Currently some of the most difficult problems in computational science involve moving interfaces between flowing or deforming media. Typically partial differential equations must be satisfied on each side of the interface (often different equations on each side) and these solutions coupled through relationships or jump conditions that must hold at the interface. These conditions may be in the form of differential equations on the lower-dimensional interface. Often the movement of the interface is unknown in advance and must be determined as part of the solution. The interface shape may be geometrically complex and may change topology with time. Particularly in three space dimensions, the ability to solve such problems accurately is limited. Exciting research is currently underway in the development of better algorithms, the analysis of the accuracy and stability of such algorithms, and the application of these techniques to specific scientific and engineering problems. The primary goal of this Workshop is to bring together a number of researchers (mostly applied mathematicians) who are working on such methods, to foster interaction and the exchange of ideas.
A wide variety of different approaches are being studied in the context of many different applications areas, and we cannot hope to cover everything in a workshop of this nature. Instead, we have identified several of the more promising approaches in which the organizers each have their distinct expertise. There are obvious common features of the approaches and some overlap of methodology has been developing of late, making now is an ideal time for a selection of international experts to come together and learn from each others' experience. We believe that a BIRS Workshop at this time provides a unique opportunity for the participants to make significant progress and to initiate new, long-term research interactions.
We briefly describe the level set, interface fitting, and immersed interface methods below.
For "level set" methods, one represents an interface as a level set of some function for which an evolution equation must then be derived and solved. In some cases this evolution equation is the only equation that must be solved and this can be done on a uniform grid, with the interface then captured by computing the location of the appropriate level set. Complex geometries and topological changes can often be easily handled with this approach. However, in many cases the evolution equation for the level set must be coupled with other systems of equations being solved on each side of the interface, and issues arise in how best to solve these equations and couple them across the interface. One might also use a fixed grid for these computations, with the interface cutting through the grid, or use a deforming grid.
"Interface fitting" is an approach in which the computational grid deforms in order to follow the motion of an interface. This generally requires the global movement of grid points and raises a variety of issues: how best to deform the grid, how to couple this motion with interface conditions, and how to accurately solve the differential equations on the moving grids, for example.
In "immersed interface" methods, on the other hand, a fixed grid is used, often a uniform Cartesian grid, with the interface cutting through the grid. Standard numerical methods can then generally be used away from the interface, but they require careful modification near the interface to maintain good accuracy and stability in these regions.
A variety of different applications require the solution of interface problems and have motivated considerable work on these methods in recent years. Multiphase flow problems such as interface stability or bubble dynamics give fluid dynamic problems of this nature. Biological fluid dynamics is a rich source of problems with complex geometry and frequently the interaction of fluids with moving elastic structures. The study of blood flow in flexible tubes, cell dynamics and motility, and the functioning of various physiological mechanisms require solving interface problems. Phase change problems such as dendritic solidification or crystal growth lead to problems of this type. Numerical methods of the types described above have been used for all of these problems and others. We believe that this would be a very good time for a workshop which, in addition to engendering broad discussion on numerical methods for general interface problems, would allow people to share their practical experiences and facilitate solution of their specific problems.
Final Report (in PDF format)