An efficient off-step spline algorithm for wave simulation of nonlinear Kuramoto-Sivashinsky and Korteweg-de Vries equations

In this research, we present an advanced numerical technique developed by adhering to the consistency condition, leveraging the characteristics of polynomial cubic spline approximations. Our method is specifically tailored for tackling nonlinear equations like the Kuramoto-Sivashinsky and Korteweg-de Vries equations. Unlike conventional finite difference methods, which encounter difficulties in addressing singular biharmonic problems directly, our approach overcomes this hurdle. By incorporating off-step points in the spatial dimension, our method facilitates the direct resolution of singular problems. We assess stability using von Neumann stability analysis on a linear model. Chaotic simulations of the nonlinear Kuramoto-Sivashinsky equation are conducted across various time stages. Single-soliton of the Korteweg-de Vries equation is solved numerically at different time levels. Our proposed technique exhibits high accuracy, demonstrating its efficacy compared to prior research. Additionally, we delve into solving diverse benchmark problems numerically, evaluating global relative and maximum absolute errors in comparison with previous studies.