Gauge Method for Viscous Incompressible Flows

Weinan E
Courant Institute of Mathematical Sciences
New York University

Abstract: We explore the gauge formulation of the incompressible Navier-Stokes equation in terms of an auxiliary field that differs from the velocity by a gauge transformation. The gauge freedom allows us to assign simple and specific boundary conditions for both the auxiliary field and the gauge field, thus eliminating the issue of pressure boundary condition in the usual primitive variable formulation. Although it closely resembles the projection method, the gauge method has the advantage that the gauge Poisson equation (GPE) in the projection step can be solved by standard Poisson solvers including the five-point formula. This is in contrast to the projection method in which the pressure Poisson equation (PPE) has to be solved using special Poisson solvers. We also give applications to non-local variational problems.

This is joint work with Jian-Guo Liu of Temple University.