Multiscale methods and analysis for the Dirac equation in the nonrelativistic limit regime
Joint CSCAMM/KI-Net Seminar
Prof. Weizhu Bao
Department of Mathematics
National University of Singapore
In this talk, I will review our recent works on numerical methods and analysis
for solving the Dirac equation in the nonrelativistic limit regime,
involving a small dimensionless parameter which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time and the energy becomes unbounded and indefinite, which bring significant difficulty in analysis and heavy burden in numerical computation. We begin with four frequently used finite difference time domain (FDTD) methods and the time splitting Fourier pseudospectral (TSFP) method and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size and time step as well as the small parameter. Then we consider a numerical method by using spectral method for spatial derivatives combined with an exponential wave integrator (EWI) in the Gautschi-type for temporal derivatives to discretize the Dirac equation. Rigorous error estimates show that the EWI spectral method has much better temporal resolution than the FDTD methods for the Dirac equation in the nonrelativistic limit regime.
Based on a multiscale expansion of the solution, we present a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation and establish
its error bound which uniformly accurate in term of the small dimensionless parameter.
Numerical results demonstrate that our error estimates are sharp and optimal.
Finally, these methods and results are then extended to the nonlinear Dirac equation in the nonrelativistic limit regime.
This is a joint work with Yongyong Cai, Xiaowei Jia, Qinglin Tang and Jia Yin.
Advances and challenges in global sensitivity analysis
What to do when the size and complexity of your model essentially
prevent you from using it? Well, get a smaller and simpler model...
At the heart of this dimension reduction process is the notion of
parameter importance which, ultimately, is part of the modeling process
itself. Global Sensitivity Analysis (GSA) aims at efficiently
identifying important and non-important parameters; non-importance is
important! We will present in this talk advances and challenges in GSA;
these will include how to deal with correlated variables, how to treat
time-dependent problems and stochastic problems and how to analyze the
robustness of GSA itself at low cost. The role played by surrogate
models will also be discussed.
The discussion will be illustrated by an application from neurovascular
Joint work with Alen Alexanderian, Tim David, Joey Hart and Ralph Smith.
Vortex filament solutions of Navier-Stokes
I will present the construction of solutions of the 3D Navier-Stokes
equations whose initial vorticity is supported on curves (vortex
filaments). This is the first instance for 3D Navier-Stokes where the
stability of asymptotically (microscopically) self-similar solutions can
be proved. This is joint work with J. Bedrossian and B. Harrop-Griffiths.