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This paper develops a robust three-level time split high-orderLeapfrog/Crank-Nicolson technique for solving the two-dimensional unsteadysobolev and regularized long wave equations arising in fluid mechanics. A deepanalysis of the stability and error estimates of the proposed approach isconsidered using the $L^{\infty}(0,T;H^{2})$-norm. Under a suitable time steprequirement, the theoretical studies indicate that the constructed numericalscheme is strongly stable (in the sense of $L^{\infty}(0,T;H^{2})$-norm),temporal second-order accurate and convergence of order $O(h^{\frac{8}{3}})$ inspace, where $h$ denotes the grid step. This result suggests that the proposedalgorithm is less time consuming, fast and more efficient than a broad range ofnumerical methods widely discussed in the literature for the consideredproblem. Numerical experiments confirm the theory and demonstrate theefficiency and utility of the three-level time split high-order formulation.

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