High-order symmetric multistep cosine methods
Multistep cosine methods constitute a subclass of exponential integrators, designed to integrate the linear and stiff part of problems of second order in time exactly. The purpose of the paper is to construct high-order symmetric methods within this family, designed for systems that arise after a pseudospectral discretization of periodic boundary problems defined by second-order partial differential equations when appropriate regularity conditions take place. As with other families of multistep methods, the phenomenon of numerical resonance also appears with these schemes: there are some particular values of the time step size \(h\) for which the error is much greater that for other values of \(h\). To avoid this effect, a simple filter technique is proposed without losing the order of consistency. Two particular multistep cosine methods of orders 8 and 10 are constructed in the paper, which are subsequently illustrated on a pair of numerical examples in comparison with Gauss methods of the same order. These examples suggest that the new methods are more efficient than Gauss integrators.
second-order partial differential equations
scientific article; zbMATH DE number 6140835