Matrix analysis of discrete functionals in compact difference method for nonlinear problems with higher derivatives and program code (I: 1D problem)

This study is dedicated to the matrix analysis method for discrete functionals in higher-order compact difference scheme. This matrix analysis is actually an analysis of the operator matrix obtained by applying the classical difference operator to the entire calculation area, so a compact scheme for high-order derivatives can be directly obtained. By applying the proposed analytical method, a compact conservative difference scheme with high accuracy for the generalized Rosenau–Kawahara–RLW equation is studied in this paper, the discrete conservation law is presented, a priori estimates are obtained. It is verified that the numerical experiments are stable and convergent, and the convergence order in the maximum norm is \(O(\tau ^{2}+h^{4})\) $O({\tau }^2+h^4)$ . According to the matrix analysis technique, we conducted numerical experiments in order to prove the theoretical analysis and provided full MATLAB code to ensure the repeatability of the algorithm. In the numerical analysis, the advantages of the discrete functional matrix analysis are shown, and it is easy to programmability.