This paper deals with numerical computation and analysis for the initial boundary problems of two dimensional (2D) Sobolev equations with piecewise continuous argument. Firstly, a two-level high-order compact difference method (HOCDM) with computational accuracy \(\mathcal{O}(\tau^2+h_x^4+h_y^4)\) is suggested, where τ; hx; hy denote the temporal and spatial stepsizes of the method, respectively. In order to improve the temporal computational accuracy of this method, the Richardson extrapolation technique is used and thus a new two-level HOCDM is derived, which is proved to be convergent of order four both in time and space. Although the new two-level HOCDM has the higher computational accuracy in time than the previous one, it will bring a larger computational cost. To overcome this deficiency, a three-level HOCDM with computational accuracy \(\mathcal{O}(\tau^4+h_x^4+h_y^4)\) is constructed. Finally, with a series of numerical experiments, the theoretical accuracy and computational efficiency of the above methods are further verified.
