Numerical solution of the Cauchy problem for a system of differential-algebraic equations using implicit Runge-Kutta methods with a variable integration step (Q1276000)

The author investigates the numerical solution of systems of differential algebraic equations of the type \[ x'(t) = g(x(t),y(t)),\quad y(t) = f(x(t),y(t)),\quad x(0) = x^0,\quad y(0) = y^0, \] with \(t\in[0,T]\), \(x\in \mathbb{R}^m\), \(y\in \mathbb{R}^n\), \(g\: D\subset \mathbb{R}^{m+n}\to \mathbb{R}^m\), \(f\: D\subset \mathbb{R}^{m+n}\to \mathbb{R}^n\), \(y^0 = f(x^0,y^0)\). Combined numerical methods with variable integration step are proposed in terms of implicit Runge-Kutta methods and the Newton method. Approximation convergence conditions and estimates of the solution preciseness are established. No examples are considered.