TABLE OF CONTENTS In this chapter we will introduce the history and most basic concepts associated with high-resolution methods (see also introduction in Chap. 9). These methods got their start through the parallel efforts of Boris in developing the flux-corrected transport (FCT) method and van Leer's direct extension of Godunov's work to higher order. [1] These efforts were then formalized principally through the numerical analysis of Harten whose total variation diminishing (TVD) methods codified the techniques we refer to as high-resolution methods.
Before discussing Godunov-type methods and their history it is useful to clarify what exactly a high-resolution method is (and is not). Simply put, high-resolution schemes employ some sort of nonlinear "recipe" to control oscillations in the solution. This is opposed to methods that are linear using the same differencing stencil everywhere regardless of the solution. Thus, high-resolution combines two elements: nonlinear differencing where the stencil is dependent on the local solution and the use of this nonlinearity to control oscillations. Some high-resolution schemes attempt to totally eliminate oscillations while others simply minimize them.
Historically speaking, one might consider high-resolution methods to be the second generation of numerical methods for hyperbolic PDEs. The first generation of methods were linear. They were typified by either making the choice of an oscillatory solution such as that produced by the Lax-Wendroff scheme, or a diffusive solution such as that produced by upwind or Lax-Friedrichs schemes. The second generation methods achieve high-resolution by adaptively using the first generation's methods...
