Implicit shock tracking for unsteady flows by the method of lines

Shi, A.; Persson, P.-O.; Zahr, M. J.

doi:10.1016/j.jcp.2021.110906

Implicit shock tracking for unsteady flows by the method of lines

Journal Article · Sun Jan 09 23:00:00 EST 2022 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2021.110906· OSTI ID:1883333

Shi, A. ^[1]; ^[2]; Zahr, M. J. ^[3]

Univ. of California, Berkeley, CA (United States)
Univ. of California, Berkeley, CA (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Univ. of Notre Dame, IN (United States)

Here, a recently developed high-order implicit shock tracking (HOIST) framework for resolving discontinuous solutions of inviscid, steady conservation laws is extended to the unsteady case. Central to the framework is an optimization problem which simultaneously computes a discontinuity-aligned mesh and the corresponding high-order approximation to the flow, which provides nonlinear stabilization and a high-order approximation to the solution. This work extends the implicit shock tracking framework to the case of unsteady conservation laws using a method of lines discretization via a diagonally implicit Runge-Kutta method by “solving a steady problem at each timestep”. We formulate and solve an optimization problem that produces a feature-aligned mesh and solution at each Runge-Kutta stage of each timestep, and advance this solution in time by standard Runge-Kutta update formulas. A Rankine-Hugoniot based prediction of the shock location together with a high-order, untangling mesh smoothing procedure provides a high-quality initial guess for the optimization problem at each time, which results in rapid convergence of the sequential quadratic programing (SQP) optimization solver. This method is shown to deliver highly accurate solutions on coarse, high-order discretizations without nonlinear stabilization and recover the design accuracy of the Runge-Kutta scheme. We demonstrate this framework on a series of inviscid, unsteady conservation laws in both one- and two- dimensions. We also verify that our method is able to recover the design order of accuracy of our time integrator in the presence of a strong discontinuity.

View Accepted Manuscript (DOE)

Research Organization:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), High Energy Physics (HEP); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); US Air Force Office of Scientific Research (AFOSR)

Grant/Contract Number:: AC02-05CH11231

OSTI ID:: 1883333

Alternate ID(s):: OSTI ID: 1841461

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 454; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (34)

