A Combined-Mode Fourier Analysis of DG Methods for Linear Parabolic Problems

Alhawwary, Mohammad A.; Wang, Zhi Jian

doi:10.1137/20m1316962

Cited by 9 publications

(6 citation statements)

References 32 publications

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“…57 may be used to assess the accuracy of the numerical schemes. As was found in [6], no dispersion errors were obtained when analyzing the schemes developed in this work. The reason behind the lack of dispersion errors when dealing with pure-diffusion linear equations in SEM will be studied in future works.…”

Section: Von-neumann Analysis With Diffusionsupporting

confidence: 71%

The Spectral Difference Raviart–Thomas Method for Two and Three-Dimensional Elements and Its Connection with the Flux Reconstruction Formulation

2022

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The purpose of this work is to describe in detail the development of the spectral difference Raviart-Thomas (SDRT) formulation for two and three-dimensional tensor-product elements and simplexes. Through the process, the authors establish the equivalence between the SDRT method and the flux reconstruction (FR) approach under the assumption of the linearity of the flux and the mesh uniformity. Such a connection allows building a new family of FR schemes for two and three-dimensional simplexes and also to recover the well-known FR-SD method with tensor-product elements. In addition, a thorough analysis of the numerical dissipation and dispersion of both aforementioned schemes and the nodal discontinuous Galerkin FR (FR-DG) method with two and three-dimensional elements is proposed through the use of the combined-mode Fourier approach. SDRT is shown to possess an enhanced temporal linear stability in comparison to FR-DG. On the contrary, SDRT displays larger dissipation and dispersion errors with respect to FR-DG. Finally, the study is concluded with a set of numerical experiments, the linear advection-diffusion problem, the Isentropic Euler Vortex, and the Taylor-Green Vortex (TGV). The latter test case shows that SDRT schemes present a non-linear unstable behavior with simplex elements and certain polynomial degrees. For the sake of completeness, the matrix form of the SDRT method is developed and the computational performance of SDRT with respect to FR schemes is evaluated using GPU architectures.

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Section: Von-neumann Analysis With Diffusionsupporting

confidence: 71%

The Spectral Difference Raviart–Thomas Method for Two and Three-Dimensional Elements and Its Connection with the Flux Reconstruction Formulation

2022

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show abstract

“…57 may be used to assess the accuracy of the numerical schemes. As it was found in [5], no dispersion errors were obtained when analyzing the schemes developed in this work. The reason behind the lack of dispersion errors when dealing with pure-diffusion linear equations in SEM will be studied in future works.…”

Section: Von-neumann Analysis With Diffusionsupporting

confidence: 65%

“…On the other hand, with aliased wavenumbers the dissipation errors drastically increase for a high number of iterations. Recently, [5] indicated that the dissipation errors for high wavenumbers with pure-diffusion and SEM reduced the dissipation, i.e. the errors are related to the lack of appropriate diffusion provided by the numerical scheme.…”

Section: Von-neumann Analysis With Diffusionmentioning

confidence: 99%

The Spectral Difference Raviart-Thomas method for two and three-dimensional elements and its connection with the Flux Reconstruction formulation

Sáez-Mischlich¹,

Sierra²,

Gressier³

2021

Preprint

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The purpose of this work is to describe in detail the development of the Spectral Difference Raviart-Thomas (SDRT) formulation for two and three-dimensional tensor-product elements and simplexes. Through the process, the authors establish the equivalence between the SDRT method and the Flux-Reconstruction (FR) approach under the assumption of the linearity of the flux and the mesh uniformity. Such a connection allows to build a new family of FR schemes for two and three-dimensional simplexes and also to recover the well-known FR-SD method with tensor-product elements. In addition, a thorough analysis of the numerical dissipation and dispersion of both aforementioned schemes and the nodal Discontinuous Galerkin FR (FR-DG) method with two and three-dimensional elements is proposed through the use of the combined-mode Fourier approach. SDRT is shown to posses an enhanced temporal linear stability regarding the FR-DG. On the contrary, SDRT displays larger dissipation and dispersion errors with respect to FR-DG. Finally, the study is concluded with a set of numerical experiments, the linear advection-diffusion problem, the Isentropic Euler Vortex and the Taylor-Green Vortex (TGV). The latter test case shows that SDRT schemes present a non-linear unstable behavior with simplex elements and certain polynomial degrees. For the sake of completeness, the matrix form of the SDRT method is developed and the computational performance of SDRT with respect to FR schemes is evaluated using GPU architectures.

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“…This approach has provided considerable insight into the behaviour of spatial methods. However, an alternative 'holistic' approach, in which the combined effect of all the modes is considered, has recently been presented [1,3,17].…”

Section: 4mentioning

confidence: 99%

On the Use of RBF Interpolation for Flux Reconstruction

Watson¹,

Trojak²

2022

Preprint

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Flux reconstruction provides a framework for solving partial differential equations in which functions are discontinuously approximated within elements. Typically, this is done by using polynomials.Here, the use of radial basis functions as a methods for underlying functional approximation is explored in one dimension, using both analytical and numerical methods. At some mesh densities, RBF flux reconstruction is found to outperform polynomial flux reconstruction, and this range of mesh densities becomes finer as the width of the RBF interpolator is increased. A method which avoids the poor conditioning of flat RBFs is used to test a wide range of basis shapes, and at very small values, the polynomial behaviour is recovered. Changing the location of the solution points is found to have an effect similar to that in polynomial FR, with the Gauss-Legendre points being the most effective. Altering the location of the functional centres is found to have only a very small effect on performance. Similar behaviours are determined for the non-linear Burgers' equation.

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A Combined-Mode Fourier Analysis of DG Methods for Linear Parabolic Problems

Cited by 9 publications

References 32 publications

The Spectral Difference Raviart–Thomas Method for Two and Three-Dimensional Elements and Its Connection with the Flux Reconstruction Formulation

The Spectral Difference Raviart–Thomas Method for Two and Three-Dimensional Elements and Its Connection with the Flux Reconstruction Formulation

The Spectral Difference Raviart-Thomas method for two and three-dimensional elements and its connection with the Flux Reconstruction formulation

On the Use of RBF Interpolation for Flux Reconstruction

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