Relaxation Runge--Kutta Methods: Fully Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier--Stokes Equations
Abstract
The framework of inner product norm preserving relaxation Runge--Kutta methods [D. I. Ketcheson, SIAM J. Numer. Anal., 57 (2019), pp. 2850--2870] is extended to general convex quantities. Conservation, dissipation, or other solution properties with respect to any convex functional are enforced by the addition of a relaxation parameter that multiplies the Runge--Kutta update at each step. Moreover, other desirable stability (such as strong stability preservation) and efficiency (such as low storage requirements) properties are preserved. The technique can be applied to both explicit and implicit Runge--Kutta methods and requires only a small modification to existing implementations. The computational cost at each step is the solution of one additional scalar algebraic equation for which a good initial guess is available. The effectiveness of this approach is proved analytically and demonstrated in several numerical examples, including applications to high order entropy-conservative and entropy-stable semidiscretizations on unstructured grids for the compressible Euler and Navier--Stokes equations.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
S. Abhyankar, J. Brown, E. M. Constantinescu, D. Ghosh, B. F. Smith, and H. Zhang, PETSc/TS: A Modern Scalable ODE/DAE Solver Library, https://arxiv.org/abs/1806.01437, 2018.
2.
S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, R. T. Mills, T. Munson, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, H. Zhang, and H. Zhang, PETSc Users Manual, Technical report ANL-95/11 (revision 3.10), Argonne National Laboratory, 2018.
3.
P. Bogacki and L. F. Shampine, A 3 (2) pair of Runge--Kutta formulas, Appl. Math. Lett., 2 (1989), pp. 321--325, https://doi.org/10.1016/0893-9659(89)90079-7.
4.
P. Bogacki and L. F. Shampine, An efficient Runge--Kutta (4, 5) pair, Comput. Math. Appl., 32 (1996), pp. 15--28, https://doi.org/10.1016/0898-1221(96)00141-1.
5.
R. P. Brent, Algorithms for Minimization Without Derivatives, Ser. Automat. Comput., Prentice-Hall, Englewood Cliffs, NJ, 1973.
6.
J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, Chichester, 2008.
7.
M. Calvo, D. Hernández-Abreu, J. I. Montijano, and L. Rández, On the preservation of invariants by explicit Runge--Kutta methods, SIAM J. Sci. Comput., 28 (2006), pp. 868--885, https://doi.org/10.1137/04061979X.
8.
M. Calvo, M. Laburta, J. Montijano, and L. Rández, Projection methods preserving Lyapunov functions, BIT, 50 (2010), pp. 223--241, https://doi.org/10.1007/s10543-010-0259-3.
9.
M. Calvo, M. Laburta, J. I. Montijano, and L. Rández, Runge--Kutta projection methods with low dispersion and dissipation errors, Adv. Comput. Math., 41 (2015), pp. 231--251, https://doi.org/10.1007/s10444-014-9355-2.
10.
M. H. Carpenter, T. C. Fisher, E. J. Nielsen, and S. H. Frankel, Entropy stable spectral collocation schemes for the Navier-Stokes equations: Discontinuous interfaces, SIAM J. Sci. Comput., 36 (2014), pp. B835--B867, https://doi.org/10.1137/130932193.
11.
M. H. Carpenter and C. A. Kennedy, Fourth-Order 2N Runge--Kutta Schemes, Technical report NASA-TM-109112, NASA, 1994, https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19940028444.pdf.
12.
M. H. Carpenter, M. Parsani, E. J. Nielsen, and T. C. Fisher, Towards an entropy stable spectral element framework for computational fluid dynamics, in Proceedings of the 54th AIAA Aerospace Sciences Meeting, AIAA 2016-1058, American Institute of Aeronautics and Astronautics, 2016.
13.
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, 2010.
14.
L. Dalcin, D. B. Rojas, S. Zampini, D. C. Del Rey Fernández, M. H. Carpenter, and M. Parsani, Conservative and entropy stable solid wall boundary conditions for the compressible Navier--Stokes equations: Adiabatic wall and heat entropy transfer, J. Comput. Phys., 397 (2019).
15.
