Discrete conservation laws and the convergence of long time simulations of the mKdV equation

TL;DR

This paper investigates how discrete conservation laws influence the long-term accuracy of numerical simulations of the mKdV equation, emphasizing the importance of invariant-preserving schemes for reliable soliton evolution approximation.

Contribution

It demonstrates that numerical schemes conserving discrete invariants improve long-time simulation accuracy of the mKdV equation compared to standard methods.

Findings

Invariant-preserving schemes outperform non-conservative methods in long-term accuracy.

Pseudospectral methods are most robust for simulating mKdV solutions.

Finite difference schemes help analyze the role of invariants in convergence.

Abstract

Pseudospectral collocation methods and finite difference methods have been used for approximating an important family of soliton like solutions of the mKdV equation. These solutions present a structural instability which make difficult to approximate their evolution in long time intervals with enough accuracy. The standard numerical methods do not guarantee the convergence to the proper solution of the initial value problem and often fail by approaching solutions associated to different initial conditions. In this frame the numerical schemes that preserve the discrete invariants related to some conservation laws of this equation produce better results than the methods which only take care of a high consistency order. Pseudospectral spatial discretization appear as the most robust of the numerical methods, but finite difference schemes are useful in order to analyze the rule played by the conservation of the invariants in the convergence.