Riemann Invariant Manifolds for the Multidimensional Euler Equations Part I: Theoretical Development Part II: A Multidimensional Godunov Scheme and Applications

Abstract

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 characteristics 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 Invariant 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 to the traditional method of characteristics in one-dimensional flow. Explicit espressions for the local differential geometry of these manifolds can be found directly from the equations of motion. The local direction and speed of propagation of the waves that these manifolds represent, can be found as a function of the local spatial gradients of the flow. Their geometry is examined, and in particular, their relation to the characteristic surfaces. It turns out that they can be space-like or time-like, depending on the flow gradients. Wave propagation can be viewed as a superposition of these Riemann Invariant waves, whenever appropriate conditions of smoothness are met. This provides a means for decomposing the equations into a set of convective scalar fields in a way which is different and potentially more useful than the characteristic decomposition. The two decompositions become identical in the special case of one-dimellsional flow. This different approach can be used for computational purposes by discretizing the equivalent set of scalar equations. Such a computational application of this theory leads to the possibility of determining the solution at points in spacetime using information that propagates faster than the local characteristic speed, i.e., using information outside the domain of dependence. This possibility and its relation to the uniqueness theorems is discussed.

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