Construction of additive semi-implicit Runge-Kutta methods with low-storage requirements

The final publication is available at Springer via http://dx.doi.org/ 10.1007/s10915-015-0116-2Space discretization of some time-dependent partial differential equations gives rise to systems of ordinary differential equations in additive form whose terms have different stiffness properties. In these cases, implicit methods should be used to integrate the stiff terms while efficient explicit methods can be used for the non-stiff part of the problem. However, for systems with a large number of equations, memory storage requirement is also an important issue. When the high dimension of the problem compromises the computer memory capacity, it is important to incorporate low memory usage to some other properties of the scheme. In this paper we consider Additive Semi-Implicit Runge-Kutta (ASIRK) methods, a class of implicitexplicit Runge-Kutta methods for additive differential systems. We construct two second order 3-stage ASIRK schemes with low-storage requirements. Having in mind problems with stiffness parameters, besides accuracy and stability properties, we also impose stiff accuracy conditions. The numerical experiments done show the advantages of the new methods.Supported by Ministerio de Economía y Competividad, project MTM2011-23203

Postprocessed integrators for the high order integration of ergodic SDEs

The concept of effective order is a popular methodology in the deterministic literature for the construction of efficient and accurate integrators for differential equations over long times. The idea is to enhance the accuracy of a numerical method by using an appropriate change of variables called the processor. We show that this technique can be extended to the stochastic context for the construction of new high order integrators for the sampling of the invariant measure of ergodic systems. The approach is illustrated with modifications of the stochastic $\theta$-method applied to Brownian dynamics, where postprocessors achieving order two are introduced. Numerical experiments, including stiff ergodic systems, illustrate the efficiency and versatility of the approach.Comment: 21 pages, to appear in SIAM J. Sci. Compu

Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators

Dozens of exponential integration formulas have been proposed for the high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries and Ginzburg-Landau equations. We report the results of extensive comparisons in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and higher order methods, and periodic semilinear stiff PDEs with constant coefficients. Our conclusion is that it is hard to do much better than one of the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews