# A frequency-dependent p-adaptive technique for spectral methods

@article{Xia2021AFP, title={A frequency-dependent p-adaptive technique for spectral methods}, author={Mingtao Xia and Sihong Shao and Tom Chou}, journal={J. Comput. Phys.}, year={2021}, volume={446}, pages={110627} }

When using spectral methods, a question arises as how to determine the expansion order, especially for time-dependent problems in which emerging oscillations may require adjusting the expansion order. In this paper, we propose a frequency-dependent $p$-adaptive technique that adaptively adjusts the expansion order based on a frequency indicator. Using this $p$-adaptive technique, combined with recently proposed scaling and moving techniques, we are able to devise an adaptive spectral method in…

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Efficient Scaling and Moving Techniques for Spectral Methods in Unbounded Domains

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## References

SHOWING 1-10 OF 34 REFERENCES

Efficient Scaling and Moving Techniques for Spectral Methods in Unbounded Domains

- Computer Science, MathematicsSIAM J. Sci. Comput.
- 2021

A scaling technique and a moving technique to adaptively cluster enough collocation points in a region of interest in order to achieve a fast spectral convergence and to track the blowup of average cell sizes in a model for cell proliferation.

Computing nearly singular solutions using pseudo-spectral methods

- Mathematics, Computer ScienceJ. Comput. Phys.
- 2007

It is demonstrated that the pseudo-spectral method with the high order Fourier smoothing gives a much better performance than the Pseudo-spectrals with the 2/3 dealiasing rule, and that the error produced by the high orders is highly localized near the region where the solution is most singular.

A finite-difference method for the one-dimensional time-dependent schrödinger equation on unbounded domain

- Mathematics
- 2005

A finite-difference scheme is proposed for the one-dimensional time-dependent Schrodinger equation. We introduce an artificial boundary condition to reduce the original problem into an…

Stability and Error Analysis for a Second-Order Fast Approximation of the One-dimensional Schrödinger Equation Under Absorbing Boundary Conditions

- Computer Science, MathematicsSIAM J. Sci. Comput.
- 2018

A second-order Crank--Nicolson finite difference method, integrating a fast approximation of an exact discrete absorbing boundary condition, is proposed for solving the one-dimensional Schrodinger…

On p-robust saturation for hp-AFEM

- Mathematics, Computer ScienceComput. Math. Appl.
- 2017

It is shown that the analysis can be transferred from the patches to a reference triangle, and therein it is provided clear-cut computational evidence that any increment proportional to the polynomial degree yields the desired reduction.

Spectral Methods: Algorithms, Analysis and Applications

- Computer Science
- 2011

Along with finite differences and finite elements, spectral methods are one of the three main methodologies for solving partial differential equations on computers. This book provides a detailed…

An Adaptive hp–DG–FE Method for Elliptic Problems: Convergence and Optimality in the 1D Case

- MathematicsCommunications on Applied Mathematics and Computation
- 2019

We propose and analyze an hp-adaptive DG–FEM algorithm, termed \(\varvec {hp}\)-ADFEM, and its one-dimensional realization, which is convergent, instance optimal, and h- and p-robust. The procedure…

An Iterative Grid Redistribution Method for Singular Problems in Multiple Dimensions

- Mathematics
- 2000

We introduce an iterative grid redistribution method based on the variational approach. The iterative procedure enables us to gain more precise control of the grid distribution near the regions of…

Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2020

In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the…

Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation

- Mathematics
- 1984

Abstract Various numerical methods are employed in order to approximate the nonlinear Schrodinger equation, namely: (i) The classical explicit method, (ii) hopscotch method, (iii) implicit-explicit…