Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions

2019 ◽  
Vol 357 ◽  
pp. 103-122 ◽  
Author(s):  
Mahmoud A. Zaky
Keyword(s):  
High Order ◽  
Collocation Methods ◽  
High Order Accuracy ◽  
Smooth Solutions ◽  
Spectral Collocation ◽  
Order Accuracy ◽  
Spectral Collocation Methods

scholarly journals Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations

Computation ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 2
Author(s):  
Călin-Ioan Gheorghiu
Keyword(s):  
High Order ◽  
Collocation Methods ◽  
Benchmark Problems ◽  
Special Technique ◽  
Boundary Singularity ◽  
Spectral Collocation ◽  
Mixed Spectrum ◽  
Multiple Eigenvalues ◽  
Domain Truncation ◽  
Spectral Collocation Methods

We are concerned with the study of some classical spectral collocation methods, mainly Chebyshev and sinc as well as with the new software system Chebfun in computing high order eigenpairs of singular and regular Schrödinger eigenproblems. We want to highlight both the qualities as well as the shortcomings of these methods and evaluate them in conjunction with the usual ones. In order to resolve a boundary singularity, we use Chebfun with domain truncation. Although it is applicable with spectral collocation, a special technique to introduce boundary conditions as well as a coordinate transform, which maps an unbounded domain to a finite one, are the special ingredients. A challenging set of “hard”benchmark problems, for which usual numerical methods (f. d., f. e. m., shooting, etc.) fail, were analyzed. In order to separate “good”and “bad”eigenvalues, we have estimated the drift of the set of eigenvalues of interest with respect to the order of approximation and/or scaling of domain parameter. It automatically provides us with a measure of the error within which the eigenvalues are computed and a hint on numerical stability. We pay a particular attention to problems with almost multiple eigenvalues as well as to problems with a mixed spectrum.

scholarly journals High-Order Hybrid WCNS-CPR Scheme for Shock Capturing of Conservation Laws

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Jia Guo ◽  
Huajun Zhu ◽  
Zhen-Guo Yan ◽  
Lingyan Tang ◽  
Songhe Song
Keyword(s):  
Computational Cost ◽  
High Order ◽  
Shock Capturing ◽  
High Order Accuracy ◽  
Hybrid Technique ◽  
Hybrid Scheme ◽  
Order Accuracy ◽  
Order Of Accuracy ◽  
Reconstruction Scheme ◽  
High Gradients

By introducing hybrid technique into high-order CPR (correction procedure via reconstruction) scheme, a novel hybrid WCNS-CPR scheme is developed for efficient supersonic simulations. Firstly, a shock detector based on nonlinear weights is used to identify grid cells with high gradients or discontinuities throughout the whole flow field. Then, WCNS (weighted compact nonlinear scheme) is adopted to capture shocks in these areas, while the smooth area is calculated by CPR. A strategy to treat the interfaces of the two schemes is developed, which maintains high-order accuracy. Convergent order of accuracy and shock-capturing ability are tested in several numerical experiments; the results of which show that this hybrid scheme achieves expected high-order accuracy and high resolution, is robust in shock capturing, and has less computational cost compared to the WCNS.

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