## Solving Navier-Stokes Equations with Sequential Regularization Methods

**Abstract:**

We introduce the notion of partial differential-algebraic equations (PDAEs) by discussing the Navier-Stokes equations for an incompressible viscous fluid. We note that there are two concerns for solving such problems: the involvment of PDEs rather than ODEs, and the presense of algebraic constraints on these PDEs.

To deal with the first concern, we introduce solving PDEs numerically with the method of lines and finite differences with the help of an example based on the onedimensional heat equation.

After discussing some of the difficulties involved in the numerical solution to differential-algebraic equations (DAEs) , we introduce the sequential regularization method (SRM) as a method for solving certain types of DAE problems. We present a SRM for Navier-Stokes equations as a possible solution to the second concern above. A possible spatial discretization of the SRM for Navier-Stokes equations is then demonstrated.

We introduce a predicted sequential regularization method (PSRM) which significantly improves computational time over the SRM.

We test the SRM and PSRM on two problems: a problem with an exact solution and a physically motivated cavity problem. Based on these results, a discussion of the various CFL restrictions of these methods is presented.

**Author:** Colin B. Macdonald

**Advisor:** Raymond J. Spiteri

**Download:** math3743_project_solving_NS_with_SRM