An exponential time-integrator scheme for steady and unsteady inviscid flows

Li, Shu-Jie; Luo, Li-Shi; Wang, Z. J.; Ju, Lili

doi:10.1016/j.jcp.2018.03.020

An exponential time-integrator scheme for steady and unsteady inviscid flows

Journal Article · Tue Mar 20 00:00:00 EDT 2018 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2018.03.020· OSTI ID:1538432

Li, Shu-Jie ^[1]; Luo, Li-Shi ^[2]; ^[3]; Ju, Lili ^[4]

Beijing Computational Science Research Center, Beijing (China); DOE/OSTI
Beijing Computational Science Research Center, Beijing (China); Old Dominion University, Norfolk, VA (United States)
University of Kansas, Lawrence, KS (United States)
University of South Carolina, Columbia, SC (United States)

In this report an exponential time-integrator scheme of second-order accuracy based on the predictor–corrector methodology, denoted PCEXP, is developed to solve multi-dimensional nonlinear partial differential equations pertaining to fluid dynamics. The effective and efficient implementation of PCEXP is realized by means of the Krylov method. The linear stability and truncation error are analyzed through a one-dimensional model equation. The proposed PCEXP scheme is applied to the Euler equations discretized with a discontinuous Galerkin method in both two and three dimensions. The effectiveness and efficiency of the PCEXP scheme are demonstrated for both steady and unsteady inviscid flows. The accuracy and efficiency of the PCEXP scheme are verified and validated through comparisons with the explicit third-order total variation diminishing Runge–Kutta scheme (TVDRK3), the implicit backward Euler (BE) and the implicit second-order backward difference formula (BDF2). For unsteady flows, the PCEXP scheme generates a temporal error much smaller than the BDF2 scheme does, while maintaining the expected acceleration at the same time. Moreover, the PCEXP scheme is also shown to achieve the computational efficiency comparable to the implicit schemes for steady flows.

View Accepted Manuscript (DOE)

Research Organization:: University of South Carolina, Columbia, SC (United States)

Sponsoring Organization:: Beijing Computational Science Research Center (CSRC); National Natural Science Foundation of China (NSFC); National Science Foundation (NSF); USDOE; USDOE Office of Science (SC)

Grant/Contract Number:: SC0016540

OSTI ID:: 1538432

Alternate ID(s):: OSTI ID: 1548545
OSTI ID: 23087776

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: C Vol. 365; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (33)

