Riemann invariant manifolds for the multidimensional Euler equations

A new approach for studying wave propagation phenomena in an inviscid gas i s presented. This approach can be viewed as the extension of the method of characteristics to the general case of unsteady multidimensional flow. A fa mily of spacetime manifolds is found on which an equivalent one-dimensional (1-D) problem holds. Their geometry depends on the spatial gradients of th e flow, and they provide, locally, a convenient system of coordinate surfac es for spacetime. In the case of zero-entropy gradients, functions analogou s to the Riemann invariants of 1-D gas dynamics can be introduced. These ge neralized Riemann invariants are constant on these manifolds and, thus, the manifolds are dubbed Riemann invariant manifolds (RIM). Explicit expressio ns for the local differential geometry of these manifolds can be found dire ctly from the equations of motion. They can be space-like or time-like, dep ending on the flow gradients. This theory is used to develop a second-order unsplit monotonic upstream-centered scheme for conservation laws (MUSCL)-t ype scheme for the compressible Euler equations. The appropriate RIM are tr aced back in time, locally, in each cell. This procedure provides the state s that are connected with equivalent 1-D problems. Furthermore, by assuming a linear variation of all quantities in each computational cell, it is pos sible to derive explicit formulas for the states used in the 1-D characteri stic problem.