Introducing a new method for calculating the spatial and temporal distribution of pollutants in rivers Amiri, S.; Mazaheri, M.; Bavandpouri Gilan, N. International Journal of Environmental Science and Technology, Vol. 18, Issue 12 https://doi.org/10.1007/s13762-020-03096-y	journal	January 2021
A least-squares formulation of the Moving Discontinuous Galerkin Finite Element Method with Interface Condition Enforcement Kercher, Andrew D.; Corrigan, Andrew Computers & Mathematics with Applications, Vol. 95 https://doi.org/10.1016/j.camwa.2020.09.012	journal	August 2021
Convergence of the Moving Discontinuous Galerkin Method with Interface Condition Enforcement in the Presence of an Attached Curved Shock Corrigan, Andrew T.; Kercher, Andrew D.; Kessler, David A. AIAA Aviation 2019 Forum https://doi.org/10.2514/6.2019-3207	conference	June 2019
An Extended Discontinuous Galerkin Method for High-Order Shock-Fitting Geisenhofer, Markus; Kummer, Florian; Oberlack, Martin arXiv https://doi.org/10.48550/arxiv.2012.08860	preprint	January 2020
A discontinuoushp finite element method for the Euler and Navier-Stokes equations Baumann, Carlos Erik; Oden, J. Tinsley International Journal for Numerical Methods in Fluids, Vol. 31, Issue 1 https://doi.org/10.1002/(SICI)1097-0363(19990915)31:1<79::AID-FLD956>3.0.CO;2-C	journal	September 1999
High-order CFD methods: current status and perspective: HIGH-ORDER CFD METHODS Wang, Z. J.; Fidkowski, Krzysztof; Abgrall, Rémi International Journal for Numerical Methods in Fluids, Vol. 72, Issue 8 https://doi.org/10.1002/fld.3767	journal	January 2013
Unsteady shock-fitting for unstructured grids: Unsteady shock-fitting for unstructured grids Bonfiglioli, A.; Paciorri, R.; Campoli, L. International Journal for Numerical Methods in Fluids, Vol. 81, Issue 4 https://doi.org/10.1002/fld.4183	journal	October 2015
A moving discontinuous Galerkin finite element method for flows with interfaces: A moving discontinuous Galerkin finite element method for flows with interfaces Corrigan, Andrew; Kercher, Andrew D.; Kessler, David A. International Journal for Numerical Methods in Fluids, Vol. 89, Issue 9 https://doi.org/10.1002/fld.4697	journal	December 2018
Output error estimation strategies for discontinuous Galerkin discretizations of unsteady convection-dominated flows Fidkowski, Krzysztof J. International Journal for Numerical Methods in Engineering, Vol. 88, Issue 12 https://doi.org/10.1002/nme.3224	journal	May 2011
Optimization of a regularized distortion measure to generate curved high-order unstructured tetrahedral meshes: OPTIMIZATION OF A REGULARIZED DISTORTION MEASURE FOR CURVED MESHING Gargallo-Peiró, A.; Roca, X.; Peraire, J. International Journal for Numerical Methods in Engineering, Vol. 103, Issue 5 https://doi.org/10.1002/nme.4888	journal	February 2015
A distortion measure to validate and generate curved high-order meshes on CAD surfaces with independence of parameterization: A DISTORTION TO VALIDATE AND GENERATE CURVED MESHES ON CAD SURFACES Gargallo-Peiró, A.; Roca, X.; Peraire, J. International Journal for Numerical Methods in Engineering, Vol. 106, Issue 13 https://doi.org/10.1002/nme.5162	journal	December 2015
Weighted Essentially Non-oscillatory Schemes Liu, Xu-Dong; Osher, Stanley; Chan, Tony Journal of Computational Physics, Vol. 115, Issue 1 https://doi.org/10.1006/jcph.1994.1187	journal	November 1994
Efficient Implementation of Weighted ENO Schemes Jiang, Guang-Shan; Shu, Chi-Wang Journal of Computational Physics, Vol. 126, Issue 1 https://doi.org/10.1006/jcph.1996.0130	journal	June 1996
A Conservative Shock Fitting Method on Unstructured Grids Trépanier, J. -Y.; Paraschivoiu, M.; Reggio, M. Journal of Computational Physics, Vol. 126, Issue 2 https://doi.org/10.1006/jcph.1996.0147	journal	July 1996
A Problem-Independent Limiter for High-Order Runge–Kutta Discontinuous Galerkin Methods Burbeau, A.; Sagaut, P.; Bruneau, Ch. -H. Journal of Computational Physics, Vol. 169, Issue 1 https://doi.org/10.1006/jcph.2001.6718	journal	May 2001
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
Steady shock tracking and Newton's method applied to one-dimensional duct flow Shubin, G. R.; Stephens, A. B.; Glaz, H. M. Journal of Computational Physics, Vol. 39, Issue 2 https://doi.org/10.1016/0021-9991(81)90157-1	journal	February 1981
