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- Giving to Math

Event Type:

Applied Mathematics and Computation Seminar

Date/Time:

Friday, May 11, 2018 - 12:00 to 13:00

Location:

STAG 210

Event Link:

Local Speaker:

Abstract:

During the development of a Galerkin numerical method, a beginning step is to multiply by a test function and integrate by parts. The goal of the present talk is to show that this process can be used to overcome a standard difficulty that is encountered with the shallow water equations, a system of partial differential equations that arises in various circumstances in geophysical fluid dynamics. In the case of variable bottom topography, the usual formulation of this system includes a static forcing term that must be implemented carefully in order to avoid spurious forcing. Such forcing could, for example, cause the fluid to start moving in circumstances when it should remain at rest, and in that event the forcing is not “well-balanced”. With an alternative that is developed here for Discontinuous Galerkin (DG) methods, the plan is to re-visit the derivation of the shallow water equations from physical principles. At a certain step in that derivation, proceed directly to a weak Galerkin form by multiplying by a test function and integrating over the water column that lies on a given horizontal grid cell. The resulting formulation is automatically well-balanced. This result requires a modest hypothesis about the method that is used to interpolate the pressure at cell edges, such as via a Riemann problem; this hypothesis is satisfied by a method that will be described in the talk. The ideas developed here can be generalized to a multi-layer stratified fluid.