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Session Overview
MS 7b: Next generation numerical methods: Advances in discretizations with the summation-by-parts property
Tuesday, 13/July/2021:
4:00pm - 6:00pm

Session Chair: David César Del Rey Fernández
Session Chair: Matteo Parsani
Session Chair: Mark H. Carpenter
Virtual location: Zoom 2

Session Abstract

The past decade has seen an explosion in popularity of developing high-order methods with the

summation-by-parts (SBP) property. This is because the SBP framework offers a simple, yet

powerful methodology for the design and analysis of modern algorithms for the solution of par-

tial differential equations (PDEs). While SBP has its origins in finite-differences, the concept has

developed into a general purpose analysis tool applicable to general possibly nonconforming in p

and h unstructured meshes and now spans many discretization approaches including discontinu-

ous/continuous/hybrid Galerkin, flux-reconstruction, finite volume, and finite-difference methods.

The simple and strong theoretical underpinnings of SBP allows for the straightforward construction

of high-order methods with inherent flexibility that result in discretizations that are nonlinearly

robust. Starting from the continuous analysis of the PDE, in a one-to-one manner, SBP formula-

tions result in provably conservative and nonlinearly stable formulations that can be crafted to be

flexible and element local, making the resulting methods well suited for cache-efficient programming

on massively parallel supercomputers.

The purpose of this minisymposium is to bring together researchers working to extend the frontiers

of the SBP framework; some of the potential topics that will be covered are:

• Application of SBP methods to various meshing approaches

• Strategies for adaptivity that retain stability and conservation

• Extensions to the SBP concept

• Novel applications of the SBP concept

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4:00pm - 4:30pm

A discontinuous Galerkin method for elastic waves with physically motivated numerical fluxes

Kenneth Duru

The Australian National University, Australia

Elastic wave propagation in complex media supports several wave types, including dissipative surface and interface waves. The simultaneous presence of multiple wave types and physical phenomena pose a significant challenge for numerical fluxes. When modelling surface or interface waves an incompatibility of the numerical flux with the physical boundary condition leads to numerical artefacts. We present a stable and accurate DG method for elastic waves with a physically motivated numerical flux. Our numerical flux is compatible with all well-posed, internal and external, boundary conditions, including linear and nonlinear frictional constitutive equations for modelling spontaneously propagating shear ruptures in elastic solids and dynamic earthquake rupture processes.

By construction our choice of penalty parameters yield an upwind scheme and a discrete energy estimate analogous to the continuous energy estimate. We derive a priori error estimate for the DG method proving optimal convergence to discontinuous and nearly singular exact solutions.

We present numerical experiments verifying high order accuracy and asymptotic numerical stability. We demonstrate the potential of our approach for computational seismology in a regional wave propagation scenario in a geologically constrained 3D model including the geometrically complex free-surface topography of Mount Zugspitze, Germany.

4:30pm - 5:00pm

Adapting the Resolution Capacity of Finite Difference Methods via Filtering

Ayaboe K. Edoh1, Charles L. Merkle2, Venkateswaran Sankaran3

1Jacobs Engineering Inc, United States of America; 2Purdue University, United States of America; 3Air Force Research Laboratory, United States of America

The efficient computation of flows exhibiting large disparities in spatial scales (e.g. turbulence) places high demands on the resolution capacity of discretization schemes. While the performance of finite difference stencils can be designed relative to specific wavenumber ranges, the dynamic nature of simulations motivates the ability to employ tailored schemes based on the local resolution properties of the solution [1,2]. To this end, an adaptive filtering approach is proposed for recovering target centered difference stencils from a baseline method. Focusing on the convective terms in the compressible equations (e.g., the Euler system), a residual filtering procedure is used to reconstruct maximum-order stencils (i.e., optimal for low wavenumber signals) versus Dispersion Relation Preserving stencils [3] (i.e., optimal for under-resolved signals) from a standard second-order scheme. The result is a spectral relation enhancement of the overall computation via a spatially varying deconvolution. An attenuating solution filtering step is then included for reducing lingering small-scale errors (e.g., aliasing) [4] and for providing regularization in the presence of sharp gradients. Blended transitions between the target methods are naturally accommodated with adaptive filters that are able to preserve the conservation and energy-stability properties of a baseline method [5,6] when viewed from a summation-by-parts type perspective. Details on the proper use of residual filtering in multi-dimensions are highlighted with respect to impacts on numerical anisotropy and the desire to maintain a normed energy estimate.


[1] D. Fauconnier, C. De Langhe and E. Dick, “Construction of explicit and implicit dynamic finite difference schemes and application to the large-eddy simulation of the Taylor-Green vortex,” Journal of Computational Physics, Vol. 228, 2009, pp. 8053-8084.

[2] V. Linders, M. H. Carpenter and J. Nordstrom, “Accurate solution-adaptive finite difference schemes for coarse and fine grids,” Journal of Computational Physics, Vol. 410, 2020.

[3] V. Linders and J. Nordstrom, “Uniformly best wavenumber approximations by spatial central difference operators,” Journal of Computational Physics, Vol. 300, 2015, 695-709.

[4] A. K. Edoh, N. L. Mundis, A. R. Karagozian and V. Sankaran, “Balancing aspects of numerical dissipation, dispersion, and aliasing in time-accurate simulations,” International Journal of Numerical Methods in Fluids, Vol. 92, 2020, pp. 1506-1527.