K. Dekker and J. G. Verwer, Stability of Runge--Kutta Methods for Stiff Nonlinear Differential Equations, CWI Monogr. 2, North-Holland, Amsterdam, 1984.
16.
N. Del Buono and C. Mastroserio, Explicit methods based on a class of four stage fourth order Runge--Kutta methods for preserving quadratic laws, J. Comput. Appl. Math., 140 (2002), pp. 231--243, https://doi.org/10.1016/S0377-0427(01)00398-3.
17.
D. C. D. R. Fernández, M. H. Carpenter, L. Dalcin, L. Fredrich, D. Rojas, A. R. Winters, G. J. Gassner, S. Zampini, and M. Parsani, Entropy stable $p$-nonconforming discretizations with the summation-by-parts property for the compressible Euler equations, SIAM J. Sci. Comput., 2019, submitted.
18.
D. C. D. R. Fernández, M. H. Carpenter, L. Dalcin, L. Fredrich, A. R. Winters, G. J. Gassner, and M. Parsani, Entropy stable $p$-nonconforming discretizations with the summation-by-parts property for the compressible Navier--Stokes equations, Comput. & Fluids, 2019, submitted.
19.
L. Friedrich, G. Schnücke, A. R. Winters, D. C. D. R. Fernández, G. J. Gassner, and M. H. Carpenter, Entropy stable space-time discontinuous Galerkin schemes with summation-by-parts property for hyperbolic conservation laws, J. Sci. Comput., 80 (2019), pp. 175--222, https://doi.org/10.1007/s10915-019-00933-2.
20.
S. Gottlieb, D. I. Ketcheson, and C.-W. Shu, Strong Stability Preserving Runge--Kutta and Multistep Time Discretizations, World Scientific, Singapore, 2011.
21.
V. Grimm and G. Quispel, Geometric integration methods that preserve Lyapunov functions, BIT, 45 (2005), pp. 709--723, https://doi.org/10.1007/s10543-005-0034-z.
22.
E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Ser. Comput. Math. 31, Springer-Verlag, Berlin, 2006, https://doi.org/10.1007/3-540-30666-8.
23.
E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Ser. Comput. Math. 8, Springer-Verlag, Berlin, 2008, https://doi.org/10.1007/978-3-540-78862-1.
24.
I. Higueras, Monotonicity for Runge--Kutta methods: Inner product norms, J. Sci. Comput., 24 (2005), pp. 97--117, https://doi.org/10.1007/s10915-004-4789-1.
25.
E. Jones, T. Oliphant, P. Peterson, et al., SciPy: Open Source Scientific Tools for Python, 2001, http://www.scipy.org.
26.
D. I. Ketcheson, Highly efficient strong stability-preserving Runge--Kutta methods with low-storage implementations, SIAM J. Sci. Comput., 30 (2008), pp. 2113--2136, https://doi.org/10.1137/07070485X.
27.
D. I. Ketcheson, Relaxation Runge--Kutta methods: Conservation and stability for inner-product norms, SIAM J. Numer. Anal., 57 (2019), pp. 2850--2870.
28.
M. G. Knepley and D. A. Karpeev, Mesh algorithms for PDE with Sieve I: Mesh distribution, Sci. Program., 17 (2009), pp. 215--230, https://doi.org/10.3233/SPR-2009-0249.
29.
H. Kojima, Invariants preserving schemes based on explicit Runge--Kutta methods, BIT, 56 (2016), pp. 1317--1337, https://doi.org/10.1007/s10543-016-0608-y.
30.
M. Laburta, J. I. Montijano, L. Rández, and M. Calvo, Numerical methods for non conservative perturbations of conservative problems, Comput. Phys. Commun., 187 (2015), pp. 72--82, https://doi.org/10.1016/j.cpc.2014.10.012.
31.
P. G. LeFloch, J.-M. Mercier, and C. Rohde, Fully discrete, entropy conservative schemes of arbitrary order, SIAM J. Numer. Anal., 40 (2002), pp. 1968--1992, https://doi.org/10.1137/S003614290240069X.
32.
