The solutions to the pre-averaged shock normal schemes are obtained in a reasonably straight forward manner and will not be discussed here.
The Vinas-Scudder approach attempts to minimize the jump conditions in the the Rankine-Hugoniot across a shock. The local plasma number density, momentum density, and energy density can be shown to be constant across a shock a large range of conditions. With inputs of the upstream and downstream magnetic field, plamsa density, bulk flow velocity, and total temperature, the jump condition can be minimized through varience of the shock normal. The actual shock normal should give the optimum minimization
The minimization is performed using a Levenberg-Marquardt nonlinear fitting algorithm to fit the 7 Rankine-Hugoniot jump relations, 8 if the total temperature is known and the pressure equation is included. These are solved for the shock normal polar angles theta and phi. The first step in the solution is to create a surface mapping of the jump conditions across the shock as a function of the shock normal polar angles. The jump is represented as the magnetude of the jumps in the individual equations. This surface mapping surves two purposes. It allows the number and location of local minima to be determined and it allows for a reasonable initial guess at the shock normal.
If the total complement of the field and particle input data is available and all 8 Rankine-Hugoniot equations are used in the solution there is generally only a single minima in the surface plot. However, as is often times the case the total temperature is not known. In this case the pressure jump equation is omitted from the solution which can cause the appearance of two minima in the surface plot. Which represents the correct shock normal can be determined from the pressure jump equation formed from the asymtotic solutions to the Rankine-Hugoniot equations which should be positive in moving from the upstream (low entropy) to downstream (high entropy) side of the shock. The program will produce the pressure jump as a part of the solution.