Abstract
This paper follows up on the author’s recent paper “Entropy Production by Explicit Runge–Kutta schemes” (Lozano in J Sci Comput 76(1):521–564, 2018. https://doi.org/10.1007/s10915-017-0627-0), where a formula for the production of entropy by fully discrete schemes with explicit Runge–Kutta time integrators was presented. In this paper, the focus is on implicit Runge–Kutta schemes, for which the fully discrete numerical entropy evolution scheme is derived and tested.
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Notes
In its original form, this generalized Crank–Nicolson scheme requires an intermediate temporal state that unfortunately does not generally have a closed form and requires quadrature. An explicit construction has been recently derived in [25].
If A is not invertible we can consider the enlarged matrix \( \tilde{A} = \left( {\frac{A}{{b^{T} }}} \right) \) (where \( b = (b_{1} , \ldots ,b_{s} )^{T} \)). If \( {\text{rank}}(\tilde{A}) = s \), we can invert (7) in the least-squares sense as \( R_{i} = - (\tilde{A}^{T} \tilde{A})^{ - 1} \tilde{A}^{T} \Delta U_{i} \), (where \( \Delta U_{i} = \left( {U_{i}^{(1)} - U_{i}^{n} , \ldots ,U_{i}^{(s)} - U_{i}^{n} } \right)^{T} \) and \( R_{i} = (R_{i}^{1} , \ldots ,{\kern 1pt} R_{i}^{s} )^{T} \)), so in (14) we should replace \( A^{ - 1} \) with \( (\tilde{A}^{T} \tilde{A})^{ - 1} \tilde{A}^{T} . \).
An entropy conservative scheme is one for which the numerical flux verifies \( \Delta {\text{v}}_{{i + \tfrac{1}{2}}}^{T} \tilde{F}_{{i + \tfrac{1}{2}}} = \Delta \varTheta_{{i + \tfrac{1}{2}}} \) and thus the entropy production \( \varPi_{{i + \tfrac{1}{2}}} = 0 \). In the scalar case, the entropy conservative flux is unique for each choice of entropy function and can be computed as \( \tilde{F}_{{i + \tfrac{1}{2}}} = \Delta \varTheta_{{i + \tfrac{1}{2}}} /\Delta {\text{v}}_{{i + \tfrac{1}{2}}} \). As explained in [12], entropy stable schemes can be constructed by coupling an entropy conservative scheme to a suitable diffusion operator.
A Runge–Kutta method is algebraically stable if the matrices B and \( M = BA + A^{T} B - bb^{T} \) are both non-negative definite ([32], p. 275). If A is non-singular, then M and Q are congruent, \( Q = A^{ - T} MA^{ - 1} \), and thus M is non-negative definite iff Q is.
References
Lozano, C.: Entropy production by explicit Runge–Kutta Schemes. J. Sci. Comput. 76(1), 521–564 (2018). https://doi.org/10.1007/s10915-017-0627-0
Butcher, J.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, Chichester (2008)
Kennedy, C.A., Carpenter, M.H.: Diagonally implicit Runge–Kutta methods for ordinary differential equations. A review. NASA/TM–2016–219173 (2016)
Gottlieb, S., Ketcheson, D.: Time discretization techniques, chapter 21. In: Abgrall, R., Shu, C.W. (eds.) Handbook of numerical methods for hyperbolic problems, vol. 17, pp. 549–583. Elsevier, Amsterdam (2016)
Persson, P.-O., Willis, D., Peraire, J.: The numerical simulation of flapping wings at low Reynolds numbers. In: AIAA Paper 2010-724 (2010)
Bijl, H., Carpenter, M.H., Vatsa, V.N., Kennedy, C.A.: Implicit time integration schemes for the unsteady compressible Navier-Stokes equations: laminar flow. J. Comput. Phys. 179(1), 313–329 (2002). https://doi.org/10.1006/jcph.2002.7059
Jameson, A.: Evaluation of fully implicit Runge Kutta schemes for unsteady flow calculations. J. Sci. Comput. 73(2–3), 819–852 (2017). https://doi.org/10.1007/s10915-017-0476-x
