Time-Dependent Duhamel Renormalization method with Multiple Conservation and Dissipation Laws

TL;DR

This paper extends the time-dependent spectral renormalization method to include multiple conservation laws and dissipation equations, enabling more accurate and versatile numerical solutions for complex physical evolution equations.

Contribution

It introduces a generalized TDSR method that incorporates multiple conservation/dissipation laws, versatile boundary conditions, and higher order time integration.

Findings

Successfully applied to KdV, NLS, and Allen-Cahn equations.

Enhanced numerical stability and accuracy.

Demonstrated ability to handle multiple physical constraints.

Abstract

The time dependent spectral renormalization (TDSR) method was introduced by Cole and Musslimani as a novel way to numerically solve initial boundary value problems. An important and novel aspect of the TDSR scheme is its ability to incorporate physics in the form of conservation laws or dissipation rate equations. However, the method was limited to include a single conserved or dissipative quantity. The present work significantly extends the computational features of the method with the (i) incorporation of multiple conservation laws and/or dissipation rate equations, (ii) ability to enforce versatile boundary conditions, and (iii) higher order time integration strategy. The TDSR method is applied on several prototypical evolution equations of physical significance. Examples include the Korteweg-de Vries (KdV), multi-dimensional nonlinear Schr\"odinger (NLS) and the Allen-Cahn equations.