The θ-methods in numerical solution of systems of differential equations with two delay terms

This paper deals with the numerical solution of initial value problems for systems of differential equations with two delay terms. We investigate the stability of adaptations of the (θ-methods in the numerical solution of test equations
$\cases{u'(t)=a_{11}u(t)+a_{12}v(t)+b_{11} u(t-\tau_1)+b_{12}v(t-\tau_2,\cr v'(t)=a_{12}u(t)+a_{22}v(t)+b_{21} u(t-\tau_1)+b_{22}v(t-\tau_2),}\quad t > 0$
, with initial conditions
$\cases{u(t)=u_0(t),\cr v(t)=v_0(t),}\quad t \leqslant 0.$
where a
ij
, b
ij
ε C, τ
j
> 0, i, j = 1,2, and u
0
(t), v
0
(t)are continuous and complex valued. Sufficient conditionsfor the asymptotic stability of test equation are derived. Furthermore, with respect to an appropriate definition of stability for the numerical method, it is proved that the linear θ-method is stable if and only if 1/2 < θ < 1 and the one-leg θ-method is stable if and only if θ = 1.