This directory contains codes that were used for the "new temporal
integration schemes" appearing in

  Ian M. Mitchell, "The Flexible, Extensible and Efficient Toolbox of
  Level Set Methods," submitted to the Journal of Scientific Computing
  in February 2007; revised July 2007.

This directory contains Matlab m-files implementing and testing a
variety of CFL constrained explicit Runge-Kutta schemes for time
integration of PDE systems using version 1.1 of the Toolbox of Level
Set Methods by Ian M. Mitchell, which can be downloaded from

	 http://www.cs.ubc.ca/~mitchell/ToolboxLS

The files perform the following tasks.

1) odeCFLsp: Specifies the alpha-beta parameters for a number of
   different Runge-Kutta schemes that have appeared in the literature.
   Included are the standard schemes from Shu & Osher 1988 (which are
   also part of the standard ToolboxLS download in the form of odeCFLn
   for n = 1, 2, 3) as well as several other schemes.  See the help
   entry for the sources and details.

2) odeCFLab: Implements a general CFL constrained Runge-Kutta
   time-dependent ODE integrator specified by alpha-beta parameters.
   Normally this code is not called directly, but is called by
   odeCFLsp (which provides the alpha-beta parameters).

3) frontTestRK: A modified version of convectionDemo from ToolboxLS
   which is designed to exercise the various RK schemes and determine
   their error on several related simple implicit surface
   computations.  This file was used to generate the growing and
   spinning rectangle shown in figure 5 of the paper.

4) convergeFrontTestRK: A script file to test convergence rates of the
   various RK schemes using frontTestRK on a variety of different grid
   sizes.  This file (run multiple times with different settings in
   frontTestRK) was used to generate the convergence plots shown in
   figure 6 of the paper, as well as collect the timings in table 1.

5) convergeODERK: Essentially a script file to test convergence rates
   of the various RK schemes using two simple ODEs.  The benefit of
   using ODEs is that it is quick and there is no confounding factor
   from the spatial approximation scheme in a PDE.  This code was
   written to conclusively demonstrate that odeCFLab was achieving the
   appropriate rates of convergence on both time-dependent and
   state-dependent ODEs.  Note that because of the type of
   time-dependence and choice of timestep size in frontTestRK, that
   example cannot demonstrate time-dependent convergence rates by
   itself.

The first two routines (odeCFLsp and odeCFLab) will likely migrate
into the kernel of ToolboxLS in future versions.  The other routines
are for testing and examples.

The examples require the presence of an <addPathToKernel> script
file TWO directories above the one in which this README exists.
Normally this script can be a copy of the script
<Examples/addPathToKernel> from the toolbox download (after it has
been modified to point to your own Toolbox installation).

Questions, comments or bug reports can be sent to mitchell@cs.ubc.ca

Ian Mitchell
University of British Columbia
July 2007
