The effect of nonlinear transformations on the computation of weak solutions

For a nonlinear hyperbolic system, computational methods yield different weak solutions for different forms of the system. An explanation is given of the numerical mechanism by which a scheme selects a particular weak solution and why this mechanism depends not only on the scheme but also on the form of the equations. For the Lax-Friedrichs and Lax-Wendroff schemes, it is shown how a correction term can be added to a transformed system so as to preserve the weak solution. This analysis is illustrated by numerical shock-like solutions of the equations of shallow fluid flow over a ridge.