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Session Overview
Session
MS 7a: Next generation numerical methods: Advances in discretizations with the summation-by-parts property
Time:
Monday, 12/July/2021:
1:50pm - 3:50pm

Session Chair: David César Del Rey Fernández
Session Chair: Mark H. Carpenter
Session Chair: Matteo Parsani
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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Presentations
1:50pm - 2:20pm

Summation by parts methods for nonlinear dispersive wave equations

Hendrik Ranocha1, Manuel Quezada de Luna2, Dimitrios Mitsotakis3, David I. Ketcheson2

1University of Münster, Germany; 2King Abdullah University of Science and Technology, Saudi Arabia; 3Victoria University of Wellington, New Zealand

We present a general framework for designing conservative numerical methods based on summation by parts operators in space, combined with relaxation methods in time. We apply this framework to create new classes of fully-discrete conservative methods for several nonlinear dispersive wave equations. These full discretizations conserve all linear invariants and one nonlinear invariant for each system. The spatial semidiscretizations include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. The time discretization is essentially explicit for the non-stiff problems, using relaxation Runge-Kutta methods. We implement some specific schemes from among the derived classes, and demonstrate their favorable properties through numerical tests. In particular, we present numerical evidence that these conservative numerical methods result in a linear error growth in time for solitary wave solutions while standard methods yield a quadratically growing error. These findings are in accordance with analytical results available for a subset of the dispersive wave equations under consideration.



2:20pm - 2:50pm

On a linear stability issue of split form schemes for compressible flows

Vikram Singh1, Praveen Chandrashekar2

1The Ocean in the Earth System, Max Planck Institute for Meteorology, Germany; 2Center for Applicable Mathematics, Tata Institute of Fundamental Research

Split form schemes for Euler and Navier-Stokes equations are useful for computation of turbulent flows due to their better robustness. This is because they satisfy additional conservation properties of the governing equations like kinetic energy preservation leading to a reduction in aliasing errors at high orders. Recently, linear stability issues have been pointed out for these schemes for a density wave problem and we investigate this behaviour for some standard split forms. By deriving linearized equations of split form schemes, we show that most existing schemes do not satisfy a perturbation energy equation that holds at the continuous level. A simple modification to the energy flux of some existing schemes is shown to yield a scheme that is consistent with the energy perturbation equation. Numerical tests are given using a discontinuous Galerkin method to demonstrate these results.



2:50pm - 3:20pm

Numerical treatment of discontinuous material properties in acoustic and elastic wave equations

Martin Almquist

Uppsala University

We study acoustic and elastic wave equations in discontinuous media. In such settings, summation-by-parts (SBP) methods are often combined with simultaneous approximation terms (SATs) that impose interface conditions weakly. However, existing SBP-SAT methods exhibit the following three undesirable traits. First, the acoustic-acoustic coupling becomes unnecessarily stiff as the impedance contrast between the two media becomes large. Second, the acoustic-acoustic coupling becomes inconsistent in the limit where the impedance contrast tends to infinity. Mathematically, the interface conditions turn into a boundary condition in this limit, but the numerical treatment fails to mimic this property. Third, the elastic-elastic coupling fails to be consistent with the elastic-acoustic interface conditions in the limit where the shear modulus tends to zero in one of the media. We resolve these issues by identifying a free parameter in the numerical interface treatment and using appropriate averages of the material properties to set this parameter.



3:20pm - 3:50pm

Physics-informed neural-network for the compressible Navier-Stokes model

Mohammed Sayyari, Shyma Alhuwaider, Matteo Parsani

King Abdullah University of Science and Technology (KAUST), Computer Electrical and Mathematical Science and Engineering Division (CEMSE), Extreme Computing Research Center (ECRC), 23955-6900, Thuwal, Saudi Arabia

In this presentation, we use a Physics-Informed Neural-Network (PINN) with

derivatives computed using Auto-Grad (AD) for the training and define the

loss using the compressible Navier-Stokes equations. We use Direct

Numerical Simulation (DNS) Navier-Stokes solutions as our reference and

training data for our model. We highlight that the model can predict the

solution well and generate low-resolution solutions suitable as initial

data for higher resolution simulations. This step is verified using an

in-house compressible solver based on collocated discontinuous Galerkin

discretizations with the summation-by-parts property and relaxation

Runge--Kutta schemes. In addition, we use the DNS data to optimize the

parameters of the compressible Navier-Stokes equations.



 
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