Analysis of nonquadratic energy-conservative schemes for KdV type-equations

TL;DR

The paper presents a new analytical framework for energy-preserving numerical schemes applied to KdV-type equations, addressing the challenge of non-quadratic energy functionals. It introduces an induction-based three-step strategy that works with $L^{\infty}$ bounds and a compensating modified energy $\mathcal{E}_d'$ to handle negative cubic terms, leading to local solvability, conditional convergence, and global-in-time solvability for energy-preserving discretizations. The approach is extended to Ostrovsky and generalized KdV equations by adjusting the discrete energy and associated bounds, yielding analogous global solvability and convergence results. The work highlights the practical significance of energy-preserving schemes for stable long-time integration of nonlinear dispersive PDEs and broadens the mathematical understanding of their behavior beyond norm-preserving analyses.

Abstract

Numerical schemes that conserve invariants have demonstrated superior performance in various contexts, and several unified methods have been developed for constructing such schemes. However, the mathematical properties of these schemes remain poorly understood, except in norm-preserving cases. This study introduces a novel analytical framework applicable to general energy-preserving schemes. The proposed framework is applied to Korteweg-de Vries (KdV)-type equations, establishing global existence and convergence estimates for the numerical solutions.

Theorems & Definitions (22)

• Lemma 2.1
• Lemma 2.2: Discrete Sobolev Lemma; Lemma 3.2 in DVDM
• Lemma 2.3
• proof
• Theorem 3.1: local existence
• proof
• Theorem 3.2: conditional convergence
• Lemma 3.3
• proof
• Lemma 3.4