Adaptive mesh strategies for the spectral element method

An adaptive spectral method was developed for the efficient solution of time dependent partial differential equations. Adaptive mesh strategies that include resolution refinement and coarsening by three different methods are illustrated on solutions to the 1-D viscous Burger equation and the 2-D Navier-Stokes equations for driven flow in a cavity. Sharp gradients, singularities, and regions of poor resolution are resolved optimally as they develop in time using error estimators which indicate the choice of refinement to be used. The adaptive formulation presents significant increases in efficiency, flexibility, and general capabilities for high order spectral methods

Enhancement of shock-capturing methods via machine learning

In recent years, machine learning has been used to create data-driven solutions to problems for which an algorithmic solution is intractable, as well as fine-tuning existing algorithms. This research applies machine learning to the development of an improved finite-volume method for simulating PDEs with discontinuous solutions. Shock-capturing methods make use of nonlinear switching functions that are not guaranteed to be optimal. Because data can be used to learn nonlinear relationships, we train a neural network to improve the results of a fifth-order WENO method. We post-process the outputs of the neural network to guarantee that the method is consistent. The training data consist of the exact mapping between cell averages and interpolated values for a set of integrable functions that represent waveforms we would expect to see while simulating a PDE. We demonstrate our method on linear advection of a discontinuous function, the inviscid Burgers’ equation, and the 1-D Euler equations. For the latter, we examine the Shu–Osher model problem for turbulence–shock wave interactions. We find that our method outperforms WENO in simulations where the numerical solution becomes overly diffused due to numerical viscosity

Hydrodynamics of cold atomic gases in the limit of weak nonlinearity, dispersion and dissipation

Dynamics of interacting cold atomic gases have recently become a focus of both experimental and theoretical studies. Often cold atom systems show hydrodynamic behavior and support the propagation of nonlinear dispersive waves. Although this propagation depends on many details of the system, a great insight can be obtained in the rather universal limit of weak nonlinearity, dispersion and dissipation (WNDD). In this limit, using a reductive perturbation method we map some of the hydrodynamic models relevant to cold atoms to well known chiral one-dimensional equations such as KdV, Burgers, KdV-Burgers, and Benjamin-Ono equations. These equations have been thoroughly studied in literature. The mapping gives us a simple way to make estimates for original hydrodynamic equations and to study the interplay between nonlinearity, dissipation and dispersion which are the hallmarks of nonlinear hydrodynamics.Comment: 18 pages, 3 figures, 1 tabl