260 research outputs found
Flux Splitting for stiff equations: A notion on stability
For low Mach number flows, there is a strong recent interest in the
development and analysis of IMEX (implicit/explicit) schemes, which rely on a
splitting of the convective flux into stiff and nonstiff parts. A key
ingredient of the analysis is the so-called Asymptotic Preserving (AP)
property, which guarantees uniform consistency and stability as the Mach number
goes to zero. While many authors have focussed on asymptotic consistency, we
study asymptotic stability in this paper: does an IMEX scheme allow for a CFL
number which is independent of the Mach number? We derive a stability criterion
for a general linear hyperbolic system. In the decisive eigenvalue analysis,
the advective term, the upwind diffusion and a quadratic term stemming from the
truncation in time all interact in a subtle way. As an application, we show
that a new class of splittings based on characteristic decomposition, for which
the commutator vanishes, avoids the deterioration of the time step which has
sometimes been observed in the literature
From Heisenberg matrix mechanics to EBK quantization: theory and first applications
Despite the seminal connection between classical multiply-periodic motion and
Heisenberg matrix mechanics and the massive amount of work done on the
associated problem of semiclassical (EBK) quantization of bound states, we show
that there are, nevertheless, a number of previously unexploited aspects of
this relationship that bear on the quantum-classical correspondence. In
particular, we emphasize a quantum variational principle that implies the
classical variational principle for invariant tori. We also expose the more
indirect connection between commutation relations and quantization of action
variables. With the help of several standard models with one or two degrees of
freedom, we then illustrate how the methods of Heisenberg matrix mechanics
described in this paper may be used to obtain quantum solutions with a modest
increase in effort compared to semiclassical calculations. We also describe and
apply a method for obtaining leading quantum corrections to EBK results.
Finally, we suggest several new or modified applications of EBK quantization.Comment: 37 pages including 3 poscript figures, submitted to Phys. Rev.
Sharp L^p bounds on spectral clusters for Lipschitz metrics
We establish L^p bounds on L^2 normalized spectral clusters for self-adjoint
elliptic Dirichlet forms with Lipschitz coefficients. In two dimensions we
obtain best possible bounds for all p between 6<p\leq 8$. In higher dimensions we obtain best
possible bounds for a limited range of p.Comment: 28 page
Convergence of second-order in time numerical discretizations for the evolution Navier-Stokes equations
We prove the convergence of certain second-order numerical methods to weak solutions of the Navier-Stokes equations satisfying in addition the local energy inequality, and therefore suitable in the sense of Scheffer and Caffarelli-Kohn-Nirenberg. More precisely, we treat the space-periodic case in three space-dimensions and we consider a full discretization in which the the classical Crank-Nicolson method (-method with ) is used to discretize the time variable, while in the space variables we consider finite elements. The convective term is discretized in several implicit, semi-implicit, and explicit ways. In particular, we focus on proving (possibly conditional) convergence of the discrete solutions towards weak solutions (satisfying a precise local energy balance), without extra regularity assumptions on the limit problem. We do not prove orders of convergence, but our analysis identifies some numerical schemes providing also alternate proofs of existence of ``physically relevant'' solutions in three space dimensions
On the convergence of a shock capturing discontinuous Galerkin method for nonlinear hyperbolic systems of conservation laws
In this paper, we present a shock capturing discontinuous Galerkin (SC-DG)
method for nonlinear systems of conservation laws in several space dimensions
and analyze its stability and convergence. The scheme is realized as a
space-time formulation in terms of entropy variables using an entropy stable
numerical flux. While being similar to the method proposed in [14], our
approach is new in that we do not use streamline diffusion (SD) stabilization.
It is proved that an artificial-viscosity-based nonlinear shock capturing
mechanism is sufficient to ensure both entropy stability and entropy
consistency, and consequently we establish convergence to an entropy
measure-valued (emv) solution. The result is valid for general systems and
arbitrary order discontinuous Galerkin method.Comment: Comments: Affiliations added Comments: Numerical results added,
shortened proo
Approximating the Derivative of Manifold-valued Functions
We consider the approximation of manifold-valued functions by embedding the
manifold into a higher dimensional space, applying a vector-valued
approximation operator and projecting the resulting vector back to the
manifold. It is well known that the approximation error for manifold-valued
functions is close to the approximation error for vector-valued functions. This
is not true anymore if we consider the derivatives of such functions. In our
paper we give pre-asymptotic error bounds for the approximation of the
derivative of manifold-valued function. In particular, we provide explicit
constants that depend on the reach of the embedded manifold.Comment: 25 pages, 5 figure
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