Navier-Stokes solver using Green's functions
- Series
- PDE Seminar
- Time
- Tuesday, April 12, 2011 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 255
- Speaker
- Prof. Divakar Viswanath – University of Michigan – divakar@umich.edu
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].