3.3 Roe's scheme

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3.3 Roe's scheme

A very pragmatic and successful approach has been taken by Roe [9]. The exact solution to the linearized Riemann problem

is constructed. This solution consists of two simple waves since both characteristic fields are now linearly degenerate (i.e. the relation becomes an equality). The interface flux can be expressed by incrementing across each wave either from the right or from the left state. Blending both formulas leads to the well known flux function:

where

and R,R^-1 are the diagonalization matrices and Lambda the diagonal eigenvalue matrix.

The key point of Roe's scheme is the definition of a sensible linearization. The consistency requirements can be put forth as three conditions

Condition (iii), in particular, assures that the numerical flux is exact in the case of a stationary shock situated in between nodes ()_L and ()_R.

In order to find the matrix A according to criteria (i)-(iii), we define the following parameter vector

which lets us rewrite the variable vector as

and the flux vector as

Using the mean value theorem,

the jump of variables can be expressed as jump(Q)=B*jump(z), where

Analogous, the jump of the fluxes is written as jump(F)=C*jump(z), where

Inserting matrix and matrix into condition (iii) gives:

By comparing the relation with equation we notice that Roe's linear matrix for the shallow water equations is equivalent to the jacobian J of the continous system under the following change of variables:

where Q^Roe is the vector of variables averaged as follows:

The above result has been obtained by Glaister [10]. The situation is similar in the case of the Euler equations, as noted in the original paper by Roe [9]. For more complex systems. e.g. those arising in the context of two-phase flows [11] or resulting from a statistical turbulence model [12], Roe's linear matrix cannot be obtained from the jacobian by a simple variable transformation. In these latter cases, it is sometimes appropriate to resort to a scheme not relying upon the somewhat restrictive condition (iii) of Roe. One prominent example is the following "VFRoe" scheme.

3.3 Roe's scheme

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