Comparison of finite‐difference schemes for the Gross‐Pitaevskii equation

Abstract

A conservative finite‐difference scheme for numerical solution of the Gross‐Pitaevskii equation is proposed. The scheme preserves three invariants of the problem: the L ₂ norm of the solution, the impulse functional, and the energy functional. The advantages of the scheme are demonstrated via several numerical examples in comparison with some other well‐known and widely used methods. The paper is organized as follows. In Section 2 we consider three main conservation laws of GPE and derive the evolution equations for first and second moments of a solution of GPE. In Section 3 we define the conservative finite‐difference scheme and prove the discrete analogs of conservation laws. The remainder of Section 3 consists of a brief description of other finite‐difference schemes, which will be compared with the conservative scheme. Section 4 presents the results of numerical solutions of three typical problems related to GPE, obtained by different methods. Comparison of the results confirms the advantages of conservative scheme. And finally we summarize our conclusions in Section 5.

First published online: 14 Oct 2010