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Randall J. LeVeque
Applied Mathematics and Mathematics
University of Washington
rjl@amath.washington.edu
http://www.amath.washington.edu/~rjl/
Many biophysical problems give rise to fluid dynamics problems in regions which have very complicated geometry. Moreover the boundaries are often moving in a manner which is coupled with the fluid motion. Numerical methods which require a computational grid that conforms to the boundary are difficult to use. It is often much more efficient to use "Cartesian grid" methods in which the computational grid is rectangular and the boundary cuts through the grid. For biophysical problems the best known method of this type is Peskin's Immersed Boundary Method. I will give a brief summary of some other Cartesian grid approaches that may also be useful for such problems. I will discuss both finite difference methods, in which the solution is represented pointwise by values at a set of grid points, and finite-volume methods, in which the solution is represented as integrals over grid cells. The latter class of methods is particularly valuable for problems where exact conservation of some quantity is important.
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