Eventos Anais de eventos
COBEM 2021
26th International Congress of Mechanical Engineering
Verification of symmetry and convergence towards steady-state of WENO schemes
Submission Author:
Rafael Henrique Olindo de Oliveira , PR
Co-Authors:
Rafael Henrique Olindo de Oliveira , Luciano Araki, Nicholas Dicati Pereira da Silva, Rafael Brandão de Rezende Borges
Presenter: Rafael Henrique Olindo de Oliveira
doi://10.26678/ABCM.COBEM2021.COB2021-1351
Abstract
Challenging engineering problems may require high resolution and accuracy, as in the case of solving hyperbolic partial differential equations whose solutions may contain discontinuities, even when initial conditions are smooth, or when dealing with the interface between fluids. To deal with such complex flows, WENO schemes are broadly used in CFD simulations. This class of schemes aims to avoid the stencils with a discontinuity, reaching a high resolution. However, the higher the accuracy order, the lower the false diffusion is, which can intensify the round-off error perturbation effects, mainly in Euler’s equations, because there is no source of diffusion in the model. These perturbations can lead to symmetry breaking in symmetric problems, explained by the arithmetic floating-point non-associativity of algorithms. In this sense, the goal of this work is to evaluate the residuum and the symmetry preserving behavior for recent WENO schemes: WENO-Z+, WENO-EJS, WENO-EZ, WENO-MR, and WENO-MZ. They were implemented along with the positivity preserving Lax-Friedrichs splitting and the 3rd-order strong stability preserving Runge-Kutta in five test problems. The first one is a smooth quasi-1D nozzle flow to check the accuracy, and the second one was performed to test those schemes facing discontinuities produced by a normal shock wave. The third one is 2D and aims to contrast the performance with an oblique shock reflection problem. The last two are the Rayleigh-Taylor instability and the implosion problems, which present complex flow structures and are interesting test cases for symmetry. For the first two, the L^{1}, L^{2} and L^{\infty} residuum norms were evaluated and, particularly for the first one, the accuracy order is presented based on the exact solution. For the last two, those same norms were evaluated for the difference between conservative vectors on each side of the symmetry line to verify solution symmetry. As a result, only WENO-MR and WENO-EJS settled down in the nozzle flow presenting a normal shock wave. For the oblique shock reflection, only the WENO-MR was able to settle down. For the Rayleigh-Taylor problem, one can see a good hold of symmetry to every scheme but WENO-Z+. For the implosion problem, the WENO-MR and WENO-EJS presented a good hold of symmetry, while WENO-Z+ and WENO-MZ showed the highest residuum’s norms. Even though not all WENO schemes reached the machine error level, all of them were able to get the steady-state solution.
Keywords
CFD, WENO, Symmetry, High Resolution
