An Introduction to Spectral Convergence: Challenges and Paths to Solutions

2021 ◽  
Author(s):  
Andrew Herschfelt ◽  
Alex R. Chiriyath ◽  
Sharanya Srinivas ◽  
Daniel W. Bliss
Keyword(s):  
Spectral Convergence

Spectral convergence of the Hermite basis function solution of the Vlasov equation: The free-streaming term

2006 ◽  
Vol 219 (2) ◽  
pp. 477-488 ◽  
Author(s):  
Livio Gibelli ◽  
Bernie D. Shizgal
Keyword(s):  
Basis Function ◽  
Vlasov Equation ◽  
Function Solution ◽  
Spectral Convergence

Solving Vlasov-Poisson-Fokker-Planck Equations using NRxx method

2017 ◽  
Vol 21 (3) ◽  
pp. 782-807 ◽  
Author(s):  
Yanli Wang ◽  
Shudao Zhang
Keyword(s):  
Splitting Method ◽  
Linear Convergence ◽  
Mass Conservation ◽  
System A ◽  
Spectral Convergence ◽  
The Stability ◽  
The Moment ◽  
Stability And Accuracy ◽  
Fokker Planck Equations ◽  
Fokker Planck

AbstractWe present a numerical method to solve the Vlasov-Poisson-Fokker-Planck (VPFP) system using the NRxx method proposed in [4, 7, 9]. A globally hyperbolic moment system similar to that in [23] is derived. In this system, the Fokker-Planck (FP) operator term is reduced into the linear combination of the moment coefficients, which can be solved analytically under proper truncation. The non-splitting method, which can keep mass conservation and the balance law of the total momentum, is used to solve the whole system. A numerical problem for the VPFP system with an analytic solution is presented to indicate the spectral convergence with the moment number and the linear convergence with the grid size. Two more numerical experiments are tested to demonstrate the stability and accuracy of the NRxx method when applied to the VPFP system.

scholarly journals Spectral convergence of multiquadric interpolation

1993 ◽  
Vol 36 (2) ◽  
pp. 319-333 ◽  
Author(s):  
Martin Buhmann ◽  
Nira Dyn
Keyword(s):  
Fourier Transform ◽  
Nonnegative Integer ◽  
Compact Domain ◽  
Spectral Convergence

In this paper, we consider interpolants on h·ℤn from the closure of the space spanned by translates of the function (‖·‖2 + 1)β/2 (β>−n and not an even nonnegative integer) along h·ℤn. We show that these interpolants approximate a function, whose Fourier transform satisfies certain asymptotic conditions, up to an error of order hp, on any compact domain in ℝn, where p is only restricted by the smoothness of the function.

scholarly journals Spectral convergence under bounded Ricci curvature

2017 ◽  
Vol 273 (5) ◽  
pp. 1577-1662 ◽  
Author(s):  
Shouhei Honda
Keyword(s):  
Ricci Curvature ◽  
Spectral Convergence

scholarly journals Spectral convergence of Riemannian manifolds, II

1996 ◽  
Vol 48 (1) ◽  
pp. 71-120 ◽  
Author(s):  
Atsushi Kasue ◽  
Hironori Kumura
Keyword(s):  
Riemannian Manifolds ◽  
Spectral Convergence

ON VIBRATIONS OF A BODY WITH MANY CONCENTRATED MASSES NEAR THE BOUNDARY

1993 ◽  
Vol 03 (02) ◽  
pp. 249-273 ◽  
Author(s):  
MIGUEL LOBO ◽  
EUGENIA PEREZ
Keyword(s):  
Asymptotic Behavior ◽  
Small Parameter ◽  
Critical Size ◽  
Local Problem ◽  
Boundary Values ◽  
Micro Structure ◽  
Spectral Convergence ◽  
Fourier And Laplace Transforms ◽  
Concentrated Masses ◽  
The Masses

We consider the asymptotic behavior of the vibration of a body occupying a region Ω⊂ℝ3. The density, which depends on a small parameter ε, is of order O(1) out of certain regions where it is O(ε–m) with m>2. These regions, the concentrated masses with diameter O(ε), are located near the boundary, at mutual distances O(η), with η=η(ε)→0. We impose Dirichlet (respectively Neumann) conditions at the points of ∂Ω in contact with (respectively, out of) the masses. For the critical size ε=O(η2), the asymptotic behavior of the eigenvalues of order O(εm−2) is described via a Steklov problem, where the ‘mass’ is localized on the boundary, or through the eigenvalues of a local problem obtained from the micro-structure of the problem. We use the techniques of the formal asymptotic analysis in homogenization to determine both problems. We also use techniques of convergence in homogenization, Semigroups theory, Fourier and Laplace transforms and boundary values of analytic functions to prove spectral convergence. In the same framework we study the case m=2 as well as the case when other boundary conditions are imposed on ∂Ω.

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