C. Lozano, Entropy production by explicit Runge--Kutta schemes, J. Sci. Comput., 76 (2018), pp. 521--565, https://doi.org/10.1007/s10915-017-0627-0.
33.
J. J. Moré, B. S. Garbow, and K. E. Hillstrom, User Guide for MINPACK-1, Technical report ANL-80-74, Argonne National Laboratory, Argonne, Illinois, 1980.
34.
P. Öffner, J. Glaubitz, and H. Ranocha, Analysis of artificial dissipation of explicit and implicit time-integration methods, Int. J. Numer. Anal. Model., to appear.
35.
M. Parsani, M. H. Carpenter, T. C. Fisher, and E. J. Nielsen, Entropy stable staggered grid discontinuous spectral collocation methods of any order for the compressible Navier--Stokes equations, SIAM J. Sci. Comput., 38 (2016), pp. A3129--A3162.
36.
M. Parsani, M. H. Carpenter, and E. J. Nielsen, Entropy stable discontinuous interfaces coupling for the three-dimensional compressible Navier--Stokes equations, J. Comput. Phys., 290 (2015), pp. 132--138.
37.
M. Parsani, M. H. Carpenter, and E. J. Nielsen, Entropy stable wall boundary conditions for the three-dimensional compressible Navier--Stokes equations, J. Comput. Phys., 292 (2015), pp. 88--113.
38.
H. Ranocha, Shallow water equations: Split-form, entropy stable, well-balanced, and positivity preserving numerical methods, GEM Int. J. Geomath., 8 (2017), pp. 85--133, https://doi.org/10.1007/s13137-016-0089-9.
39.
H. Ranocha, On Strong Stability of Explicit Runge--Kutta Methods for Nonlinear Semibounded Operators, 2018, https://arxiv.org/abs/1811.11601.
40.
H. Ranocha, J. Glaubitz, P. Öffner, and T. Sonar, Stability of artificial dissipation and modal filtering for flux reconstruction schemes using summation-by-parts operators, Appl. Numer. Math., 128 (2018), pp. 1--23, https://doi.org/10.1016/j.apnum.2018.01.019.
41.
H. Ranocha and D. I. Ketcheson, ConvexRelaxationRungeKutta. Relaxation Runge--Kutta Methods for Convex Functionals, 2019\ https://github.com/ranocha/ConvexRelaxationRungeKutta.
42.
H. Ranocha and P. Öffner, $L_2$ stability of explicit Runge--Kutta schemes, J. Sci. Comput., 75 (2018), pp. 1040--1056, https://doi.org/10.1007/s10915-017-0595-4.
43.
C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), pp. 439--471, https://doi.org/10.1016/0021-9991(88)90177-5.
44.
J. Slotnick, A. Khodadoust, J. Alonso, D. Darmofal, W. Gropp, E. Lurie, and D. Mavriplis, CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences, NASA CR-2014-218178 (2014).
45.
E. Süli and D. F. Mayers, An Introduction to Numerical Analysis, Cambridge University Press, Cambridge, 2003, https://doi.org/10.1017/CBO9780511801181.
46.
Z. Sun and C.-W. Shu, Stability of the fourth order Runge--Kutta method for time-dependent partial differential equations, Ann. Math. Sci. Appl., 2 (2017), pp. 255--284, https://doi.org/10.4310/AMSA.2017.v2.n2.a3.
47.
Z. Sun and C.-W. Shu, Strong stability of explicit Runge--Kutta time discretizations, SIAM J. Numer. Anal., 57 (2019), pp. 1158--1182.
48.
M. Svärd and H. Özcan, Entropy-stable schemes for the Euler equations with far-field and wall boundary conditions, J. Sci. Comput., 58 (2014), pp. 61--89.
49.
E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws. I, Math. Comp., 49 (1987), pp. 91--103, https://doi.org/10.1090/S0025-5718-1987-0890255-3.
50.
E. Tadmor, From semidiscrete to fully discrete: Stability of Runge--Kutta schemes by the energy method II, in Collected Lectures on the Preservation of Stability under Discretization, D. J. Estep and S. Tavener, eds., Proc. Appl. Math. 109, SIAM, Philadelphia, 2002, pp. 25--49.
