One dimensional characteristic codes enjoy a very special
simplicity due to the two preferred sets (ingoing and outgoing)
of characteristic null hypersurfaces. This eliminates a source of
gauge freedom that otherwise exists in either two or three
dimensional characteristic codes. However, the manner in which
characteristics of a hyperbolic system determine domains of
dependence and lead to propagation equations for shock waves is
exactly the same as in the one dimensional case. This makes it
desirable for the purpose of numerical evolution to enforce
propagation along characteristics as extensively as possible. In
basing a Cauchy algorithm upon shooting along characteristics,
the infinity of characteristic rays (technically,
bicharacteristics) at each point leads to an arbitrariness which, for a practical
numerical scheme, makes it necessary either to average the
propagation equations over the sphere of characteristic
directions or to select out some preferred finite subset of
propagation equations. The latter approach has been successfully
applied by Butler [48] to the Cauchy evolution of 2dimensional fluid flow but there
seems to have been very little followup along these lines.
The formal ideas behind the construction of two or three
dimensional characteristic codes are the same, although there are
some additional technical complications in three dimensions
associated with a nonsingular choice of angular coordinates for
the null rays. Historically, most characteristic work graduated
first from 1D to 2D because of the available computing power.

Characteristic Evolution and Matching
Jeffrey Winicour
http://www.livingreviews.org/lrr19985
© MaxPlanckGesellschaft. ISSN 14338351
Problems/Comments to
livrev@aeipotsdam.mpg.de
