The talk will give an overview of a recently proposed general methodology
for the discretization of elliptic and hyperbolic equations on arbitrary
domains, possibly with moving boundaries, embedded in a rectangular region
which is discretized on a Cartesian mesh. The domain D, embedded in the
unit cube in two or three dimensions, is suitably defined by a level set
function. The unit cube is then discretized by a regular Cartesian grid.
Three kinds of grid points are identified: interior points (grid points
inside D), ghost points (grid points outside of D, with at least a neighbor
inside D), and exterior points (the rest of the grid points). The
(stationary or time dependent) equations are discretized on the interior
points, while boundary conditions are used to define equations on the ghost
additional points.
Elliptic equations. This approach provides an innovative techniques for a
unified treatment of elliptic problems with mixed BC (Dirichlet, Neumann,
Robin), using ghost-point approach and level set, in conjunction with
multigrid. Applications to volcanology will be shown.
Hyperbolic equations. Euler equations of gas dynamics and shallow water
equations are solved in domains with fixed and moving boundary. In
particular, boundary conditions for Euler equations are discussed.
Note: the speaker intends to make the talk accessible to a general
math audience.
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