GPS Applied Mathematics Seminar
The GPS (Graduate Participation Seminar) is a weekly seminar by and for graduate students. If you're interested in presenting a topic or your own research, contact the organizers, Qin Li and Sarah Tumasz.
All seminars are on Mondays from 2:25pm to 3:15pm in B211 Van Vleck.
|Sept 19||Qin Li||AP scheme for multispecies Boltzmann equation|
|Sept 26||Sarah Tumasz||An Introduction to Topological Mixing|
|Oct 3||Zhennan Zhou||Perturbation Theory and Molecular Dynamics|
|Oct 10||Li Wang||A class of well balanced scheme for hyperbolic system with source term|
|Oct 17||E. Alec Johnson||Boundary Integral Positivity Limiters|
|Oct 24||Bokai Yan||TBA|
|Nov 7||David Seal||TBA|
Monday, Sept 19: Qin Li
AP scheme for multispecies Boltzmann equation
It is well-known that the Euler equation and the Navier–Stokes equation are 1st and 2nd order asymptotic limit of the Boltzmann equation when the Knudsen number goes to zero. Numerically the solution to the Boltzmann equation should converge to the Euler limit too. However, when the Knudsen number is small, one has to resolve the mesh to avoid instability, which causes tremendous computational cost. Asymptotic preserving scheme is a type of schemes that only uses coarse mesh but preserves the asymptotic limits of the Boltzmann equation in a discrete setting when Knudsen number vanishes. I'm going to present an AP scheme -- the BGK penalization method to solve the multispecies Boltzmann equation. New difficulties for this multispecies system come from: 1. the accurate definition of BGK term, 2. the different time scaling needed for different species to achieve the equilibrium.
Monday, Sept 26: Sarah Tumasz
An Introduction to Topological Mixing
What does topology have to do with mixing fluids? I will give an introduction to topological mixing from the bottom up. The talk will include a description of the basic theory, and demonstration of how to apply the techniques to a specific system. No prior knowledge of topology is needed!
Monday, Oct 3: Zhennan Zhou
Perturbation Theory and Molecular Dynamics
I would like to give a brief introduction to quantum molecular dynamics with the method of adiabatic perturbation theory.In the framework of Quantum Mechanics, the dynamics of a molecule is governed by the (time-dependent) Schr\"odinger equation, involving nuclei and electrons coupled through electromagnetic interactions. In recent years, Born-Oppenheimer approximation with many applications in mathematics, physics and chemistry, turns out to be a very successful approximation scheme, which is a prototypical example of adiabatic decoupling, and plays a fundamental role in the understanding of complex molecular systems.
Monday, Oct 10: Li Wang
A class of well balanced scheme for hyperbolic system with source term
In many physical problems one encounters source terms that are balanced by internal forces, and this kind of problem can be described by a hyperbolic system with source term. In comparison with the homogeneous system, a significant difference is that this system encounters non-constant stationary sloutions. So people want to preserve the steay state solutions, or some discrete versions at least, with enough accuracy. This is the so called well balanced scheme. I will give some basic idea of the scheme through a typical example, the Saint-Venant system for shallow water flows with nonuniform bottom. This talk is based on the paper [E.Audusse, etc SIAM J. Sci. Comput. 2004].
Monday, Oct 17: E. Alec Johnson
Boundary Integral Positivity Limiting
We consider positivity-preserving discontinuous Galerkin (DG) schemes for hyperbolic PDEs. For simplicity we focus on scalar PDEs with flux functions that may be spatially varying. We assume that physical solutions maintain positivity of the solution.
The DG method evolves a piecewise polynomial representation. Specifically, the representation is typically discontinuous at mesh cell interfaces and when restricted to a mesh cell is a polynomial. The coefficients of the representation are evolved using an ODE solver, which for simplicity we take to be the explicit Euler method.
Positivity limiters maintain positivity of the cell average by after each time step damping the deviation from the cell average just enough so that a cell positivity condition is satisfied.
The question we consider is how the cell positivity condition ought to be defined. The positivity condition should at least require positivity at the boundary nodes (where Riemann problems must be solved) and should at most require positivity everywhere in the cell (lest order of accuracy be violated).
Testing whether a higher-order polynomial with extremum arbitrarily close to zero is everywhere positive is NP-hard. We therefore seek a less stringent positivity indicator which is inexpensive to compute.
The time until an Euler step violates positivity of the cell average is the ratio of the amount of stuff in the cell to the rate at which it is flowing out of the boundary. This immediately suggests a simple positivity indicator which we call the boundary integral positivity indicator. Enforcing positivity of the boundary integral positivity indicator is computationally no more expensive than enforcing positivity at a single point and guarantees the same positivity-preserving time step as if positivity were enforced everywhere in the mesh cell.
This is joint work with James Rossmanith.