Numerical Boundary Conditions for Wave Propagation and FluidDynamics

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The principal investigator will study various questions involving open boundary conditions for problems in wave propagation and fluid dynamics. The topics of investigation include absorbing boundary conditions for dispersive waves, open boundary conditions for the shallow water equations in a rotating reference frame, and absorbing boundary conditions for the velocity-stress formulation of the elastic wave equation. This work is motivated by the fact that many physical problems are defined on spatial domains that either are unbounded or are so large that it is not feasible to compute numerical solutions over the entire region. In such situations, it is necessary to truncate the domain. One then encounters the problem of finding boundary conditions to impose at the artificial computational boundary. The development of such boundary conditions is the subject of this research. This work is expected to have applications to computer modeling of several kinds of physical phenomena, including large scale motions of the atmosphere and oceans, seismic waves within the earth, and waves on the surface of water. For example, suppose that an oceanographer wants to model the circulation in an ocean. One possibility is to compute the relevant physical quantities on an array of grid points distributed over the entire ocean. If the circulation is needed for the entire ocean, this is what must be done. On the other hand, if the circulation is needed only on a relatively small portion, such as the continental shelf, then the above procedure is far too costly. Instead, one would prefer to compute the circulation only on the region of real interest. In such a computation, the region is bounded partly or entirely by a boundary that is "open" instead of physical. Telling the solution what to do at such a boundary is rather tricky, and an incorrect choice of boundary conditions can seriously corrupt the solution that is computed. The goal of the present research is to develop effective open boundary conditions for the physical situations mentioned above. Open boundary conditions for seismic waves would be useful to geophysicists who use seismic methods to search for geologic structures that could contain petroleum, and open boundary conditions for water waves would be useful to engineers who study the effects of wave actions on coastal and offshore structures.