Projects

The Navier-Stokes equations, through the conservation of mass and momentum, form a system of differential equations that describe the evolution of a fluid. In many cases, such as water, the behaviour of the fluid is essentially incompressible, and consequently the conservation of mass reduces to a divergence free condition on the fluid velocity. Even in simple domains, the numerical solution of the resulting incompressible Navier-Stokes equations is difficult, and there is often a trade off between accuracy and computational efficiency. In work with R. Rosales [arxiv:1011.3589], we develop a method for solving the Navier-Stokes equations, both efficiently and accurately. The review paper by Guermond, Minev and Shen, (Comput. Methods Appl. Mech. Engrg. 195 (2006) 6011-6045) provides a nice background on the difficulties associated with projection and fractional-step methods. For instance they illustrate numerical boundary layers in the computation of the pressure field even for simple geometries such as a square.
The above plot shows the error in the pressure produced by our scheme for the same test as in the paper by Guermond et al. Note the error in the pressure field converges uniformly with no numerical boundary layers. The total amplitude of the error may be reduced by decreasing the grid spacing.
In addition, we perform similar tests in more complicated domains, such as flow around a circle. Here the domain is immersed in a regular grid. The above figure shows the pressure, and does not have a numerical boundary layer.


Another advantage with the approach, is that the resulting equations may be discretized (in theory) to any order in space and time, provided that the discretization scheme is stable and consistent. In our tests we take a finite difference scheme which is second order in space. The above plot shows the error (in L-infinity) for the pressure (circles) and velocity field (squares) in a domain with a hole removed.

Lastly, there is an additional advantage when analyzing and developing semi-implicit schemes. In simple time splitting schemes with an implicit treatment of viscosity and explicit treatment of the Stokes pressure, the current approach exactly conserves the velocity divergence condition. For instance, no projection step onto the space of divergence free fields is required since the velocity is always divergence free.

Although the simplicity of the approach is comparable to the projection method, two practical problems arise. Specifically, future work includes:

A weak (finite element) framework.

Developing spatial discretization methods for the implicit treatment of the viscosity term in arbitrary geometries.