A Robust Three-Level Time Split High-Order Leapfrog/Crank-Nicolson Scheme for Two-Dimensional Sobolev and Regularized Long Wave Equations Arising in Fluid Mechanics (Advances in Applied Mathematics and Mechanics, 2025)
This paper develops a robust three-level time split high-order Leapfrog/Crank-Nicolson technique for solving the two-dimensional unsteady Sobolev and regularized long wave equations arising in fluid mechanics. A deep analysis of the stability and error estimates of the proposed approach is considered using the $L^∞(0,T;H^2)$-norm. Under a suitable time step requirement, the theoretical studies indicate that theconstructed numerical scheme is strongly stable (in the sense of $L^∞(0,T;H^2)$-norm),temporal second-order accurate and convergence of order $\mathcal{O}(h^{8/3})$ in space, where $h$ denotes the grid step. This result suggests that the proposed algorithm is less timeconsuming, faster and more efficient than a broad range of numerical methods widelydiscussed in the literature for the considered problem. Numerical experiments confirm the theory and demonstrate the efficiency and utility of the three-level time splithigh-order formulation.