51.
E. Tadmor, Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta Numer., 12 (2003), pp. 451--512, https://doi.org/10.1017/S0962492902000156.
52.
E. Tadmor and W. Zhong, Entropy stable approximations of Navier-Stokes equations with no artificial numerical viscosity, J. Hyperbolic Differ. Equ., 3 (2006), pp. 529--559, https://doi.org/10.1142/S0219891606000896.
53.
V. A. Titarev and E. F. Toro, Finite volume WENO schemes for three-dimensional conservation laws, J. Comput. Phys., 201 (2014), pp. 238--260.
54.
J. H. Verner, Explicit Runge--Kutta methods with estimates of the local truncation error, SIAM J. Numer. Anal., 15 (1978), pp. 772--790, https://doi.org/10.1137/0715051.
55.
N. Wintermeyer, A. R. Winters, G. J. Gassner, and D. A. Kopriva, An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry, J. Comput. Phys., 340 (2017), pp. 200--242.
56.
A. R. Winters and G. J. Gassner, Affordable, entropy conserving and entropy stable flux functions for the ideal MHD equations, J. Comput. Phys., 304 (2016), pp. 72--108.
57.
H. Zakerzadeh and U. S. Fjordholm, High-order accurate, fully discrete entropy stable schemes for scalar conservation laws, IMA J. Numer. Anal., 36 (2016), pp. 633--654, https://doi.org/10.1093/imanum/drv020.
58.
X. Zhang and C.-W. Shu, Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments, Proc. Roy. A, 467 (2011), pp. 2752--2776, https://doi.org/10.1098/rspa.2011.0153.
Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: A612 - A638
DOI: 10.1137/19M1263480
ISSN (online): 1095-7197
Copyright
© 2020, Society for Industrial and Applied Mathematics.
History
Submitted: 22 May 2019
Accepted: 6 November 2019
Published online: 12 March 2020
Keywords
MSC codes
Authors
Funding Information
King Abdullah University of Science and Technology https://doi.org/10.13039/501100004052
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Arbitrarily high-order explicit energy-conserving methods for the generalized nonlinear fractional Schrödinger wave equationsMathematics and Computers in Simulation, Vol. 216 | 1 Feb 2024
- Resolving entropy growth from iterative methodsBIT Numerical Mathematics, Vol. 63, No. 4 | 15 September 2023
- Efficient GPU parallelization of adaptive mesh refinement technique for high-order compressible solver with immersed boundaryComputers & Fluids, Vol. 266 | 1 Nov 2023
- Multiple-Relaxation Runge Kutta Methods for Conservative Dynamical SystemsJournal of Scientific Computing, Vol. 97, No. 1 | 14 August 2023
- Applications of Limiters, Neural Networks and Polynomial Annihilation in Higher-Order FD/FV SchemesJournal of Scientific Computing, Vol. 97, No. 1 | 7 September 2023
- Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High OrderCommunications on Applied Mathematics and Computation, Vol. 35 | 11 September 2023
- Functional‐preserving predictor‐corrector multiderivative schemesPAMM, Vol. 11 | 10 September 2023
- High Order Conservative Schemes for the Generalized Benjamin–Ono Equation on the Unbounded DomainJournal of Scientific Computing, Vol. 96, No. 2 | 15 June 2023
- Stability of structure-aware Taylor methods for tentsMathematics of Computation, Vol. 92, No. 341 | 27 January 2023
- Entropy-stable discontinuous Galerkin difference methods for hyperbolic conservation lawsJournal of Computational and Applied Mathematics, Vol. 422 | 1 Apr 2023
- Bounds preserving temporal integration methods for hyperbolic conservation lawsComputers & Mathematics with Applications, Vol. 135 | 1 Apr 2023
- Global entropy stability for a class of unlimited second-order schemes for 2D hyperbolic systems of conservation laws on unstructured meshesJournal of Computational Physics | 1 Apr 2023
- A simple and general framework for the construction of thermodynamically compatible schemes for computational fluid and solid mechanicsApplied Mathematics and Computation, Vol. 440 | 1 Mar 2023
