M12: Numerical fluid flow on dynamic implicit surfaces

Partial differential equations (PDEs) are essential for modelling and understanding processes in all areas of science.  Fluid flow is one such process and has been modelled by PDEs since the days of Leonhard Euler.  Our work examines fluid flow on curved surfaces, i.e. the PDEs feature differential operators intrinsic to a surface. The study of fluid flow on curved surfaces can be applied to the simulation of water waves on Earth, or thin liquid films on a curved substrate, such as in industrial coating.

The Closest Point Method (CPM) is a set of mathematical principles and associated numerical techniques for solving PDEs on surfaces.  Part of this project is investigating the analytical foundations of the method, as well as its consistency and stability.  On the application side, we explore the CPM regarding surface fluid flow.

Results

We have developed a theoretical foundation of a general closest point framework and have proven the basic principles underlying the CPM. Beyond that, we have defined the entire class of closest point functions that can support these principles and be used in numerical methods.  Regarding fluid flow, we have focused on inviscid hyperbolic test problems.  We have implemented the CPM using two different conservative schemes (for hyperbolic problems): the first-order accurate Lax–Friedrichs scheme and the second-order Nessyahu–Tadmor scheme. We have generated numerical evidence on simple curves in two dimensions, which shows that the resulting schemes are first- and second-order accurate and that shocks move at the correct speed.  We have also demonstrated that advection and diffusion on codimension 2 surfaces with the CPM using non-Euclidean closest point functions converge at the rates expected from the discrete schemes.

The Future         

On the applications side, we will tackle viscous flow problems, and on the theory side, we will further investigate the consistency and stability of the numerical techniques in the CPM.

References

[12/19] März T., Macdonald C.B.: Calculus on surfaces with general closest point functions.

Ruuth S., Merriman B.: A Simple Embedding Method for Solving Partial Differential Equations on Surfaces, Journal of Computational Physics, 2008

Macdonald C.B., Ruuth S.: Level set equations on surfaces via the Closest Point Method, Journal of Scientific Computing, 2008