scholarly journals Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis

2012 ◽  
Vol 5 (3) ◽  
pp. 563-581 ◽  
Author(s):  
Hongyun Peng ◽  
◽  
Lizhi Ruan ◽  
Changjiang Zhu ◽  
Keyword(s):  
Conservation Laws ◽  
Large Amplitude ◽  
Convergence Rates ◽  
Diffusion Limit ◽  
Zero Diffusion Limit

Gravito-inertial waves in a rotating stratified sphere or spherical shell

1999 ◽  
Vol 398 ◽  
pp. 271-297 ◽  
Author(s):  
B. DINTRANS ◽  
M. RIEUTORD ◽  
L. VALDETTARO
Keyword(s):  
Periodic Orbit ◽  
Spherical Shell ◽  
Periodic Orbits ◽  
Point Spectrum ◽  
Boussinesq Approximation ◽  
Diffusion Limit ◽  
Order Partial Differential Equation ◽  
Inertial Waves ◽  
Zero Diffusion Limit ◽  
Hyperbolic Domain

The properties of gravito-inertial waves propagating in a stably stratified rotating spherical shell or sphere are investigated using the Boussinesq approximation. In the perfect fluid limit, these modes obey a second-order partial differential equation of mixed type. Characteristics propagating in the hyperbolic domain are shown to follow three kinds of orbits: quasi-periodic orbits which cover the whole hyperbolic domain; periodic orbits which are strongly attractive; and finally, orbits ending in a wedge formed by one of the boundaries and a turning surface. To these three types of orbits, our calculations show that there correspond three kinds of modes and give support to the following conclusions. First, with quasi-periodic orbits are associated regular modes which exist at the zero-diffusion limit as smooth square-integrable velocity fields associated with a discrete set of eigenvalues, probably dense in some subintervals of [0, N], N being the Brunt–Väisälä frequency. Second, with periodic orbits are associated singular modes which feature a shear layer following the periodic orbit; as the zero-diffusion limit is taken, the eigenfunction becomes singular on a line tracing the periodic orbit and is no longer square-integrable; as a consequence the point spectrum is empty in some subintervals of [0, N]. It is also shown that these internal shear layers contain the two scales E1/3 and E1/4 as pure inertial modes (E is the Ekman number). Finally, modes associated with characteristics trapped by a wedge also disappear at the zero-diffusion limit; eigenfunctions are not square-integrable and the corresponding point spectrum is also empty.

scholarly journals Arbitrary-Order Bernstein–Bézier Functions for DGFEM Transport on 3D Polygonal Grids

2021 ◽  
Vol 2 (3) ◽  
pp. 239-245
Author(s):  
Michael Hackemack
Keyword(s):  
Convergence Rates ◽  
Arbitrary Order ◽  
Diffusion Limit ◽  
Element Discretization ◽  
Lagrange Polynomials ◽  
Galerkin Finite Element ◽  
Polynomial Degree ◽  
Numerical Tests ◽  
Full Resolution ◽  
Discontinuous Galerkin Finite Element

In this paper, we present an arbitrary-order discontinuous Galerkin finite element discretization of the SN transport equation on 3D extruded polygonal prisms. Basis functions are formed by the tensor product of 2D polygonal Bernstein–Bézier functions and 1D Lagrange polynomials. For a polynomial degree p, these functions span {xayb}(a+b)≤p⊗{zc}c∈(0,p) with a dimension of np(p+1)+(p+1)(p−1)(p−2)/2 on an extruded n-gon. Numerical tests confirm that the functions capture exactly monomial solutions, achieve expected convergence rates, and provide full resolution in the thick diffusion limit.

Sign in / Sign up

Export Citation Format

Share Document