On differentiable local bounds preserving stabilization for Euler equations

Badia, Santiago; Bonilla, Jesús; Mabuza, Sibusiso; Shadid, John N.

doi:10.1016/j.cma.2020.113267

On differentiable local bounds preserving stabilization for Euler equations

Journal Article · Thu Oct 01 00:00:00 EDT 2020 · Computer Methods in Applied Mechanics and Engineering

DOI:https://doi.org/10.1016/j.cma.2020.113267· OSTI ID:1639088

Badia, Santiago ^[1]; Bonilla, Jesús ^[2]; Mabuza, Sibusiso ^[3]; Shadid, John N. ^[4]

Monash Univ., Clayton, VIC (Australia); Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Castelldefels (Spain)
Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Castelldefels (Spain); Univ. Politecnica de Catalunya, Barcelona (Spain)
Clemson Univ., SC (United States)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research; Univ. of New Mexico, Albuquerque, NM (United States)

This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in the case of the transient form. A differentiable local bounds preserving method has been developed, which combines a Rusanov artificial diffusion operator and a differentiable shock detector. Nonlinear stabilization schemes are usually stiff and highly nonlinear. This issue is mitigated by the differentiability properties of the proposed method. Moreover, in order to further improve the nonlinear convergence, we also propose a continuation method for a subset of the stabilization parameters. The resulting method has been successfully applied to steady and transient problems with complex shock patterns. Numerical experiments show that it is able to provide sharp and well resolved shocks. Furthermore, the importance of the differentiability is assessed by comparing the new scheme with its non-differentiable counterpart. Numerical experiments suggest that, for up to moderate nonlinear tolerances, the method exhibits improved robustness and nonlinear convergence behavior for steady problems. Additionally, in the case of transient problem, we also observe a reduction in the computational cost.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: La Caixa Foundation; USDOE; USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

Grant/Contract Number:: AC04-94AL85000; NA0003525

OSTI ID:: 1639088

Alternate ID(s):: OSTI ID: 1637576

Report Number(s):: SAND--2020-6831J; 687134

Journal Information:: Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering Vol. 370; ISSN 0045-7825

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (30)

