2,947 research outputs found
Spurious frequencies as a result of numerical boundary treatments
The stability theory for finite difference Initial Boundary-Value approximations to systems of hyperbolic partial differential equations states that the exclusion of eigenvalues and generalized eigenvalues is a sufficient condition for stability. The theory, however, does not discuss the nature of numerical approximations in the presence of such eigenvalues. In fact, as was shown previously, for the problem of vortex shedding by a 2-D cylinder in subsonic flow, stating boundary conditions in terms of the primitive (non-characteristic) variables may lead to such eigenvalues, causing perturbations that decay slowly in space and remain periodic time. Characteristic formulation of the boundary conditions avoided this problem. A more systematic study of the behavior of the (linearized) one-dimensional gas dynamic equations under various sets of oscillation-inducing legal boundary conditions is reported
Cohomology fractals
We introduce cohomology fractals; these are certain images associated to a
cohomology class on a hyperbolic three-manifold. They include images made
entirely from circles, and also images with no geometrically simple features.
They are closely related to limit sets of kleinian groups, but have some key
differences. As a consequence, we can zoom in almost any direction to arbitrary
depth in real time. We present an implementation in the setting of ideal
triangulations using ray-casting.Comment: 8 pages, 30 figures and subfigure
Effective distance between nested Margulis tubes
We give sharp, effective bounds on the distance between tori of fixed
injectivity radius inside a Margulis tube in a hyperbolic 3-manifold.Comment: 25 pages, 3 figures. v3 contains minor revisions. To appear in
Transactions of the AM
Splitting methods for low Mach number Euler and Navier-Stokes equations
Examined are some splitting techniques for low Mach number Euler flows. Shortcomings of some of the proposed methods are pointed out and an explanation for their inadequacy suggested. A symmetric splitting for both the Euler and Navier-Stokes equations is then presented which removes the stiffness of these equations when the Mach number is small. The splitting is shown to be stable
Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to high-order compact schemes
We present a systematic method for constructing boundary conditions (numerical and physical) of the required accuracy, for compact (Pade-like) high-order finite-difference schemes for hyperbolic systems. First, a roper summation-by-parts formula is found for the approximate derivative. A 'simultaneous approximation term' (SAT) is then introduced to treat the boundary conditions. This procedure leads to time-stable schemes even in the system case. An explicit construction of the fourth-order compact case is given. Numerical studies are presented to verify the efficacy of the approach
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