded systems; ETD; Exponential time-differencing; KdV; KORTEWEG-DE VRIES EQUATION; Linear operators; NLS; Nonlinear dynamics; RUNGE-KUTTA METHOD; SCALARS; SOLITONS; STABILITY; Stability analysis; TWO-DIMENSIONAL CALCULATIONS; Exponential time-differencing; ETD; NLS; KdV; Embedded Runge–Kutta; Burgers
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2014.09.038

We have presented the first embedded Runge–Kutta exponential time-differencing (RKETD) methods of fourth order with third order embedding and fifth order with third order embedding for non-Rosenbrock type nonlinear systems. A procedure for constructing RKETD methods that accounts for both order conditions and stability is outlined. In our stability analysis, the fast time scale is represented by a full linear operator in contrast to particular scalar cases considered before. An effective time-stepping strategy based on reducing both ETD function evaluations and rejected steps is described. Comparisons of performance with adaptive-stepping integrating factor (IF) are carried out on a set of canonical partial differential equations: the shock-fronts of Burgers equation, interacting KdV solitons, KS controlled chaos, and critical collapse of two-dimensional NLS.