Method of lines approximations of delay differential equations

We consider a special type of numerical methods for delay differential equations (DDEs). By introducing a new independent variable, an initial value problem for DDEs is converted into an initial-boundary value problem for the convection equation. Thus, it is also possible to get an approximate solution to the DDE problem by solving the initial-boundary value problem with a suitable numerical method instead of solving the original problem. In this paper, we study a family of method of lines (MOL) approximations to the problem, which is obtained by applying Runge-Kutta (RK) methods for space discretization, and prove their convergence under the assumption that the RK methods satisfy a condition, known as an algebraic characterization of A-stability. The result is also confirmed by numerical experiments. Moreover, we show that the condition derives several stability properties of the MOL approximations.