## Stability of Runge-Kutta methods in the numerical solution of equation $$u'(t)=au(t)+a_{0} u([t])+a_{1} u([t-1])$$.(English)Zbl 1063.65070

The authors discuss the numerical solution of the initial value problem $$u'(t) = a u(t) + a_0 u([t]) + a_1 u([t-1])$$, $$u(0) = u_0$$, $$u(-1) = u_{-1}$$, where $$[\cdot]$$ denotes the floor function (round down to nearest integer). This is a special case of a delay differential equation with piecewise continuous argument. The numerical methods under consideration are of Runge-Kutta type. The authors first explain how standard Runge-Kutta methods can be applied to this class of problems. Then, the asymptotic stability of various special types of Runge-Kutta methods (e.g., Gauss, Lobatto, and Radau) is investigated.

### MSC:

 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L10 Numerical solution of boundary value problems involving ordinary differential equations 34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)

RODAS
Full Text:

### References:

  Busenberg, S.; Cooke, K.L., Models of vertically transmitted diseases with sequential-continuous dynamics, (), 179-187  Butcher, J.C., The numerical analysis of ordinary differential equations: runge – kutta and general linear methods, (1987), John Wiley New York · Zbl 0616.65072  Cooke, K.L.; Wiener, J., Retarded differential equations with piecewise constant delays, J. math. anal. appl, 99, 265-297, (1984) · Zbl 0557.34059  Dekker, K.; Verwer, J.G., Stability of runge – kutta methods for stiff nonlinear differential equations, (1984), North-Holland Amsterdam · Zbl 0571.65057  Hairer, E.; Nørsett, S.P.; Wanner, G., Solving ordinary differential equations II, stiff and differential algebraic problems, (1993), Springer-Verlag New York  Iserles, A.; Nørsett, S.P., Order stars and rational approximations to exp(z), Appl. numer. math, 5, 63-70, (1989) · Zbl 0674.65043  Kocic, V.L.; Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, (1993), Kluwer Academic Publishers Dordrecht · Zbl 0787.39001  Liu, P.; Gopalsamy, K., Global stability and chaos in a population model with piecewise constant arguments, Appl. math. comput, 101, 63-88, (1999) · Zbl 0954.92020  Miller, J.J.H., On the location of zeros of certain classes of polynomials with applications to numerical analysis, J. inst. math. appl, 8, 397-406, (1971) · Zbl 0232.65070  Shah, S.M.; Wiener, J., Advanced differential equations with piecewise constant argument deviations, Internat. J. math. math. sci, 6, 671-703, (1983) · Zbl 0534.34067  Wanner, G.; Hairer, E.; Nørsett, S.P., Order stars and stability theorems, Bit, 18, 475-489, (1978) · Zbl 0444.65039  Wiener, J., Differential equations with piecewise constant delays, (), 547-552  Wiener, J., Generalized solutions of differential equations, (1993), World Scientific Singapore  Wiener, J., Pointwise initial-value problems for functional differential equations, (), 571-580
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.