3 - The wave-propagation algorithm

The multi-dimentional wave-propagation algorithm described in [2] is a high-resolution method that is second order accurate on smooth solutions while maintaining sharp discontinuities through the use of slope-limiters; While based on ideas developed for hyperbolic systems of conservation laws

in the context of shock capturing, these methods apply in more general framework that allows their application to other hyperbolic system which are not in conservation form. Variable-coefficient hyperbolic systems of the form

arise, for example, in studying acoustics or elasticity in heterogeneous materials with varying material properties.

In one space dimentional, the wave-propagation algorithm is based on solving a Riemann problem at each interface between grid cells, and using the resulting wave structure to update the solution in the grid cell to each side. This is, of course, the basis for a host of methods for conservation law, dating back to godunov method.

The wave-propagation algorithms are based on using the waves directly to update cell values, including second order corrections with wave limiters. For conservation laws these methods can be rewritten in conservation form by defining flux functions in terms of the waves, but they are implemented in a way that allows their application to hyperbolic problems not in conservation form, for which there is still a well-defined wave structure but not flux function. As an example, we have the advection equation wih variable velocity,

This equation, sometimes called the "color equation", is not in conservation form. The value of q is constant along characteristics but the integral of q is not conserved.