Invariant preserving schemes for double dispersion equations

Natalia Kolkovska,Veselina Vucheva

doi:10.1186/s13662-019-2152-y

Abstract

We propose and study two finite difference schemes (FDSs) for the double dispersion equations. The first FDS is symplectic, while the second one preserves the discrete momentum exactly. Both FDS conserve the discrete energy approximately with O(h^{2}+tau ^{2}) global error. The extensive numerical experiments agree well with the theoretical results for single solitary wave as well as for the interaction between two solitary waves.

Highlights

1 Introduction The aim of this paper is to develop and analyze two finite difference schemes (FDSs) for the solution of the double dispersion equations (DDEs)
In addition we prove that this scheme satisfies equalities, which connect the discrete momentum on every two consecutive time levels; the same holds true for the discrete energy
We prove that the second FDS, i.e. FDS-M, preserves exactly the discrete momentum and approximately, up to O(h2 + τ 2) global error, the discrete Hamiltonian

Summary

Introduction

The aim of this paper is to develop and analyze two finite difference schemes (FDSs) for the solution of the double dispersion equations (DDEs). We show that FDS-S preserves the discrete momentum and the discrete Hamiltonian in time approximately, up to O(h2 + τ 2) global error. Theorem 6 (Discrete identity for momentum) The solution of FDS-S (10) satisfies the following discrete identities between every two consecutive time levels k + 1 and k: Mhk+1 Uk+1, V k+1/2 – Mhk Uk, V k–1/2 = –τ hUxk,if Uik , k = 1, 2, . Remark 1 Following the lines of the proof of Theorem 6, we can prove that the solution to the FDS-E conserves the discrete momentum approximately with O(h2) global error. We prove in Theorem 6 that the solution to the symplectic scheme FDS-S approximately preserves the discrete momentum (17) with O(h2) global error

Conclusion

Methods and Applications

Methods