## 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)

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### References:

 [1] Busenberg, S.; Cooke, K.L., Models of vertically transmitted diseases with sequential-continuous dynamics, (), 179-187 [2] Butcher, J.C., The numerical analysis of ordinary differential equations: runge – kutta and general linear methods, (1987), John Wiley New York · Zbl 0616.65072 [3] Cooke, K.L.; Wiener, J., Retarded differential equations with piecewise constant delays, J. math. anal. appl, 99, 265-297, (1984) · Zbl 0557.34059 [4] Dekker, K.; Verwer, J.G., Stability of runge – kutta methods for stiff nonlinear differential equations, (1984), North-Holland Amsterdam · Zbl 0571.65057 [5] Hairer, E.; Nørsett, S.P.; Wanner, G., Solving ordinary differential equations II, stiff and differential algebraic problems, (1993), Springer-Verlag New York [6] Iserles, A.; Nørsett, S.P., Order stars and rational approximations to exp(z), Appl. numer. math, 5, 63-70, (1989) · Zbl 0674.65043 [7] Kocic, V.L.; Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, (1993), Kluwer Academic Publishers Dordrecht · Zbl 0787.39001 [8] 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 [9] 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 [10] 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 [11] Wanner, G.; Hairer, E.; Nørsett, S.P., Order stars and stability theorems, Bit, 18, 475-489, (1978) · Zbl 0444.65039 [12] Wiener, J., Differential equations with piecewise constant delays, (), 547-552 [13] Wiener, J., Generalized solutions of differential equations, (1993), World Scientific Singapore [14] Wiener, J., Pointwise initial-value problems for functional differential equations, (), 571-580
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