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Van Leer Flux Splitting Scheme

    The Van Leer (1979) flux vector splitting is one of a large body of similar techniques.  Since a general fluid flow contains wave speeds that are both positive and negative (so that eigenvalue information can pass both upstream and downstream), the basic idea behind all of these techniques is that the flux can be split into two components so that each may be properly discretized using relatively upwind stencils to maintain stability and accuracy.  There are many possible ways to split the flux term by basing the splitting on the eigenvalue structure or some similar representation of the flow.
    From Tannehill, Anderson, and Pletcher (1997), the particular Van Leer flux splitting sought to correct some problems found at sonic and stagnation points for an earlier splitting called Steger-Warming.  The Van Leer splitting offers both zero and first order continuity through sonic and stangation points, thus correcting this problem.  This particular variation of flux splitting is based on Mach number splitting (Laney 1998).  The Mach number for supersonic flow is simply the full scalar Mach number in the downwind direction, and zero in the upwind direction.  For subsonic flow, the Mach number is slightly more complex, and is given in eqn. (21).

(21)

Here, the Mach number is approximated using a second order polynomial.  This insures zero and first order continuity at the sonic points, M = +1 and M = -1.  This can be found by adding the "plus" and "minus" contributions together at each of these two points.  From this, the flux can be defined as a result.  In a supersonic flow case, the flux is likewise equal to the full flux from the upwind side (with 0 contribution from the downwide side).  Although notation can be somewhat confusing here, a simple convention is given which will be used from this point on.  For a flow moving from left to right, a positive sense flux will also move from left to right.  A negative sense flux will move from right to left.  Thus, to preserve proper upwind stenciling, a variable or flux term from the left should be stenciled from the left, and a variable or flux from the right should be evaluated using points from the right.  Thus, the term "left" and "+" will be used interchangeably, and the term "right" and "-" will be used interchageably.
    From this, for positive supersonic flow from left to right (M > 1.0), the full flux will be a function of variables from the left or plus.  For a right to left supersonic flow (M < -1.0), the full flux will be a function of variables solely from the right, or minus.  For a subsonic flow, the flux is a summation of both plus and minus flux contributions, with each component appropriately stenciled.  This arrangement is given in eqn. (22).

(22)

The flux components are defined as given in eqn. (23), which just result from the Mach number splitting above.  Van Leer put these components in terms of Mach number to make the analysis more consistent.

(23)

It is worthwile to note for implementation purposes that there is a lot of repitition in the split flux definition.  It is a simple substitution to define some parameters that can be used to avoid redundant calculations and increase calculation efficiency.  This abbreviated form is given in eqn. (24).

with...

(24)

Likewise, the full flux, used in the supersonic conditions, can be defined in terms of Mach number to match the split fluxes.  This is given in eqn. (25).

(25)

    Explicit time integration methods may simply utilize eqns. (24) and (25) to calculate the flux and corresponding residual dependent on the condition given in eqn. (21).  The remaining topic to consider is that for implicit methods, a Jacobian matrix must be calculated for the left hand side matrix.  This Jacobian requires the derivatives of the flux values with respect to each conservative variable at a given cell center (solution) point.  This Jacobian is required for all the solution points that exist, forming, in general, a penta-diagonal system of matrices.  This is somewhat difficult directly since the flux is calculated in terms of a reconstructed conservative variable set at each individual face.  Hence, the chain rule is applied to get the final derivative for the flux with respect to the appropriate solution point conservative variable.  This construction is shown in eqn. (26).

(26)

    First, the Jacobian of the full flux term is given since it is relatively straight forward and simple.  The full flux is given  in eqn. (27) completely in terms of conservative variables since these are the derivatives that must be known.  

(27)

The Jacobian matrix follows by taking the relatively straight-forward derivative of each term with respect to each variable.  This is given in eqn. (28).

(28)

    Taking a derivative in the split flux case is somewhat more complicated.  It is easiest to take derivatives of fa and fb, and then use the chain rule on eqn. (24) to get the face Jacobian.  It happens that a relatively common form appears in all the derivative terms, and thus is defined purely to make the problem more manageable.  Calculating it once in the implementation of the Jacobian programming also offers a small increase in efficiency.  This commonly seen term is given in eqn. (29) before providing the derivative terms.

(29)

The derivatives of fa with respect to the conservative variables are given in eqn. (30).

(30)

Similarly, the derivatives of fb with respect to the conservative variables are given in eqn. (31).

(31)

This can be put all together by using the chain rule on eqn. (24) to get the Jacobian matrix with respect to the face variable for either the plus or minus split flux in terms of derivatives of fa and fb.  The needed Jacobian for the implicit schemes can be found by using eqn. (32) coupled with eqns. (30) and (31) as well as eqn. (28).

(click image to enlarge view)

(32)

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