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In this paper, we pursue further analysis of the performance of a compact structure-preserving finite difference scheme. The convergence, stability, and accuracy of the approximate solution with respect to grid refinement are discussed. The compact difference approach to precisely preserving invariants on any time-space regions gives a three-level linear-implicit scheme with the spatial accuracy, found to be fourth order on a uniform grid. The method is verified by comparison with a solution of the Rosenau-Kawahara equation just obtained with second-order finite difference schemes recently. Also, the efficiency of the present algorithm is confirmed by simulations of the problem at a long time. Details of CPU time are examined in order to assess the usefulness of the compact scheme for determining an approximate solution.

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We introduce a new technique, a three-level average linear-implicit finite difference method, for solving the Rosenau-Burgers equation. A second-order accuracy on both space and time numerical solution of the Rosenau-Burgers equation is obtained using a five-point stencil. We prove the existence and uniqueness of the numerical solution. Moreover, the convergence and stability of the numerical solution are also shown. The numerical results show that our method improves the accuracy of the solution significantly.