An Unconditionally Stable Space-Time FE Method for the Korteweg-de Vries Equation

TL;DR

This paper introduces an unconditionally stable finite element method for the Korteweg-de Vries equation, enabling stable space-time solutions without CFL constraints, with demonstrated optimal convergence and adaptive refinement capabilities.

Contribution

The paper presents a novel AVS-FE method combining global stability with classical FE solutions for the KdV equation, using a saddle point system and adaptive mesh refinement.

Findings

Unconditionally stable for linear and nonlinear KdV equations

Achieves optimal convergence rates in numerical tests

Supports adaptive mesh refinement in space and time

Abstract

We introduce an unconditionally stable finite element (FE) method, the automatic variationally stable FE (AVS-FE) method for the numerical analysis of the Korteweg-de Vries (KdV) equation. The AVS-FE method is a Petrov-Galerkin method which employs the concept of optimal discontinuous test functions of the discontinuous Petrov- Galerkin (DPG) method. However, since AVS-FE method is a minimum residual method, we establish a global saddle point system instead of computing optimal test functions element-by-element. This system allows us to seek both the approximate solution of the KdV initial boundary value problem (IBVP) and a Riesz representer of the approximation error. The AVS-FE method distinguishes itself from other minimum residual methods by using globally continuous Hilbert spaces, such as H1, while at the same time using broken Hilbert spaces for the test. Consequently, the AVS-FE approximations are classical C0 continuous FE solutions. The unconditional stability of this method allows us to solve the KdV equation space and time without having to satisfy a CFL condition. We present several numerical verifications for both linear and nonlinear versions of the KdV equation leading to optimal convergence behavior. Finally, we present a numerical verification of adaptive mesh refinements in both space and time for the nonlinear KdV equation.