A Convergent Numerical Scheme for the Compressible Navier-Stokes Equations

In this paper, the three-dimensional compressible Navier-Stokes equations are considered on a periodic domain. We propose a semi-discrete numerical scheme and derive a priori bounds that ensures that the resulting system of ordinary differential equations is solvable for any h > 0. An a posteriori examination that density remain uniformly bounded away from 0 will establish that a subsequence of the numerical solutions converges to a weak solution of the compressible Navier-Stokes equations. 1. Background The compressible Navier-Stokes equations have received considerable attention and yet strong well-posedness results are lacking. For the incompressible counterpart, weak solutions were proven to exist in [Ler34] and for the isentropic compressible equations in [Lio98]. For the full compressible equations a particular form of weak solutions were derived in [Fei04]. In [FN12] these solutions were shown to satisfy a so-called ”weak-strong uniqueness”. The latter implies that as long as a weak solution has sufficient regularity, i.e., it is a strong solution, it is unique. The weak solutions they derive satisfy the usual continuity and momentum equations weakly. However, the energy equation is replaced with an entropy inequality and the constraint that the total energy is conserved. If the solutions, derived in [Fei04] and [FN12], are sufficiently smooth, they satisfy the energy equation in the usual sense. However, when not smooth, these weak solutions may be different from weak solutions satisfying the standard continuity, momentum, and energy equations weakly, whose existence is subject to investigation in this paper. Establishing existence of solutions is of paramount importance for numerical simulations. The information provided from well-posedness results helps in the design of effective numerical schemes. Without such knowledge, any and all simulations are uncertain. There is no way to tell whether or not the solution produced by a numerical scheme is an approximation of the true solution. Of great importance to numerical simulations is also robustness, in the sense that the scheme always produces an approximation for reasonably bounded data. Given a numerical solution, it is possible to examine if it lies within the physical range of applicability of the model. In this paper we consider the compressible Navier-Stokes equations in three space dimensions on a periodic domain. The system includes a, non-zero but possibly very small, bulk viscosity. (This condition can likely be weakened, which will be discussed later.) We propose a finite difference scheme and derive appropriate a priori estimates and we show that the discrete scheme is solvable on arbitrary fine grids producing a sequence of solutions. Furthermore, the a priori estimates ensure convergence (i.e. existence) of weak solutions, if the density remains uniformly bounded away from 0. Although we can not establish a weak solution in the presence Date: February 26, 2015. 1