Using Crank-Nikolson Scheme to Solve the Korteweg-de Vries (KdV) Equation

TL;DR

This paper implements the Crank-Nicolson finite difference scheme to solve the Korteweg-de Vries (KdV) equation, demonstrating its stability and improved accuracy for modeling wave propagation in dispersive media.

Contribution

It introduces a stable, implicit numerical method for solving the KdV equation and analyzes its convergence and error performance through various test cases.

Findings

Crank-Nicolson scheme offers enhanced stability over explicit methods.

The method achieves good convergence and low error in test cases.

Code implementation is provided for reproducibility.

Abstract

The Korteweg-de Vries (KdV) equation is a fundamental partial differential equation that models wave propagation in shallow water and other dispersive media. Accurately solving the KdV equation is essential for understanding wave dynamics in physics and engineering applications. This project focuses on implementing the Crank-Nicolson scheme, a finite difference method known for its stability and accuracy, to solve the KdV equation. The Crank-Nicolson scheme's implicit nature allows for a more stable numerical solution, especially in handling the dispersive and nonlinear terms of the KdV equation. We investigate the performance of the scheme through various test cases, analyzing its convergence and error behavior. The results demonstrate that the Crank-Nicolson method provides a robust approach for solving the KdV equation, with improved accuracy over traditional explicit methods. Code is available at the end of the paper.