A splitting approach for the Kadomtsev--Petviashvili equation

TL;DR

This paper introduces an efficient splitting method for solving the Kadomtsev--Petviashvili equation, compares its accuracy with exponential integrators, and proposes a stable extrapolation technique for high-order solutions.

Contribution

It presents a novel splitting approach with efficient interpolation, a stable extrapolation method for fourth-order accuracy, and analyzes conservation and order reduction issues.

Findings

The splitting scheme is efficiently implementable with periodic boundary conditions.

The proposed extrapolation method achieves stable fourth-order accuracy.

The scheme's conservation properties and order reduction are thoroughly analyzed.

Abstract

We consider a splitting approach for the Kadomtsev--Petviashvili equation with periodic boundary conditions and show that the necessary interpolation procedure can be efficiently implemented. The error made by this numerical scheme is compared to exponential integrators which have been shown in Klein and Roidot (SIAM J. Sci. Comput., 2011) to perform best for stiff solutions of the Kadomtsev--Petviashvili equation. Since many classic high order splitting methods do not perform well, we propose a stable extrapolation method in order to construct an efficient numerical scheme of order four. In addition, the conservation properties and the possibility of order reduction for certain initial values for the numerical schemes under consideration is investigated.