- High order entropy preserving ADER-DG schemesApplied Mathematics and Computation, Vol. 440 | 1 Mar 2023
- Discrete Adjoint Computations for Relaxation Runge–Kutta MethodsJournal of Scientific Computing, Vol. 94, No. 3 | 1 February 2023
- Linearly implicit and high-order energy-preserving relaxation schemes for highly oscillatory Hamiltonian systemsJournal of Computational Physics, Vol. 477 | 1 Mar 2023
- A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equationsJournal of Computational Physics, Vol. 475 | 1 Feb 2023
- Implicit-explicit relaxation Runge-Kutta methods: construction, analysis and applications to PDEsMathematics of Computation, Vol. 92, No. 339 | 19 August 2022
- Convergence of discontinuous Galerkin schemes for the Euler equations via dissipative weak solutionsApplied Mathematics and Computation, Vol. 436 | 1 Jan 2023
- Mimetic Relaxation Runge Kutta MethodsSpectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1 | 29 November 2022
- Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid DynamicsCommunications on Applied Mathematics and Computation, Vol. 4, No. 4 | 22 November 2021
- Inducing multistability in discrete chaotic systems using numerical integration with variable symmetryChaos, Solitons & Fractals, Vol. 165 | 1 Dec 2022
- A New Family of Thermodynamically Compatible Discontinuous Galerkin Methods for Continuum Mechanics and Turbulent Shallow Water FlowsJournal of Scientific Computing, Vol. 93, No. 2 | 10 October 2022
- Entropy stable discontinuous Galerkin methods for balance laws in non-conservative form: Applications to the Euler equations with gravityJournal of Computational Physics, Vol. 468 | 1 Nov 2022
- On Energy Laws and Stability of Runge--Kutta Methods for Linear Seminegative ProblemsSIAM Journal on Numerical Analysis, Vol. 60, No. 5 | 13 September 2022
- Entropy–Preserving and Entropy–Stable Relaxation IMEX and Multirate Time–Stepping MethodsJournal of Scientific Computing, Vol. 93, No. 1 | 4 September 2022
- On the performance of relaxation and adaptive explicit Runge–Kutta schemes for high-order compressible flow simulationsJournal of Computational Physics, Vol. 464 | 1 Sep 2022
- An Entropic Method for Discrete Systems with Gibbs EntropySIAM Journal on Numerical Analysis, Vol. 60, No. 4 | 31 August 2022
- Performance analysis of relaxation Runge–Kutta methodsThe International Journal of High Performance Computing Applications, Vol. 36, No. 4 | 12 May 2022
- Active flow control optimization with stability guaranteesAIAA AVIATION 2022 Forum | 20 June 2022
- Reinterpretation and extension of entropy correction terms for residual distribution and discontinuous Galerkin schemes: Application to structure preserving discretizationJournal of Computational Physics, Vol. 453 | 1 Mar 2022
- Limiter-based entropy stabilization of semi-discrete and fully discrete schemes for nonlinear hyperbolic problemsComputer Methods in Applied Mechanics and Engineering, Vol. 389 | 1 Feb 2022
- Conservative and Entropy-Stable Nonconformal Interfaces With Lower Accuracy Quadrature: Circumventing the Inner-Product Preservation PropertyAIAA SCITECH 2022 Forum | 29 December 2022
- Entropy stable modal discontinuous Galerkin schemes and wall boundary conditions for the compressible Navier-Stokes equationsJournal of Computational Physics, Vol. 448 | 1 Jan 2022
- Relaxation Deferred Correction Methods and their Applications to Residual Distribution SchemesThe SMAI Journal of computational mathematics, Vol. 8 | 13 October 2022
- Enforcing Strong Stability of Explicit Runge-Kutta Methods with SuperviscosityCommunications on Applied Mathematics and Computation, Vol. 3, No. 4 | 4 March 2021
- On the rate of error growth in time for numerical solutions of nonlinear dispersive wave equationsPartial Differential Equations and Applications, Vol. 2, No. 6 | 18 October 2021
