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Abstract. Quasi-random (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the s-dimensional unit cube is measured by its discrepancy, which is of size (log N) N-Ifor large N, as opposed to discrepancy of size (log log N) 1/2 N -1/2 for a random sequence (i.e., for almost any randomly chosen sequence). Several types of discrepancies, one of which is new, are defined and analyzed. A critical discussion of the theoretical bounds on these discrepancies is presented. Computations of discrepancies are presented for a wide choice of dimension s, number of points N, and different quasi-random sequences. In particular for moderate or large s, there is an intermediate regime in which the discrepancy of a quasirandom sequence is almost exactly the same as that of a randomly chosen sequence. A simplified proof is given for Woniakowski's result relating discrepancy and average integration error, and this result is generalized to other measures on function space.

The moment closure method of Levermore applied to the Boltzmann equation for rare ed gas dynamics leads to a hierarchy of symmetric hyperbolic systems of partial di erential equations. The Euler system is the rst member of this hierarchy of closures. In this paper we investigate the next member, the ten moment Gaussian closure. We rst reduce the collision term to an integral which may be explicitly evaluated for the special case of Maxwell molecular interaction. The resulting collision term for this case is shown to be equivalent to the term obtained by replacing the Boltzmann collision operator with the BGK approximation. We then analyze the Gaussian system applied to the canonical ow problem of a stationary planar shock. An analytic shock pro le for the Gaussian closure is derived and compared with the numerical solutions of the Boltzmann and Navier-Stokes equations. The results show reasonable agreement for weak shocks and close agreement between the downstream Gaussian and Navier-Stokes pro les. The results also suggest what may be expected from higher moment closure systems. In particular, the presence of discontinuities in the solution are seen not to prohibit the development of signi cant pro les.

The compressible Euler equations possess a family of generalized entropy densities of the form ρf (σ), where ρ is the mass density, σ is the specific entropy, and f is an arbitrary function. Entropy inequalities associated with convex entropy densities characterize physically admissible shocks. For polytropic gases, Harten has determined which ρf (σ) are strictly convex. In this paper we extend this determination to gases with an arbitrary equation of state. Moreover, we show that at every state where the sound speed is positive (i.e., where the Euler equations are hyperbolic) there exist ρf (σ) that are strictly convex, thereby establishing the converse of the general fact that the existence of a strictly convex entropy density implies hyperbolicity.