Fully implicit shock tracking Bell, J. B.; Shubin, G. R.; Solomon, J. M. Journal of Computational Physics, Vol. 48, Issue 2 https://doi.org/10.1016/0021-9991(82)90048-1	journal	November 1982
Self adjusting grid methods for one-dimensional hyperbolic conservation laws Harten, Ami; Hyman, James M. Journal of Computational Physics, Vol. 50, Issue 2 https://doi.org/10.1016/0021-9991(83)90066-9	journal	May 1983
Uniformly high order accurate essentially non-oscillatory schemes, III Harten, Ami; Engquist, Bjorn; Osher, Stanley Journal of Computational Physics, Vol. 71, Issue 2 https://doi.org/10.1016/0021-9991(87)90031-3	journal	August 1987
Thirty-six years of shock fitting Moretti, Gino Computers & Fluids, Vol. 31, Issue 4-7 https://doi.org/10.1016/S0045-7930(01)00072-X	journal	May 2002
Discontinuous Galerkin solution of the Navier–Stokes equations on deformable domains Persson, P. -O.; Bonet, J.; Peraire, J. Computer Methods in Applied Mechanics and Engineering, Vol. 198, Issue 17-20 https://doi.org/10.1016/j.cma.2009.01.012	journal	April 2009
Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves Johnsen, Eric; Larsson, Johan; Bhagatwala, Ankit V. Journal of Computational Physics, Vol. 229, Issue 4 https://doi.org/10.1016/j.jcp.2009.10.028	journal	February 2010
On high-order shock-fitting and front-tracking schemes for numerical simulation of shock–disturbance interactions Rawat, Pradeep Singh; Zhong, Xiaolin Journal of Computational Physics, Vol. 229, Issue 19 https://doi.org/10.1016/j.jcp.2010.05.021	journal	September 2010
An optimization-based approach for high-order accurate discretization of conservation laws with discontinuous solutions Zahr, M. J.; Persson, P. -O. Journal of Computational Physics, Vol. 365 https://doi.org/10.1016/j.jcp.2018.03.029	journal	July 2018
Implicit shock tracking using an optimization-based high-order discontinuous Galerkin method Zahr, M. J.; Shi, A.; Persson, P. -O. Journal of Computational Physics, Vol. 410 https://doi.org/10.1016/j.jcp.2020.109385	journal	June 2020
Extrapolated Shock Tracking: Bridging shock-fitting and embedded boundary methods Ciallella, Mirco; Ricchiuto, Mario; Paciorri, Renato Journal of Computational Physics, Vol. 412 https://doi.org/10.1016/j.jcp.2020.109440	journal	July 2020
A novel stabilization method for high-order shock fitting with finite element methods D'Aquila, Luke M.; Helenbrook, Brian T.; Mazaheri, Alireza Journal of Computational Physics, Vol. 430 https://doi.org/10.1016/j.jcp.2020.110096	journal	April 2021
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
Steady Shock Tracking, Newton’s Method, and the Supersonic Blunt Body Problem Shubin, G. R.; Stephens, A. B.; Glaz, H. M. SIAM Journal on Scientific and Statistical Computing, Vol. 3, Issue 2 https://doi.org/10.1137/0903009	journal	June 1982
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems Arnold, Douglas N.; Brezzi, Franco; Cockburn, Bernardo SIAM Journal on Numerical Analysis, Vol. 39, Issue 5 https://doi.org/10.1137/S0036142901384162	journal	January 2002
Conservative Front Tracking with Improved Accuracy Glimm, James; Li, Xiaolin; Liu, Yingjie SIAM Journal on Numerical Analysis, Vol. 41, Issue 5 https://doi.org/10.1137/S0036142901388627	journal	January 2003
Multidimensional Least Squares Fluctuation Distribution Schemes with Adaptive Mesh Movement for Steady Hyperbolic Equations Baines, M. J.; Leary, S. J.; Hubbard, M. E. SIAM Journal on Scientific Computing, Vol. 23, Issue 5 https://doi.org/10.1137/S1064827500370202	journal	January 2002
A Study of Several Artiﬁcial Viscosity Models within the Discontinuous Galerkin Framework Hesthaven, Jian Yu & Jan S. Communications in Computational Physics, Vol. 27, Issue 5 https://doi.org/10.4208/cicp.OA-2019-0118	journal	February 2020

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71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
Discontinuous Galerkin
High-order methods
High-speed flows
Method of lines
Shock fitting
Shock tracking

Implicit shock tracking for unsteady flows by the method of lines

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