Navier-Stokes solver using Green's functions

PDE Seminar
Tuesday, April 12, 2011 - 11:00
1 hour (actually 50 minutes)
Skiles 255
University of Michigan
The incompressible Navier-Stokes equations provide an adequate physical model of a variety of physical phenomena. However, when the fluid speeds are not too low, the equations possess very complicated solutions making both mathematical theory and numerical work challenging. If time is discretized by treating the inertial term explicitly, each time step of the solver is a linear boundary value problem. We show how to solve this linear boundary value problem using Green's functions, assuming the channel and plane Couette geometries. The advantage of using Green's functions is that numerical derivatives are replaced by numerical integrals. However, the mere use of Green's functions does not result in a good solver. Numerical derivatives can come in through the nonlinear inertial term or the incompressibility constraint, even if the linear boundary value problem is tackled using Green's functions. In addition, the boundary value problem will be singularly perturbed at high Reynolds numbers. We show how to eliminate all numerical derivatives in the wall-normal direction and to cast the integrals into a form that is robust in the singularly perturbed limit. [This talk is based on joint work with Tobasco].