Comparison of different discontinuous Galerkin methods based on various reformulations for gKdV equation: Soliton dynamics and blowup

Abstract

In this work, we explore the use of several discontinuous Galerkin (DG) methods for simulating soliton dynamics in the generalized Korteweg-de Vries (gKdV) equation. The presence of high-order nonlinearity in this model can lead to the finite-time blowup phenomenon, which challenges every aspect of a numerical scheme. The considered DG schemes encompass popular methods derived from the energy and/or Hamiltonian conservations of the gKdV equation. To evaluate the performance of these DG schemes from various angles, we present a series of numerical experiments. Through comprehensive comparisons, we aim to identify the scheme that exhibits the best performance. To further enhance the accuracy and efficiency of DG methods, particularly in the context of blowup simulations, we incorporate the arbitrary Lagrangian-Eulerian method for adaptive mesh movement.