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Session Overview
Session
MS 18b: Advances in high order methods for fluid dynamic
Time:
Wednesday, 14/July/2021:
11:50am - 1:50pm

Session Chair: Henar Herrero
Session Chair: Jezabel Curbelo
Virtual location: Zoom 2


Session Abstract

On approaching turbulence and geophysical domains, different scales appear in

the problems (Aurnou et al, 2019). Small scales and large domains are difficult to

reach with spectral methods because they are ill conditioned so that it is not

possible to increase suitably the size of the expansions. A strategy to increase

the size of the mesh avoiding the increase of the expansion are domain

decomposition (DD) techniques (Quarteroni, 1999, 2002). Among them, the

alternating Schwarz methods respect the original computation (Blayo, 2016; Xu,

2005). Herrero et al. applied this methodology to a stationary natural convection

problem (Herrero, 2018). The method is expensive in time and unstable for some

values of the parameters. A way of stabilization and optimization consists of using

two levels of mesh in the alternating Schwarz method (Axelsson, 2019).

Geophysical problems are studied using direct numerical simulations (GL Kooij

et al. 2019, Curbelo et al. 2019). The study of instabilities requires the tracking of

bifurcations varying parameters. These methods are based on the observation

that varieties calculated by means of proper orthogonal decomposition (POD)

from solutions at some values of the parameters also contain solutions to different

values of the parameters. If the equations that govern the evolution of a system

are dissipative, the behavior of the system for long times is contained in a finite-

dimensional inertial variety, which is often of a low dimension (Foias, 1988).

Herrero et al. applied this type of methods for a Rayleigh-Bénard problem in a

rectangular domain with reduced basis (Herrero, 2013) and taking as snapshots

transient states of a temporal evolution towards a stationary solution or transient

states of Newton's iteration for the treatment of non-linearity for a fixed parameter

value, managing to reproduce the diagram of bifurcations with very low

computational cost (Pla, 2015). Gutierrez-Castillo et al. analyzed the use of POD

for temporal flow patterns in non-newtonian fluids.


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Presentations
11:50am - 12:20pm

An alternating Schwarz Method for a Rayleigh-Bénard Problem

Henar Herrero

Universidad de Castilla-La Mancha, Spain

A Schwarz domain decomposition numerical method for a stationary Rayleigh–Bénard convection problem is presented. The model equations are the stationary version of the incompressible Navier–Stokes equations coupled with a heat equation under Boussinesq approximation in a rectangular domain. A Newton method is used to deal with the nonlinearities. Each step in the Newton method is solved with an alternating Schwarz domain decomposition method. Their convergence properties are studied theoretically in a simplified domain. The numerical resolution of the problem confirms the theoretical results. The convergence rate is less than one when overlap is considered. Convergence is achieved for large values of the aspect ratio, which are inabordable for the standard Legendre collocation method. Other advantages of this methodology compared with standard methods are parallelization and high order.



12:20pm - 12:50pm

High-order fully well-balanced Lagrange-Projection scheme for shallow water equations

Tomas Morales de Luna

Universidad de Cordoba, Spain

We propose a novel strategy to define high-order fully well-balanced Lagrange-Projection finite volume solvers for balance laws. In particular, we focus on the 1D shallow water system. By fully well-balanced, it is meant that the scheme is able to preserve stationary smooth solutions.

As usual in Lagrange-Projection schemes, we use the standard decomposition to naturally decouple the acoustic and transport phenomena. Such a decomposition proved to be useful and very efficient to deal with subsonic or near low-Froude number flows. In this case, the usual CFL time step limitation of Godunov-type schemes is indeed driven by the acoustic waves and can thus be very restrictive. The Lagrange-Projection strategy allows to design a very natural implicit-explicit and large time-step schemes with a CFL restriction based on the (slow) transport waves and not on the (fast) acoustic waves.

The proposed technique produces good results when applied to shallow water system. Note that, the Lagrange-Projection scheme may be extended to other systems and applications such as bedload sediment transport, turbidity currents, Ripa system, etc.



12:50pm - 1:20pm

High-order spatial and temporal schemes suitable for polyhedral unstructured grids

Pedro M.P. Costa, Duarte M.S. Albuquerque

Instituto Superior Técnico, Portugal

In a transient convection-diffusion equation the global numerical error is the sum of the spatial and the temporal errors. With the finite volume method, the temporal term of the equation must be integrated in space with the same accuracy order as the convection and diffusion schemes. This novel approach considers an operator that converts pointwise values to mean ones, which enables the solution of unsteady high-order convection-diffusion equations in the pointwise framework.

One advantage of these schemes is the low number of contributions to the coefficient matrix when compared to the mean-value counterpart. This becomes evident when extending eighth-order schemes to three-dimensional cases. This novel operator uses only even cell-centered derivatives and cell’s even momentum of inertial.

To reduce the number of required time steps, high-order backwards differentiation schemes are implemented leading to considerable time savings when using high-order spatial schemes. The numerical spatial error evolution with the solver runtime and the required memory is studied as an efficiency metrics of the implemented high-order schemes.

Acknowledgement.

The first author would like to acknowledge the current grant received by FCT – Portugal “Fundação para a Ciência e Tecnologia”, with the reference number 2020.07754.BD

This work is a contribution from the research project HIBforMBP – “High-order immersed boundary for moving body problems”, reference PTDC/EME-EME/32315/2017, which is financially supported by FCT - “Fundação para a Ciência e Tecnologia” - www.fct.pt .



 
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