ACURA: Acoustics Convection Upstream Resolution Algorithm

A novel multi-dimensional upwind bias formulation is presented. Designed to solve accurately the Euler and Navier-Stokes conservation-law systems on arbitrary grids, this new formulation is developed within the continuous conservation-law systems ahead of any discretization in space and time. This formulation utilizes shock-jump surface integrals, which emerge within integral weak statements for these conservation laws. The mean value theorem is then applied to express these surface integrals in terms of differences of gradients of differentiable entropy solutions. In the continuum, this process yields shock-capturing entropy weak statements with embedded upwind bias that are consistent with the theory of characteristics. These entropy weak statements are then discretized in space through a Galerkin finite element method, an inherently centered dissipation-free discretization. This automatically generates an intrinsically multi-dimensional upstreamed discrete system on any grid without the need for any further upwinding within the discrete equations. The computational solutions crisply capture shocks and reflect available exact solutions.