A class of rosenbrock-type schemes for second-order nonlinear systems of ordinary differential equations

AbstractWe develop a class of generalized Rosenbrock-type schemes for second-order nonlinear systems of ordinary differential equations. We convert the second-order systems to equivalent first-order form, and then employ the square of the Jacobian. These methods when applied to a linear time-invariant system Uu + AU = 0, reproduce a class of schemes given by Baker and Bramble that are derived from a particular class of rational approximations to the exponential with denominators of the form (1 − γ2z2)s for an s-stage method. For our problem, then, an s-stage scheme requires the solution of 2s linear algebraic systems at each time step, with the same real matrix. We employ the theory of Butcher [1–4] series to develop order conditions and then present specific examples of fourth-order methods which are unconditionally stable by appropriate choice of parameter γ2. Numerical results, confirming the rate of convergence, are presented