Legendre-Tau approximations for functional differential equations
SIAM Journal on Control and Optimization
Complete algebraic characterization of A-stable Runge-Kutta methods
SIAM Journal on Numerical Analysis
The Trotter-Kato theorem and approximation of PDEs
Mathematics of Computation
Proceedings of the on Numerical methods for differential equations
Asymptotic Stability Barriers for Natural Runge--Kutta Processes for Delay Equations
SIAM Journal on Numerical Analysis
Numerical solution of retarded functional differential equations as abstract Cauchy problems
Journal of Computational and Applied Mathematics
Computers & Mathematics with Applications
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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.