A pseudo-spectral Strang splitting method for linear dispersive problems with transparent boundary conditions

TL;DR

This paper introduces a second-order Strang splitting method combined with spectral spatial discretization for linear dispersive equations with transparent boundary conditions, addressing boundary compatibility issues to maintain accuracy.

Contribution

It presents a modified Strang splitting scheme that preserves second-order accuracy despite boundary compatibility challenges in dispersive problems.

Findings

The scheme achieves spectral accuracy in space.

The method maintains second-order temporal accuracy.

Numerical results confirm theoretical stability and accuracy.

Abstract

The present work proposes a second-order time splitting scheme for a linear dispersive equation with a variable advection coefficient subject to transparent boundary conditions. For its spatial discretization, a dual Petrov--Galerkin method is considered which gives spectral accuracy. The main difficulty in constructing a second-order splitting scheme in such a situation lies in the compatibility condition at the boundaries of the sub-problems. In particular, the presence of an inflow boundary condition in the advection part results in order reduction. To overcome this issue a modified Strang splitting scheme is introduced that retains second-order accuracy. For this numerical scheme a stability analysis is conducted. In addition, numerical results are shown to support the theoretical derivations.