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The Exact Iterative Riemann Solver

Let us start by presenting the exact iterative solver for one dimensional gas dynamics equations. In order to specify the one-dimensional unsteady state of a perfect gas, we need three independent variables and two constants. The two constants which are generally considered to specify the state of the gas are the universal gas constant appearing in the state equation and the ratio of the specific heats . Several different variables can be defined and several sets can be used. The one-dimensional conservative set of variables and its physical flux function are

The physical state of the gas can be described by the previous set of variables as well as by the primitive set :

Let us define a temperature from the equation of state and the sound speed from the isentropic definition . The state of the gas can still be defined by the set of variables :

The initial discontinuity in Riemann problems for Euler equations will evolve in time breaking into leftward and rightward moving waves separated by a contact surface, see Anderson (1982) [12].

These propagating waves can be either shocks or rarefaction waves depending on the initial data. The available combinations can produce four different wave patterns which are self-similar, i.e. they depend only on .

There is also a fifth pattern which is essentially of theoretical interest because it is the limit of the perfect gas equations at zero pressure and temperature and real-gas effects prevent its practical realization. This last case is defined by two rarefaction waves propagating rightward and leftward confined by two contact surfaces separated by a vacuum region in the middle.

A complete solution of the one-dimensional Riemann problem requires the determination of the types of the waves, their relative strenghts, and the flow properties in each region between the waves and the contact surface. In the formulation of Gottlieb and Groth (1978) [7], the state of the gas is described by the set of variables . The unknowns are obtained solving the three conservation laws of mass, momentum and energy and the equation of state, which reduces to the well-known Rankine-Hugoniot relations across shocks and to the well-known isentropic characteristic equations across rarefaction waves. The flow-state is described by the initial left and right states and by the six state variables and characterizing the solution on the two sides of the contact surface. The known left state is connected to the unknown state by the non-linear algebraic relations describing the leftward moving wave. In the case of a compression wave (or a shock), i.e. , the shock-relations are employed; in the case of a rarefaction wave, i.e. , rarefaction-waves relations are employed. The right state is similarly connected to the unknown state to the right of the contact surface. Finally, across the contact surface, the following relations hold:

Appling this relations, the set of equations describing the Riemann problem may be reduced to a single non-linear algebraic equation in one unknown, the common flow velocity in the intermediate states , for any particular wave pattern. This equation is generally implicit in the unknown or and an iterative solution is provided by a standard Newton iterative method. The initial guess and is provided by considering compression waves instead of shocks. This solution is exactly the same as for the pattern where two isentropic rarefaction waves separated by a contact discontinuity are present. The values of and are given by the following relations:

where is a useful ratio of pressures and sound speeds and are the Riemann invariants for left- and right-ward moving waves:



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