Wednesdays at 1:10-2:00 PM in Surge Hall 268.
Hydrodynamical evolution in a gravitational field arises in many astrophysical and atmospheric problems. Improper treatment of the gravitational force can lead to a solution which oscillates around the equilibrium. In this presentation, we propose a recently developed well-balanced discontinuous Galerkin method for the Euler equations under gravitational fields, which can maintain the hydrostatic equilibrium state exactly. Some numerical tests are performed to verify the well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions.
We propose the frozen Gaussian approximation (FGA) for the computation of high frequency wave propagation. This method approximates the solution to the wave equation by an integral representation. We also present a systematic introduction on applying FGA to compute synthetic seismograms in three-dimensional earth models. In the method, seismic wavefield is decomposed into frozen (fixed-width) Gaussian functions, which propagate along ray paths. Rather than the coherent state solution to the wave equation, this method is rigorously derived by asymptotic expansion on phase plane, with analysis of its accuracy determined by the ratio of short wavelength over large domain size. Similar to other ray-based beam methods (e.g. Gaussian beam methods), one can use relatively small number of Gaussians to get accurate approximations of high-frequency wavefield. The algorithm is embarrassingly parallel, which can drastically speed up the computation with a multico re-processor computer station. Furthermore, we incorporate the Snell's law into the FGA formulation, and asymptotically derive reflection, transmission and free surface conditions for FGA to compute high-frequency seismic wave propagation in high contrast media. We numerically test these conditions by computing traveltime kernels of different phases in the 3D crust-over-mantle model.
In this talk, we present two types of novel gradient recovery methods for elliptic interface problem: 1. Finite element methods based on body-fitted mesh; 2. Immersed finite element methods. Due to the lack of regularity of solution at interface, standard gradient recovery methods fail to give superconvergent results, and thus will lead to over-refinement when served as a posteriori error estimator. This drawback is overcome by designing an immersed gradient recovery operator in our methods. We analyze the superconvergence of these methods, and provide several numerical examples to verify the superconvergence and its robustness as a posteriori error estimator.
In this talk, we will investigate the problem of recovering the conductivity of an electrical network from the knowledge of the magnitude of the current along the edges and either the knowledge of the voltage on the boundary of the network or the current flowing in or out of the network. We will discuss how this problem corresponds to finding the minimizers of a l^1 minimization problem. I will also talk about the dual of this minimization problem which provides valuable information about the minimizers. We will also present a numerical algorithm that finds solutions to both the dual and primal problem. I will also discuss the applications of the results to random walks on graphs.
We develop direct discontinuous Galerkin (DDG) methods to solve Keller-Segel Chemotaxis equations. Different to available DG methods or other numerical methods in literature, we introduce no extra variable to approximate the chemical density gradients and solve the system directly. With P^k polynomial approximations, we observe no order loss and optimal (k+1)th order convergence is obtained. The reason that DDG methods have better convergence is because that DDG methods have super convergence phenomena on approximating solution gradients. With Fourier (Von Neumann) analysis technique, we prove the DDG solution’s spatial derivative is super convergent with at least (k+1)th order under momentum norm or in weak sense. We show the cell density approximations are strictly positive with at least third order of accuracy. Blow up features are captured well.
There have been considerable recent developments on fluid dynamics equations with partial or fractional dissipation. These partial differential equations (PDEs) are not only mathematically important but also physically relevant. This talk focuses on fundamental mathematical problems on several such PDEs with fractional partial dissipation including the 3D Navier-Stokes, the surface quasi-geostrophic equation, the 2D Boussinesq and the 2D MHD equations with partial dissipation. When there is only partial fractional dissipation, some of the classical tools such as the maximal regularity estimates for the heat operator no longer apply. New techniques have been developed to fully exploit the regularization effects of such incomplete dissipation.We present several global regularity results based on these techniques.
For questions, contact Dr. Yulong Xing, email@example.com