Browsing by Subject "Entropy Viscosity"
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Item Discontinuous Galerkin Finite Element Method for the Nonlinear Hyperbolic Problems with Entropy-Based Artificial Viscosity Stabilization(2012-07-16) Zingan, Valentin NikolaevichThis work develops a discontinuous Galerkin finite element discretization of non- linear hyperbolic conservation equations with efficient and robust high order stabilization built on an entropy-based artificial viscosity approximation. The solutions of equations are represented by elementwise polynomials of an arbitrary degree p > 0 which are continuous within each element but discontinuous on the boundaries. The discretization of equations in time is done by means of high order explicit Runge-Kutta methods identified with respective Butcher tableaux. To stabilize a numerical solution in the vicinity of shock waves and simultaneously preserve the smooth parts from smearing, we add some reasonable amount of artificial viscosity in accordance with the physical principle of entropy production in the interior of shock waves. The viscosity coefficient is proportional to the local size of the residual of an entropy equation and is bounded from above by the first-order artificial viscosity defined by a local wave speed. Since the residual of an entropy equation is supposed to be vanishingly small in smooth regions (of the order of the Local Truncation Error) and arbitrarily large in shocks, the entropy viscosity is almost zero everywhere except the shocks, where it reaches the first-order upper bound. One- and two-dimensional benchmark test cases are presented for nonlinear hyperbolic scalar conservation laws and the system of compressible Euler equations. These tests demonstrate the satisfactory stability properties of the method and optimal convergence rates as well. All numerical solutions to the test problems agree well with the reference solutions found in the literature. We conclude that the new method developed in the present work is a valuable alternative to currently existing techniques of viscous stabilization.Item Numerical Simulations of Bouncing Jets(2014-07-18) Lee, SanghyunThe Kaye effect is a fascinating phenomenon of a leaping shampoo stream which was first described by Alan Kaye in 1963 as a property of non-Newtonian fluid. It manifest itself when a thin stream of non-Newtonian fluid is poured into a dish of fluid. As pouring proceeds, a small stream of liquid occasionally leaps upward from the heap. We investigate numerically the impact of the experimental setting as well as the fluid rheology on the apparition of bouncing jets. In particular, we observe the importance of the creation of a thin lubricating layer of air between the jet and the rest of the liquid. The numerical method consists of a projection method coupled with a level set formulation for the interface representation. Adaptive finite element methods are advocated to capture the different length scales inherent to this context. In addition, we design and study two modifications of the first order standard pressure correction projection scheme for the Stokes system. The first scheme improves the existing schemes in the case of open boundary condition by modifying the pressure increment boundary condition, thereby minimizing the pressure boundary layer and recovering the optimal first order decay. The second scheme allows for variable time stepping. It turns out that the straightforward modification to variable time stepping leads to unstable schemes. The proposed scheme is not only stable but also exhibits the optimal first order decay. Numerical computations illustrating the theoretical estimates are provided for both new schemes.