Exact particle methods and gradient-augmented level set approaches
B. Seibold (Math, Temple)

We present two examples of computational approaches in which we deviate from "classical" methods, by using information that is specifically useful in the considered class of problems:

Nonlinear flows in networks (traffic, pipeline, information) are frequently described by a collection of scalar hyperbolic conservation laws or balance laws. We present a class of characteristic particle methods that solve scalar conservation laws exactly, even in the presence of shocks. In various more complex problems, these exact methods can be applied to the convective parts of the equations, thus leading to accurate solution schemes. In stiff models of reaction kinetics, the particle approach leads to correct propagation velocities of detonation waves, without explicitly resolving their small scale dynamics.

In the context of level set methods, we present a gradient-augmented approach, in which the method of characteristic is combined with a suitable Hermite interpolation. The approach advects interfaces with third order accuracy, and admits a second order accurate approximation of curvature, both with optimally local stencils, i.e. information from only a single grid cell is used. In addition, the presence of gradient information gives rise to a certain level of subgrid resolution, making the approach particularly able at tracking small structures, such as thin films or droplets.