Advance in compact structure-preserving manner to the Rosenau–Kawahara model of shallow-water wave

Abstract

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.