Numerical methods for advection equations with delay

We consider the equation in partial derivatives of the first order with effect of heredity ∂u(x,t)∂t+a∂u(x,t)∂x = f(x,t,u(x,t),ut(x,⋅)),ut(x,⋅) = {u(x,t+s),−τ≤s<0}. For such equation from positions of the principle of separating finite-dimensional and infinite-dimensional parts of state, the grid methods are constructed: an analog of the running count scheme, analog of the Crank-Nicolson scheme, an approximation method on the middle of square. The piecewise linear one-dimensional and double interpolation and extrapolation by continuation are applied for accounting the effect of heredity. It is proved that the considered methods have orders of local error respectively O(h+Δ), O(h+Δ2) and O(h2+Δ2), where h is a step of discretization on spatial variable, Δ is a step of discretization on time variable. Properties of double piecewise linear interpolation are investigated. Using results of the general theory of difference schemes, stability conditions of the offered methods are obtained. By means of inclusion in the general scheme of numerical methods for the functional-differential equations the theorems on convergence orders of the designed algorithms are obtained. Test examples on comparison of the methods errors are given.