Arbitrarily high-order conservative schemes for the generalized Korteweg-de Vries equation

TL;DR

This paper introduces a high-order conservative numerical scheme for the generalized KdV equation using the SAV method, ensuring exact invariance preservation and spectral accuracy, demonstrated through efficient breather simulations.

Contribution

It develops a novel high-order conservative scheme based on the SAV method for the generalized KdV equation, preserving multiple invariants exactly and achieving spectral accuracy.

Findings

Exact conservation of momentum and energy in space-time discretization

Spectral accuracy in mass conservation

Efficient simulation of breathers in mKdV equation

Abstract

This paper proposes a new class of arbitrarily high-order conservative numerical schemes for the generalized Korteweg-de Vries (KdV) equation. This approach is based on the scalar auxiliary variable (SAV) method. The equation is reformulated into an equivalent system by introducing a scalar auxiliary variable, and the energy is reformulated into a sum of two quadratic terms. Therefore, the quadratic preserving Runge-Kutta method will preserve all the three invariants (momentum, mass and the reformulated energy) in the discrete time flow (assuming the spatial variable is continuous). With the Fourier pseudo-spectral spatial discretization, the scheme conserves the first and third invariant quantities (momentum and energy) exactly in the space-time full discrete sense. The discrete mass possesses the precision of the spectral accuracy. Our numerical experiment shows the great efficiency of this scheme in simulating the breathers for the mKdV equation.