A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation Codina, Ramon Computer Methods in Applied Mechanics and Engineering, Vol. 110, Issue 3-4 https://doi.org/10.1016/0045-7825(93)90213-h	journal	December 1993
Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems Mabuza, Sibusiso; Shadid, John N.; Kuzmin, Dmitri Journal of Computational Physics, Vol. 361 https://doi.org/10.1016/j.jcp.2018.01.048	journal	May 2018
A linearity preserving nodal variation limiting algorithm for continuous Galerkin discretization of ideal MHD equations Mabuza, Sibusiso; Shadid, John N.; Cyr, Eric C. Journal of Computational Physics, Vol. 410 https://doi.org/10.1016/j.jcp.2020.109390	journal	June 2020
Integrationalism and resilience: a dynamic systems analysis of EU regional integration in the Mediterranean and North Africa Hierro, Lara Resilience https://doi.org/10.1080/21693293.2017.1308634	journal	April 2017
Invariant Domains and First-Order Continuous Finite Element Approximation for Hyperbolic Systems Guermond, Jean-Luc; Popov, Bojan SIAM Journal on Numerical Analysis, Vol. 54, Issue 4 https://doi.org/10.1137/16m1074291	journal	January 2016
Strong Stability-Preserving High-Order Time Discretization Methods Gottlieb, Sigal; Shu, Chi-Wang; Tadmor, Eitan SIAM Review, Vol. 43, Issue 1 https://doi.org/10.1137/s003614450036757x	journal	January 2001
Multidimensional FEM-FCT schemes for arbitrary time stepping: MULTIDIMENSIONAL FEM-FCT SCHEMES Kuzmin, D.; Möller, M.; Turek, S. International Journal for Numerical Methods in Fluids, Vol. 42, Issue 3 https://doi.org/10.1002/fld.493	journal	May 2003
Flux Correction Tools for Finite Elements Kuzmin, D.; Turek, S. Journal of Computational Physics, Vol. 175, Issue 2 https://doi.org/10.1006/jcph.2001.6955	journal	January 2002
Maps of Convex Sets and Invariant Regions¶for Finite-Difference Systems¶of Conservation Laws Frid, Hermano Archive for Rational Mechanics and Analysis, Vol. 160, Issue 3 https://doi.org/10.1007/s002050100166	journal	November 2001
Fusion Simulation Project Workshop Report Kritz, Arnold; Keyes, David Journal of Fusion Energy, Vol. 28, Issue 1 https://doi.org/10.1007/s10894-008-9151-4	journal	June 2008
FEMPAR: An Object-Oriented Parallel Finite Element Framework Badia, Santiago; Martín, Alberto F.; Principe, Javier Archives of Computational Methods in Engineering, Vol. 25, Issue 2 https://doi.org/10.1007/s11831-017-9244-1	journal	October 2017
Approximate Riemann solvers, parameter vectors, and difference schemes Roe, P. L. Journal of Computational Physics, Vol. 43, Issue 2 https://doi.org/10.1016/0021-9991(81)90128-5	journal	October 1981
The group finite element formulation Fletcher, C. A. J. Computer Methods in Applied Mechanics and Engineering, Vol. 37, Issue 2 https://doi.org/10.1016/0045-7825(83)90122-6	journal	April 1983
A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations Shakib, Farzin; Hughes, Thomas J. R.; Johan, Zdeněk Computer Methods in Applied Mechanics and Engineering, Vol. 89, Issue 1-3 https://doi.org/10.1016/0045-7825(91)90041-4	journal	August 1991
A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation Codina, Ramon Computer Methods in Applied Mechanics and Engineering, Vol. 110, Issue 3-4 https://doi.org/10.1016/0045-7825(93)90213-H	journal	December 1993
Analysis of a group finite element formulation Barrenechea, Gabriel R.; Knobloch, Petr Applied Numerical Mathematics, Vol. 118 https://doi.org/10.1016/j.apnum.2017.03.008	journal	August 2017
Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes Kuzmin, Dmitri Journal of Computational and Applied Mathematics, Vol. 236, Issue 9 https://doi.org/10.1016/j.cam.2011.11.019	journal	March 2012
Stabilization and shock-capturing parameters in SUPG formulation of compressible flows Tezduyar, Tayfun E.; Senga, Masayoshi Computer Methods in Applied Mechanics and Engineering, Vol. 195, Issue 13-16 https://doi.org/10.1016/j.cma.2005.05.032	journal	February 2006
Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization Badia, Santiago; Bonilla, Jesús Computer Methods in Applied Mechanics and Engineering, Vol. 313 https://doi.org/10.1016/j.cma.2016.09.035	journal	January 2017
Differentiable monotonicity-preserving schemes for discontinuous Galerkin methods on arbitrary meshes Badia, Santiago; Bonilla, Jesús; Hierro, Alba Computer Methods in Applied Mechanics and Engineering, Vol. 320 https://doi.org/10.1016/j.cma.2017.03.032	journal	June 2017
Maximum-principle preserving space–time isogeometric analysis Bonilla, Jesús; Badia, Santiago Computer Methods in Applied Mechanics and Engineering, Vol. 354 https://doi.org/10.1016/j.cma.2019.05.042	journal	September 2019
A tutorial-driven introduction to the parallel finite element library FEMPAR v1.0.0 Badia, Santiago; Martín, Alberto F. Computer Physics Communications, Vol. 248 https://doi.org/10.1016/j.cpc.2019.107059	journal	March 2020
Synchronized flux limiting for gas dynamics variables Lohmann, Christoph; Kuzmin, Dmitri Journal of Computational Physics, Vol. 326 https://doi.org/10.1016/j.jcp.2016.09.025	journal	December 2016
Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems Cockburn, Bernardo; Shu, Chi-Wang Journal of Scientific Computing, Vol. 16, Issue 3, p. 173-261 https://doi.org/10.1023/A:1012873910884	journal	September 2001
Invariant regions for systems of conservation laws Hoff, David Transactions of the American Mathematical Society, Vol. 289, Issue 2 https://doi.org/10.1090/S0002-9947-1985-0784005-3	journal	February 1985
A finite difference scheme for a system of two conservation laws with artificial viscosity Hoff, David Mathematics of Computation, Vol. 33, Issue 148 https://doi.org/10.1090/S0025-5718-1979-0537964-9	journal	January 1979
Convergence Analysis of Pseudo-Transient Continuation Kelley, C. T.; Keyes, David E. SIAM Journal on Numerical Analysis, Vol. 35, Issue 2 https://doi.org/10.1137/S0036142996304796	journal	April 1998
Strong Stability-Preserving High-Order Time Discretization Methods Gottlieb, Sigal; Shu, Chi-Wang; Tadmor, Eitan SIAM Review, Vol. 43, Issue 1 https://doi.org/10.1137/S003614450036757X	journal	January 2001
A tutorial-driven introduction to the parallel finite element library FEMPAR v1.0.0 Badia, Santiago Mendeley https://doi.org/10.17632/dtx487wp57	dataset	January 2019
A tutorial-driven introduction to the parallel finite element library FEMPAR v1.0.0 Badia, Santiago Mendeley https://doi.org/10.17632/dtx487wp57.1	dataset	January 2019

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