Higher-Order Numerical Solution of the KdV-BBM Equation: A Comparative Analysis of Temporal Integration Schemes in the Method of Lines Framework

Abstract

This study investigates the numerical simulation of solitary wave propagation governed by the KdV-BBM equation using a robust Method of Lines (MOL) framework. The governing nonlinear equation is transformed into a system of ordinary differential equations through spatial discretization, and the performance of three temporal integration schemes is evaluated: the first-order Euler method, the fourth-order Runge-Kutta (RK-4), and the fifth-order iRK-5 method. Quantitative analysis using Mean Absolute Error (MAE) for various time steps (Δt = 0.2, 0.1, 0.05, and 0.01) reveals that the iRK-5 scheme provides enhanced temporal precision, achieving error magnitudes as low as 10⁻⁶ and consistently aligning with the exact traveling-wave solution. Notably, the iRK-5 method demonstrates greater algorithmic efficiency, achieving an accuracy level of 4.55 × 10⁻⁶ at a coarser time step of Δt = 0.1, whereas the RK-4 scheme requires a finer time step of Δt = 0.05 to reach the same precision. Both high-order methods eventually reach a spatial error floor where further temporal refinement yields no significant reduction in MAE, emphasizing that high-order temporal integration, particularly the iRK-5 scheme, is essential for preserving the physical integrity of complex nonlinear wave phenomena while maintaining optimal computational effort.