Peta-scale computers are monstrous (literally speaking) compared to the workstations or clusters most scientists use when developing models, methods and codes. Yet, several applications that sustain a petaflop/s under production condition on general-purpose supercomputers have emerged within months of the existence of systems that could sustain a petaflop/s on the Linpack benchmark. In this presentation we will review the developments that lead to the much faster than expected advent of "sustained petaflop/s computing". The experience of integrated teams that unify domain knowledge, applied mathematics and computer science, as well as the challenge to develop applications to solve models of complex systems call for a nimble programming environment. We will discuss the use of generic libraries and the concept of just in time abstraction that enable several of today's sustained petaflop/s applications.

Many macroscopic plasma models, arising in astrophysics, solar physics, electrical and aerospace engineering are modeled with the equations of ideal Magneto-HydroDynamics (MHD). These constitute a non-linear system of conservation laws in several space dimensions, augmented with the divergence constraint. Efficient numerical simulation for the MHD equations is quite complicated on account of the lack of strict hyperbolicity or convexity of the equations and due to the divergence constraint. We design robust finite volume schemes for an equivalent Godunov-Powell form of the MHD equations, based on HLL type approximate Riemann solvers for computing numerical fluxes. The Godunov-Powell source term is discretized in an upwind manner using the local wave structure. Positivity preserving non-oscillatory high-order reconstructions are also proposed. The resulting schemes are tested on a large number of benchmark numerical experiments in two and three space dimensions. The talk is based on joint work with F. Fuchs, A. D. McMurry, N. H. Risebro (CMA, Univ. of Oslo, Norway) and K. Waagan (CSCAMM, Univ. of Maryland, U.S.A).