Colloquium

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.