An efficient space–time two-grid difference approach to symmetric regularized long waves: Enhanced efficiency and accuracy

This paper focuses on the development and analysis of a space–time two-grid (ST-TG) difference scheme for solving the symmetric regularized long wave (SRLW) equation. The scheme combines a time two-grid technique with a spatial interpolation operator to boost efficiency. Regarding the numerical results, we provide a much deeper discussion. In terms of computational efficiency, not only do we highlight that the ST-TG scheme demands merely around 20% of the CPU time of the standard nonlinear scheme, but we also present detailed graphs and tables to elucidate how this time reduction correlates with different grid sizes and problem complexities. For accuracy, the second-order convergence analysis of the ST-TG scheme is given. A step-by-step account of the convergence calculation and verification is included, along with comprehensive comparisons to other schemes. Our conservation laws analysis is also deepened, with detailed numerical examples provided to showcase how the ST-TG scheme faithfully adheres to the conservation of mass, momentum, and energy, underpinning its stability and physical consistency. Moreover, we explore the long-term behavior of the numerical solutions more thoroughly, presenting plots of wave propagation over extended time periods and dissecting any changes or oscillations. These enhancements offer potential readers a more profound and comprehensive understanding of the ST-TG scheme, making it a more valuable contribution to the field of solving the SRLW equation. Overall, the proposed ST-TG scheme presents a novel and effective approach with significant advantages in computational efficiency, accuracy, and physical consistency for dealing with problems related to SRLW equation.