High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations Bassi, F.; Rebay, S. Journal of Computational Physics, Vol. 138, Issue 2 https://doi.org/10.1006/jcph.1997.5454	journal	December 1997
A new class of exponential propagation iterative methods of Runge–Kutta type (EPIRK) Tokman, M. Journal of Computational Physics, Vol. 230, Issue 24 https://doi.org/10.1016/j.jcp.2011.08.023	journal	October 2011
Exponential Rosenbrock-Type Methods Hochbruck, Marlis; Ostermann, Alexander; Schweitzer, Julia SIAM Journal on Numerical Analysis, Vol. 47, Issue 1 https://doi.org/10.1137/080717717	journal	January 2009
Analysis and Application of an Orthogonal Nodal Basis on Triangles for Discontinuous Spectral Element Methods Deng, Shaozhong; Cai, Wei Applied Numerical Analysis & Computational Mathematics, Vol. 2, Issue 3 https://doi.org/10.1002/anac.200510007	journal	December 2005
Exponential time integration using Krylov subspaces Schulze, J. C.; Schmid, P. J.; Sesterhenn, J. L. International Journal for Numerical Methods in Fluids, Vol. 60, Issue 6 https://doi.org/10.1002/fld.1902	journal	June 2009
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
Higher‐order discontinuous Galerkin method for pyramidal elements using orthogonal bases Bergot, Morgane; Duruflé, Marc Numerical Methods for Partial Differential Equations, Vol. 29, Issue 1 https://doi.org/10.1002/num.21703	journal	March 2012
Krylov Methods for the Incompressible Navier-Stokes Equations Edwards, W. S.; Tuckerman, L. S.; Friesner, R. A. Journal of Computational Physics, Vol. 110, Issue 1 https://doi.org/10.1006/jcph.1994.1007	journal	January 1994
The Runge–Kutta Discontinuous Galerkin Method for Conservation Laws V Cockburn, Bernardo; Shu, Chi-Wang Journal of Computational Physics, Vol. 141, Issue 2 https://doi.org/10.1006/jcph.1998.5892	journal	April 1998
Exponential Time Differencing for Stiff Systems Cox, S. M.; Matthews, P. C. Journal of Computational Physics, Vol. 176, Issue 2 https://doi.org/10.1006/jcph.2002.6995	journal	March 2002
A BDF2 integration method with step size control for elasto-plasticity Eckert, S.; Baaser, H.; Gross, D. Computational Mechanics, Vol. 34, Issue 5 https://doi.org/10.1007/s00466-004-0581-1	journal	May 2004
A Class of Explicit Exponential General Linear Methods Ostermann, A.; Thalhammer, M.; Wright, W. M. BIT Numerical Mathematics, Vol. 46, Issue 2 https://doi.org/10.1007/s10543-006-0054-3	journal	May 2006
Influence of Reference-to-Physical Frame Mappings on Approximation Properties of Discontinuous Piecewise Polynomial Spaces Botti, Lorenzo Journal of Scientific Computing, Vol. 52, Issue 3 https://doi.org/10.1007/s10915-011-9566-3	journal	December 2011
Fast Explicit Integration Factor Methods for Semilinear Parabolic Equations Ju, Lili; Zhang, Jian; Zhu, Liyong Journal of Scientific Computing, Vol. 62, Issue 2 https://doi.org/10.1007/s10915-014-9862-9	journal	May 2014
Fast High-Order Compact Exponential Time Differencing Runge–Kutta Methods for Second-Order Semilinear Parabolic Equations Zhu, Liyong; Ju, Lili; Zhao, Weidong Journal of Scientific Computing, Vol. 67, Issue 3 https://doi.org/10.1007/s10915-015-0117-1	journal	October 2015
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
Implementation of exponential Rosenbrock-type integrators Caliari, Marco; Ostermann, Alexander Applied Numerical Mathematics, Vol. 59, Issue 3-4 https://doi.org/10.1016/j.apnum.2008.03.021	journal	March 2009
Comparative performance of exponential, implicit, and explicit integrators for stiff systems of ODEs Loffeld, J.; Tokman, M. Journal of Computational and Applied Mathematics, Vol. 241 https://doi.org/10.1016/j.cam.2012.09.038	journal	March 2013
High-order methods for computational fluid dynamics: A brief review of compact differential formulations on unstructured grids Huynh, H. T.; Wang, Z. J.; Vincent, P. E. Computers & Fluids, Vol. 98 https://doi.org/10.1016/j.compfluid.2013.12.007	journal	July 2014
Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods Tokman, M. Journal of Computational Physics, Vol. 213, Issue 2 https://doi.org/10.1016/j.jcp.2005.08.032	journal	April 2006
Efficient semi-implicit schemes for stiff systems Nie, Qing; Zhang, Yong-Tao; Zhao, Rui Journal of Computational Physics, Vol. 214, Issue 2 https://doi.org/10.1016/j.jcp.2005.09.030	journal	May 2006
Compact integration factor methods in high spatial dimensions Nie, Qing; Wan, Frederic Y. M.; Zhang, Yong-Tao Journal of Computational Physics, Vol. 227, Issue 10 https://doi.org/10.1016/j.jcp.2008.01.050	journal	May 2008
High order multi-moment constrained finite volume method. Part I: Basic formulation Ii, Satoshi; Xiao, Feng Journal of Computational Physics, Vol. 228, Issue 10 https://doi.org/10.1016/j.jcp.2009.02.009	journal	June 2009
A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids Wang, Z. J.; Gao, Haiyang Journal of Computational Physics, Vol. 228, Issue 21 https://doi.org/10.1016/j.jcp.2009.07.036	journal	November 2009
Krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes: Application to discontinuous Galerkin methods Chen, Shanqin; Zhang, Yong-Tao Journal of Computational Physics, Vol. 230, Issue 11 https://doi.org/10.1016/j.jcp.2011.01.010	journal	May 2011
Efficient design of exponential-Krylov integrators for large scale computing Tokman, M.; Loffeld, J. Procedia Computer Science, Vol. 1, Issue 1 https://doi.org/10.1016/j.procs.2010.04.026	journal	May 2010
Application of a Higher Order Discontinuous Galerkin Wolkov, A. V.; Hirsch, Ch.; Petrovskaya, N. B. Mathematical Modelling of Natural Phenomena, Vol. 6, Issue 3 https://doi.org/10.1051/mmnp/20116310	journal	January 2011
Analysis of Some Krylov Subspace Approximations to the Matrix Exponential Operator Saad, Y. SIAM Journal on Numerical Analysis, Vol. 29, Issue 1 https://doi.org/10.1137/0729014	journal	February 1992
On Krylov Subspace Approximations to the Matrix Exponential Operator Hochbruck, Marlis; Lubich, Christian SIAM Journal on Numerical Analysis, Vol. 34, Issue 5 https://doi.org/10.1137/S0036142995280572	journal	October 1997
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems Cockburn, Bernardo; Shu, Chi-Wang SIAM Journal on Numerical Analysis, Vol. 35, Issue 6 https://doi.org/10.1137/S0036142997316712	journal	December 1998
Exponential Integrators for Large Systems of Differential Equations Hochbruck, Marlis; Lubich, Christian; Selhofer, Hubert SIAM Journal on Scientific Computing, Vol. 19, Issue 5 https://doi.org/10.1137/S1064827595295337	journal	September 1998
A review of flux reconstruction or correction procedure via reconstruction method for the Navier-Stokes equations Wang, Z. J.; Huynh, H. T. Mechanical Engineering Reviews, Vol. 3, Issue 1 https://doi.org/10.1299/mer.15-00475	journal	January 2016
On the use of exponential time integration methods in atmospheric models Clancy, Colm; Pudykiewicz, Janusz A. Tellus A: Dynamic Meteorology and Oceanography, Vol. 65, Issue 1 https://doi.org/10.3402/tellusa.v65i0.20898	journal	May 2013

Similar Records

Inviscid-viscous coupled solution for unsteady flows through vibrating blades: Part 1--Description of the method

Journal Article · Thu Dec 31 23:00:00 EST 1992 · Journal of Turbomachinery; (United States) · OSTI ID:5975529

An Efficient, Semi-implicit Pressure-based Scheme Employing a High-resolution Finitie Element Method for Simulating Transient and Steady, Inviscid and Viscous, Compressible Flows on Unstructured Grids

Technical Report · Mon Mar 31 19:00:00 EST 2003 · OSTI ID:910726

Implicit shock tracking for unsteady flows by the method of lines

Journal Article · Sun Jan 09 19:00:00 EST 2022 · Journal of Computational Physics · OSTI ID:1883333

Related Subjects

97 MATHEMATICS AND COMPUTING
compressible flow
discontinuous Galerkin
exponential time integration
large time step
predictor–corrector method
unstructured meshes

An exponential time-integrator scheme for steady and unsteady inviscid flows

Citation Formats

References (33)

Similar Records

Related Subjects