Grid methods of solving advection equations with delay

We consider a first-order partial differential equation with heredity effect
$$ \frac{\partial u(x,t)}{\partial t}+a\frac{\partial u(x,t)}{\partial x}=f(x,t,u(x,t),u_t(x,\cdot)),\quad u_t(x,\cdot)=\{u(x,t+s),\ -\tau\leqslant s<0\}. $$
For such an equation we construct grid methods using the principle of separation of finite-dimensional and infinite-dimensional state components. These grid methods are: analog of running schemes family, analog of Crank–Nicolson scheme, an approximation method to the middle of the square. The one-dimensional and double piecewise linear interpolation and the extrapolation by continuation are applied in order to account the effect of heredity. It is shown that the considered methods have orders of a local error: $O(h+\Delta)$, $O(h+\Delta^2)$ and $O(h^2+\Delta^2)$ respectively, where $h$ is the spatial discretization interval, $\Delta$ is the time discretization interval. Properties of double piecewise linear interpolation are investigated. Using the results of the general theory of differential schemes, stability conditions of the proposed methods are established. Including them in the general scheme of numerical methods for the functional-differential equations, theorems of orders of proposed algorithms convergence are received. Test examples comparing errors of methods are given.