Nonlinear wave equations are widely used in many areas of science and engineering. This paper proposes two high-order compact (HOC) difference schemes with convergence orders of $O\left({\tau }^4+h_x^4+h_y^4\right)$ that can be used to solve nonlinear wave equations. The first scheme is a nonlinear compact difference scheme with three temporal levels. After calculating the second-order spatial derivatives of the previous time level using the Padé scheme, numerical solutions of the next time level are obtained through repeated iterations. The second scheme is a three-level linearized compact difference scheme. Unlike the first scheme, iterations are not required and it obtains numerical solutions through an explicit calculation. The two proposed schemes are applied to solutions of the coupled sine-Gordon equations. Finally, some numerical experiments are presented to confirm the effectiveness and accuracy of the proposed schemes.

In this paper, we consider the Keller-Segel chemotaxis model with self- and cross-diffusion terms and a logistic source. This system consists of a fully nonlinear reaction-diffusion equation with additional cross-diffusion. We establish some high-order finite difference schemes for solving one- and two-dimensional problems. The truncation error remainder correction method and fourth-order Padé compact schemes are employed to approximate the spatial and temporal derivatives, respectively. It is shown that the numerical schemes yield second-order accuracy in time and fourth-order accuracy in space. Some numerical experiments are demonstrated to verify the accuracy and reliability of the proposed schemes. Furthermore, the blow-up phenomenon and bacterial pattern formation are numerically simulated.