Convergence of the difference method of solving the two-dimensional wave equation with heredity

The paper presents the consideration of the wave equation with two space variables and one time variable and with heredity effect
$$ \frac{\partial^2 u}{\partial t^2}=a^2\left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) + f\big(x,y,t,u(x,y,t),u_t(x,y,\cdot)\big),\quad u_t(x,y,\cdot)=\big\{u(x,y,t+\xi),-\tau\le\xi\le0\big\}. $$
A family of grid methods is constructed for the numerical solution of this equation; the methods are based on the idea of separating the current state and the history function. A complete analog of the factorization method which is known for an equation without delay is constructed according to the current state. Influence of prehistory is taken into consideration by interpolation constructions. The local error order of the algorithm is investigated. A theorem on the convergence and on the order of convergence of methods is obtained by means of embedding into a general difference scheme with aftereffect. The results of calculating a test example with variable delay are presented.