Alternative direction implicit method for solving first order 2D hyperbolic delay differential equations

This article considers delayed two-dimensional first order hyperbolic differential equations. The propagation of the discontinuity of the solution is also established. An alternating implicit finite difference method and backward Euler finite difference methods are presented. We proved that this method is first-order convergent. Illustrating numerical examples are given for validation. We also present an application of the proposed approach to the variable delay problem.