Pacific Institute for the Mathematical Sciences - PIMS

Some New Numerical Techniques to Some Old PDE Problems

Problems involving complicated deforming time-dependent boundaries or interfaces (i.e. codim-1 surfaces) are ubiquitous in the modeling of physical systems. The resulting Partial Differential Equations (PDEs) often have irregular, and even discontinuous solutions along these surfaces. In turn, the solution of these PDEs couples back to flow and hence deform the surfaces in question. The numerical approximation of such systems isnotoriously challenging.

In a regular Cartesian grid setting, I propose to replace the original PDE by another PDE which better approximates, in the discrete setting, the exact solution to the original PDE. I will provide 3 examples of this approach.

First, in the active penalty method one does not enforce the PDE boundary conditions directly, but rather solves the PDE in a larger domain without boundary (e.g. a flat torus) and adds a carefully constructed source or penalty term that mimics boundary conditions. I will show how to systematically construct penalty terms which improve the convergence rates of the penalized PDE, thereby allowing for higher-order finite-difference or Fourier-spectral numerical schemes to be applied to problems withnon-conforming boundaries.

Second, in the correction function method I tackle the problem of solving PDEs with jump discontinuities across a codim-1 surface, i.e. an interface. This is achieved by formulating an auxiliary local PDE which solution smoothly extends across the interface while enforcing the jump conditions. I will show that this approach is general, and can achieve arbitrary order of convergence while incurring (asymptotically) no additional computational cost.

Third, I will present methods for evolving in time arbitrary geometric objects such as boundaries or interfaces, but also general open/closed surfaces with possibly no regularity (e.g. fractals). This is achieved by evolving in time the flow map and composing it with the initial conditions. This method fit naturally within the gradient-augmented level set framework and enables use of a two-grid approach to achieve arbitrary order of convergence and optimal efficiency.

Throughout the talk I will illustrate these approaches with simulations of various physical systems including problems from fluid dynamics, electromagnetism, solid mechanics for which these methods may need to be combined.

Location: ESB 2012