- Mortar-based Entropy-Stable Discontinuous Galerkin Methods on Non-conforming Quadrilateral and Hexahedral MeshesJournal of Scientific Computing, Vol. 89, No. 2 | 15 October 2021
- A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equationsESAIM: Mathematical Modelling and Numerical Analysis, Vol. 55, No. 6 | 15 November 2021
- Entropy stabilization and property-preserving limiters for ℙ 1 discontinuous Galerkin discretizations of scalar hyperbolic problemsJournal of Numerical Mathematics, Vol. 29, No. 4 | 2 December 2021
- An IoT-Based Motion Tracking System for Next-Generation Foot-Related Sports Training and Talent SelectionJournal of Healthcare Engineering, Vol. 2021 | 25 Jun 2021
- Kinetic Functions for Nonclassical Shocks, Entropy Stability, and Discrete Summation by PartsJournal of Scientific Computing, Vol. 87, No. 2 | 7 April 2021
- DeC and ADER: Similarities, Differences and a Unified FrameworkJournal of Scientific Computing, Vol. 87, No. 1 | 17 February 2021
- A New Class of A Stable Summation by Parts Time Integration Schemes with Strong Initial ConditionsJournal of Scientific Computing, Vol. 87, No. 1 | 10 March 2021
- On Entropy-Stable Discretizations and the Entropy AdjointJournal of Scientific Computing, Vol. 86, No. 3 | 18 January 2021
- Numerical analysis and applications of explicit high order maximum principle preserving integrating factor Runge-Kutta schemes for Allen-Cahn equationApplied Numerical Mathematics, Vol. 161 | 1 Mar 2021
- On the robustness and performance of entropy stable collocated discontinuous Galerkin methodsJournal of Computational Physics, Vol. 426 | 1 Feb 2021
- A Novel Robust Strategy for Discontinuous Galerkin Methods in Computational Fluid Mechanics: Why? When? What? Where?Frontiers in Physics, Vol. 8 | 29 January 2021
- Simulation of Turbulent Flows Using a Fully Discrete Explicit hp -nonconforming Entropy Stable Solver of Any Order on Unstructured GridsAIAA Scitech 2021 Forum | 4 January 2021
- A Conservative Fully Discrete Numerical Method for the Regularized Shallow Water Wave EquationsSIAM Journal on Scientific Computing, Vol. 43, No. 2 | 26 April 2021
- Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier–Stokes EquationsEfficient High-Order Discretizations for Computational Fluid Dynamics | 5 January 2021
- High-order accurate entropy-stable discontinuous collocated Galerkin methods with the summation-by-parts property for compressible CFD frameworks: Scalable SSDC algorithms and flow solverJournal of Computational Physics, Vol. 424 | 1 Jan 2021
- Positivity-preserving adaptive Runge–Kutta methodsCommunications in Applied Mathematics and Computational Science, Vol. 16, No. 2 | 2 November 2021
- General relaxation methods for initial-value problems with application to multistep schemesNumerische Mathematik, Vol. 146, No. 4 | 28 October 2020
- Highly efficient invariant-conserving explicit Runge-Kutta schemes for nonlinear Hamiltonian differential equationsJournal of Computational Physics, Vol. 418 | 1 Oct 2020
- Fully discrete explicit locally entropy-stable schemes for the compressible Euler and Navier–Stokes equationsComputers & Mathematics with Applications, Vol. 80, No. 5 | 1 Sep 2020
- Relaxation Runge–Kutta Methods for Hamiltonian ProblemsJournal of Scientific Computing, Vol. 84, No. 1 | 9 July 2020
- Optimized geometrical metrics satisfying free-stream preservationComputers & Fluids, Vol. 207 | 1 Jul 2020
- Entropy-stable Galerkin difference discretization on unstructured gridsAIAA AVIATION 2020 FORUM | 8 June 2020
- Entropy stable h/p-nonconforming discretization with the summation-by-parts property for the compressible Euler and Navier–Stokes equationsSN Partial Differential Equations and Applications, Vol. 1, No. 2 | 16 March 2020
- Energy Stability of Explicit Runge--Kutta Methods for Nonautonomous or Nonlinear ProblemsSIAM Journal on Numerical Analysis, Vol. 58, No. 6 | 24 November 2020