HOW TO RENDER 2ND-ORDER ACCURATE TIME-STEPPING ALGORITHMS 4TH-ORDER ACCURATE WHILE RETAINING THE STABILITY AND CONSERVATION PROPERTIES

For a fairly general class of second order accurate algorithms, a sub- stepping procedure is described which achieves fourth order accuracy w hile retaining the stability, conservation properties and implementati on of the underlying second order method. Two features render the prop osed approach especially attractive from a computational perspective o ver conventional fourth order methods. First, the improved (fourth ord er) accuracy is attained merely by two additional residual evaluations , with no changes in the implementation of the underlying second order method. In particular, no additional storage or extra computations of high order gradients are required. Second, for Hamiltonian systems, t he stability properties and possible conservation properties of the un derlying second order method, such as energy, momentum and symplectic character, are preserved. The new technique is applied to a number of standard algorithms for elastodynamics, structural and rigid body dyna mics. In particular, when applied to the recently proposed (second ord er accurate) energy-momentum method, this approach results in a fourth order accurate energy, linear and angular momentum conserving algorit hm for general Hamiltonian systems.