LAPPAS, T., LEONARD, A., and DIMOTAKIS, P. E. 1994 "Riemann Invariant Manifolds for the Multidimensional Euler Equations. Part I: Theoretical Development," GALCIT FM94-6 report.


A new approach for studying wave propagation phenomena in an inviscid gas is presented. This approach can be viewed as the extension of the method of charac-teristics to the general case of unsteady multidimensional flow. The general case of the unsteady compressible Euler equations in several space dimensions is examined. A family of spacetime manifolds is found on which an equivalent one-dimensional problem holds. Their geometry depends on the spatial gradients of the flow, and they provide, locally, a convenient system of coordinate surfaces for spacetime. In the case of zero entropy gradients, functions analogous to the Riemann invariants of 1-D gas dynamics can be introduced. These generalized Riemann Invariants are constant on these manifolds and, thus, the manifolds are dubbed Riemann Invari-ant Manifolds (RIM). In this special case of zero entropy gradients, the equations of motion are integrable on these manifolds, and the problem of computing the solution becomes that of determining the manifold geometry in spacetime. This situation is completely analogous to the traditional method of characteristics in one-dimensional flow.