San Yih Lin

TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems

This is the third paper in a series in which we construct and analyze a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws ut+Σi=1d(fi(u)xi=0. In this paper we present the method in a system of equations, stressing the point of how to use the weak form in the component spaces, but to use the local projection limiting in the characteristic fields, and how to implement boundary conditions. A 1-dimensional system is thus chosen as a model. Different implementation techniques are discussed, theories analogous to scalar cases are proven for linear systems, and numerical results are given illustrating the method on nonlinear systems. Discussions of handling complicated geometries via adaptive triangle elements will appear in future papers.

Relaxation and gradient methods for molecular orientation in liquid crystals

Abstract Relaxation and gradient methods have been developed during the past several years for the computation of molecular orientation in liquid crystals. These methods have contributed to the classification of stable defect structures and the description of some of the transitions that occur under the influence of electric and magnetic fields. The bulk energy of the liquid crystal depends on the orientation of the optic axis, and this gives the variational problem a nonconvex constraints. Further, minimum energy configurations can be discontinous and can have singularities. In this paper, new relaxation and gradient methods are proposed to handle this nonconvex constraint. The results of numerical experiments and error analysis are presented.

Relaxation methods for liquid crystal problems

A point relaxation method for the liquid crystal problem is proposed and analyzed. The liquid crystal problem is a minimization problem with a nonconvex local constraint. It is proved that the energy of successive iterates is nonincreasing for the point relaxation method with relaxation parameter

Discontinuous Galerkin Finite Element Method for Euler and Navier-Stokes Equations

\omega

FLOW CONTROL SIMULATIONS AROUND A CIRCULAR CYLINDER BY A FINITE VOLUME SCHEME

satisfying

Numerical study of MUSCL schemes for computational aeroacoustics

0 < \omega < 2

Computational fluid dynamics analysis of the vertical axis wind turbine blade with tubercle leading edge

for a class of material constants. It is also shown that the difference between successive iterates converges to zero, and that limit points of the iteration sequence are minima with respect to perturbations which are supported at a point and which satisfy the constraint. Numerical results are given which demonstrate the improved rate of convergence for overrelaxation.

Parametric Study of Weighted Essentially Nonoscillatory Schemes for Computational Aeroacoustics

A finite element method for the Euler and Navier-Stokes equations has been developed. The spatial discretization involves the discontinuous Galerkin finite element method and Lax-Friedrichs flux method. The temporal discretizations used are the explicit Runge-Kutta time integrations. The scheme is formally second-order accurate in space and time. A dynamic mesh algorithm is included to simulate flows over moving bodies. The inviscid flows passing through a channel with circular arc bump, through the NACA 0012 airfoil, and the laminar flows passing over a flat plate with shock interaction are investigated to confirm the accuracy and convergence of the finite element method. Also the unsteady flow through a pitching NACA 0012 airfoil is performed to prove the capability of the present method.

Computation of superconductivity in thin films

Abstract An artificial compressibility method for solving the incompressible Navier-Stokes equations has been applied to the study of flow past a circular cylinder with and without flow control devices. The Reynolds number ranges from 46 to 200. The numerical method is based on upwind finite volume methods for space discretizations and an explicit Runge-Kutta time integration with implicit residual smoothing methods for time discretizations. Two kinds of flow control devices are investigated: (1) placing an attached or a detached splitter plate and (2) placing a second small cylinder (control cylinder) behind a circular cylinder. Numerical investigations show that both are effective in the suppression of vortex shedding and the reduction of drag.

Aerodynamic performance study of flapping- Wing flowfields

Two upwind finite-volume schemes are studied for solving the solutions of 2D Euler equations. They are based on the MUSCL (monotonic upstream scheme for scalar conservation law) approach with the Roe approximate Riemann solver for the numerical flux evaluation. First, dissipation and dispersion relation and group velocity are carried out to analyze the capability of one of the proposed schemes for capturing physical waves. Then two schemes are greatly enhanced through a special treatment on the numerical dissipation to effectively handle aeroacoustic computations. The numerical results indicate that the numerical dissipation treatment allows that two schemes simulate continuous waves at fourth-order accuracy and captures discontinuous waves sharply as well as most of upwind schemes do. The tested problems include 1D group velocity analysis, propagation of discontinuous and sine waves, shock and sine wave interaction, and 2D traveling vortex in an uniform freestream. The numerical results show that the proposed dissipation treatment can reduce dispersion and dissipation errors of two upwind schemes for simulating the aeroacoustic problems. (Author)