Riemann Invariant Manifolds for the Multidimensional Euler Equations

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. A family of spacetime manifolds is found on which an equivalent one-dimensional (1-D) 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). Explicit expressions for the local differential geometry of these manifolds can be found directly from the equations of motion. They can be space-like or time-like, depending on the flow gradients. This theory is used to develop a second-order unsplit monotonic upstream-centered scheme for conservation laws (MUSCL)-type scheme for the compressible Euler equations. The appropriate RIM are traced back in time, locally, in each cell. This procedure provides the states that are connected with equivalent 1-D problems. Furthermore, by assuming a linear variation of all quantities in each computational cell, it is possible to derive explicit formulas for the states used in the 1-D characteristic problem.