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

TL;DR

This paper introduces a new analytical framework for energy-preserving numerical schemes applied to KdV-type equations, providing insights into their mathematical properties and establishing convergence and existence results.

Contribution

It develops a general framework for analyzing energy-preserving schemes, extending understanding beyond norm-preserving cases, and applies it to KdV equations to prove global existence and convergence.

Findings

Established global existence of solutions

Proved convergence estimates for numerical schemes

Extended analysis to general energy-preserving schemes

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.