Mathematical Sciences: Presidential Young Investigator Award

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LeVeque is studying numerical methods for solving nonlinear partial differential equations, particularly those arising in fluid dynamics such as the Navier-Stokes equations or the inviscid Euler equations. The Euler equations are a hyperbolic system of conservation laws with solutions that can involve shock waves and other discontinuities. Special numerical methods are required to determine highly accurate solutions with sharp dis- continuities. LeVeque is studying a class of explicit methods that can be efficiently applied with large time steps. This work involves both algorithmic aspects and extension of the mathematical theory of wave interaction in nonlinear conservation laws. One aspect of this work is the specification of boundary conditions with reasonable time steps for bodies embedded in Cartesian grids. A related question, handled by different techniques, is the proper specification of boundary conditions for numerical methods that solve the incompressible Navier-Stokes equations by fractional steps. A sequence of easier subproblems is solved in each time step that require special nonphysical boundary conditions. One application of the incompressible Navier-Stokes equations is to model wave propogation in the cochlea (inner ear). LeVeque is involved in an ongoing project in this direction with C. Peskin. LeVeque is also studying the interaction of solitary waves in nonlinear partial differential equations such as the KdV equations arising in elasticity.