[5] A. K. Edoh and V. Sankaran, “Boundary prescriptions for spectrally-tunable discrete filters,” AIAA 2019-2170, January 2019.

[6] T. Lundquist and J. Nordstrom, “Stable and accuate filtering procedures,” Journal of Scientific Computing, Vol. 82, 2020.

5:00pm - 5:30pm

Discretely Entropy Stable Split Forms for the Flux Reconstruction High-Order Method: Three-Dimensional Numerical Validation

Alexander Nicholas Walter Cicchino, Siva Nadarajah

McGill University, Canada

The flux reconstruction method has gained popularity in the research community as it recovers promising high-order methods through modally filtered correction fields, such as the Discontinuous Galerkin (DG) and spectral difference methods, on unstructured grids over complex geometries. Under a class of energy stable flux reconstruction (ESFR) schemes also known as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes [1], the flux reconstruction method allows for larger time-steps than DG while ensuring linear stability on linear elements. Alternatively, for nonlinear problems, split forms and two-point entropy conserving fluxes have emerged as the popular approach proving nonlinear stability for unsteady problems on coarse unstructured grids -- albeit only having been proved for the strong form DG scheme and numerically shown for the g2-lumped lobatto strong form ESFR scheme for the Euler equations [2,3]. In this work we show that incorporating split forms with VCJH schemes, alike Ranocha et. al. [2] and Abe et. al. [3], generally lead to unstable discretizations. This work is an extension of Cicchino, Nadarajah, and Fernández [4], where they derived a new class of general nonlinearly stable FR schemes in split form for the one-dimensional Burgers' and proved stability. This new approach simplifies to VCJH type schemes for linear problems, but for nonlinear problems, it ensures nonlinear stability and the correct orders of convergence. The stability of all uncollocated, modal nonlinearly stable FR schemes are verified with three-dimensional Euler flow in curvilinear coordinates.

[1] Vincent, P. E., Castonguay, P., and Jameson, A., “A New Class of High-Order Energy Stable Flux Reconstruction Schemes,”

Journal of Scientific Computing, Vol. 47, No. 1, 2011, pp. 50–72.

[2] Ranocha, H., Öffner, P., and Sonar, T., "Summation-by-Parts Operators for Correction Procedure via Reconstruction," Journal of Computational Physics, 311, (2016), 299–328.

[3] Abe, Y., Morinaka, I., Haga, T., Nonomura, T., Shibata, H., and Miyaji, K. "Stable, Non-Dissipative, and Conservative Flux-Reconstruction Schemes in

Split Forms," Journal of Computational Physics, 353, (2018), 193–227.

[4] A. Cicchino, S. Nadarajah, D. Del Rey Fernández, Nonlinearly stable flux reconstruction high-order methods in split form, (Submitted to the Journal of Computational Physics) (2021).

5:30pm - 6:00pm

Eigenanalysis and non-modal analysis of entropy stable discretizations with the summation by parts property

Irving Enrique Reyna Nolasco, Matteo Parsani, Lisandro Dalcin, Radouan Boukharfane

King Abdullah University of Science and Technology, Saudi Arabia

In recent years, high-order methods based on summation-by-part (SBP) operators have shown a good maturity level in efficiency and accuracy for dealing with wave propagation problems on complex geometries. In general, high-order discretization schemes have demonstrated an ability to produce accurate solutions with smaller numerical dissipation and dispersion compared to lower order methods. Moreover, numerical diffusion plays a crucial role in the robustness and accuracy of under-resolved turbulent computations. In the context of implicit large-eddy simulations (LES), the effect of numerical errors, and specifically, the numerical dissipation is expected to resemble the effect of the turbulence model. Therefore, the analysis of the numerical errors arising in any scheme is essential to understand the accuracy of a numerical solution, and which is more important, this is a step towards designing robust and accurate numerical discretization schemes.

In the present work, the dispersion and diffusion properties are analyzed for the semi-discrete equation of the one-dimensional linear advection and linear advection-diffusion equation. The spatial derivatives of the equations are discretized through a differentiation matrix collocated at the Legendre-Gauss-Lobatto (LGL) points, which satisfy the SBP property. The boundary data is weakly imposed through the simultaneous-approximation-term technique (SAT).

The analysis performed here is based on the traditional Fourier analysis or eigenanalysis tech nique and the non-modal analysis introduced by P. Fernández. The non-modal analysis aims to characterize the numerical diffusion properties of the methods, by taking into account all modes for each wavenumber. This is achieved through the study of the short-term dynamics of the semi-discretization equation of the linear advection diffusion equation. The short-term diffusion behavior is investigated as a function of the spatial discretizations and speed regimes. In the eigenanalysis presented in this work, the behavior of the so-called primary and secondary eigenmodes is analyzed. Additionally, the energy loss of the initial wave is computed in order to describe the diffusion for different spatial discretizations. The main goal of the analysis presented in this work is to characterize the numerical dispersion and diffusion properties arising in the SBP-SAT discretizations, at the same time, we get some insights on the robustness of the discretization scheme and the effects of numerical dissipation on the unresolved-scales. The analysis observations are validated with numerical results, performed on the SSDC framework, presented recently.

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