Spectral method

Summary

Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain "basis functions" (for example, as a Fourier series which is a sum of sinusoids) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible. Spectral methods and finite element methods are closely related and built on the same ideas; the main difference between them is that spectral methods use basis functions that are generally nonzero over the whole domain, while finite element methods use basis functions that are nonzero only on small subdomains (compact support). Consequently, spectral methods connect variables globally while finite elements do so locally. Partially for this reason, spectral methods have excellent error properties, with the so-called "exponential convergence" being the fastest possib

Official source

https://en.wikipedia.org/wiki/Spectral_method

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Global Fourier spectral methods are excellent tools to solve conserva- tion laws. They enable fast convergence rates and highly accurate solutions. However, being high-order methods, they suffer from the Gibbs phenomenon, which leads to spurious numerical oscillations in the vicinity of discontinuities. This can have a detrimental effect on the solution quality and lead to unphysical results. While local approxima- tion techniques allow for local limiting or reconstruction, there are no such possibilities for global methods. This thesis proposes a neural net- work based method that adds artificial viscosity around discontinuities of the solution to the conservation law. This enables the transforma- tion of discontinuities into steep but continuous jumps. Test cases in one and two spatial dimensions as well as systems of conservation laws (Euler equations) are solved. Furthermore, the method is generalized to other global basis approaches on non-uniform grids and reduced ba- sis methods. The proposed method delivers satisfactory results in all test cases. On the one hand, it is able to detect and handle discontinu- ities. On the other hand, it stays highly accurate for smooth data.

2020

Controlling oscillations in spectral methods by local artificial viscosity governed by neural networks

Jan Sickmann Hesthaven, Deep Ray, Lukas Schwander

While a nonlinear viscosity is used widely to control oscillations when solving conservation laws using high-order elements based methods, such techniques are less straightforward to apply in global spectral methods as a local estimate of solution regularity generally is required. In this work we demonstrate how to train and use a local artificial neural network to estimate the local solution regularity and demonstrate the efficiency of nonlinear artificial viscosity methods based on this in the context of Fourier spectral methods. We compare with entropy viscosity techniques and illustrate the promise of the neural network based estimators when solving one- and two-dimensional conservation laws, including the Euler equations.

2020

Controlling oscillations in spectral methods by local artificial viscosity governed by neural networks

Jan Sickmann Hesthaven, Deep Ray, Lukas Schwander

While a nonlinear viscosity is used widely to control oscillations when solving conservation laws using high-order elements based methods, such techniques are less straightforward to apply in global spectral methods since a local estimate of the solution regularity is generally required. In this work we demonstrate how to train and use a local artificial neural network to estimate the local solution regularity and demonstrate the efficiency of nonlinear artificial viscosity methods based on this, in the context of Fourier spectral methods. We compare with entropy viscosity techniques and illustrate the promise of the neural network based estimators when solving one- and two-dimensional conservation laws, including the Euler equations. (C) 2021 Elsevier Inc. All rights reserved.

ACADEMIC PRESS INC ELSEVIER SCIENCE2021