A modified Gautschi’s method without order reduction when integrating boundary value nonlinear wave problems

Abstract

In this paper we analyse the order reduction which turns up when integrating nonlinear wave problems with non-homogeneous and time-dependent boundary conditions with the well-known Gautschi method. Moreover, a technique is suggested to avoid that order reduction so that the classical local order 4 and global order 2 are recovered. On the other hand, the usual approximation for the derivative which is used together with this method is also analysed and a substantial improvement is suggested. Some numerical results are shown which corroborate the performed analysis.

Integrating Semilinear Wave Problems with Time-Dependent Boundary Values Using Arbitrarily High-Order Splitting Methods

“…However, for these second-order in-time evolutionary problems, the stability condition is acceptable, and the step size in time and space may be taken to be of a similar size. It is also possible to use implicit methods, (see, for example, [6]), where Gautschi methods are studied avoiding the order-reduction phenomenon that appears with these methods.…”