New classes of optimal splitting-up schemes for the computation of transient flows

Classes of explicit second-order splitting-up disintegration schemes are first presented for solving the three-dimensional Euler equations, and full time-dependent compressible viscous Navier-Stokes equations of fluid mechanics. Some results, obtained for transient three-dimensional flows, are given. They show that these schemes are well-adapted to compute such flows. New classes of 'optimal' second order schemes are then examined: an explicit 'dissipation-optimal' class (defined as dissipative with minimal dissipation), and 'time-optimal' classes (in the sense that they allow optimal time steps). The first class yields a shock profile without oscillations for an unsteady shock problem. The second classes use explicit, implicit or hybrid (implicit or explicit), and and hybrid disintegration-hopscotch schemes. The 'time-optimal' explicit method should be quicker than the other existing explicit splitting-up methods, and as efficient as implicit methods for solving inviscid or viscous transient flows.