Lax, P.: Shock waves and entropy. In: Zarantonello, E. (ed.) Contributions to Nonlinear Functional Analysis, pp. 603–634. Academic Press, New York (1971)
Harten, A., Hyman, J.M., Lax, P.D., Keyfitz, B.: On finite-difference approximations and entropy conditions for shocks. Commun. Pure Appl. Math. 29, 297–322 (1976)
Osher, S.: Riemann solvers, the entropy condition, and difference approximations. SIAM J. Numer. Anal. 21(2), 217–235 (1984)
Tadmor, E.: Entropy stable schemes, chapter 18. In: Abgrall, R., Shu, C.W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems, vol. 17, pp. 467–493. Elsevier, Amsterdam (2016)
Tadmor, E.: Entropy Stability Theory for Difference Approximations of Nonlinear Conservation Laws and Related Time-dependent Problems. Acta Numerica 12, 451–512 (2003). https://doi.org/10.1017/S0962492902000156
Lefloch, P.G., Mercier, J.M., Rohde, C.: Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40(5), 1968–1992 (2002)
Fjordholm, U., Mishra, S., Tadmor, E.: Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50(2), 544–573 (2012)
Jiang, G.-S., Shu, C.-W.: On a cell entropy inequality for discontinuous Galerkin method. Math. Comp. 62, 531–538 (1994)
Shu, C.W.: Discontinuous Galerkin methods: General approach and stability. In: Numerical Solutions of Partial Differential Equations, Advanced Courses in Mathematics CRM Barcelona, pp. 149–201 (2009)
Hiltebrand, A., Mishra, S.: Entropy stable shock capturing streamline diffusion space-time discontinuous Galerkin (DG) methods for systems of conservation laws. Numer. Math. 126, 103–151 (2014). https://doi.org/10.1007/s00211-013-0558-0
Qiu, J., Zhang, Q.: Stability, error estimate and limiters of discontinuous Galerkin methods. In: Abgrall, R., Shu, C.W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems, chapter 7, vol. 17, pp. 147–171. Elsevier, Amsterdam (2016)
Chen, T., Shu, C.-W.: Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys. 345, 427–461 (2017). https://doi.org/10.1016/j.jcp.2017.05.025
Carpenter, M., Fisher, T., Nielsen, E., Parsan, M., Svärd, M., Yamaleev, N.: Entropy stable summation-by-parts formulations for compressible computational fluid dynamics, chapter 19. In: Abgrall, R., Shu, C.W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems, vol. 17, pp. 495–524. Elsevier, Amsterdam (2016)
Jameson, A.: The construction of discretely conservative finite volume schemes that also globally conserve energy or entropy. J. Sci. Comput. 34, 152–187 (2008)
Jameson, A.: Formulation of kinetic energy preserving conservative schemes for gas dynamics and direct numerical simulation of one-dimensional viscous compressible flow in a shock tube using entropy and kinetic energy preserving schemes. J. Sci. Comput. 34, 188–208 (2008)
Ismail, F., Roe, P.L.: Affordable, entropy-consistent Euler flux functions II: entropy production at shocks. J. Comput. Phys. 228(15), 5410–5436 (2009)
Diosdady, L., Murman, S.: Higher-order methods for compressible turbulent flows using entropy variables. In: AIAA Paper 2015-0294, 53rd AIAA Aerospace Sciences Meeting, Kissimmee, FL, Jan 5, 2015
Gouasmi, A., Murman, S., Duraisamy, K.: Entropy conservative schemes and the receding flow problem. Preprint arXiv:1801.10132v1 [math.NA], January 30, 2018
Gouasmi, A., Duraisamy, K., Murman, S.: On entropy stable temporal fluxes. Preprint arXiv:1807.03483v2 [math.NA], July 21, 2018
Fjordholm, U., Mishra, S., Tadmor, E.: Energy preserving and energy stable schemes for the shallow water equations. In: Cucker, F., Pinkus, A., Todd, M. (eds.), Proceedings of Foundations of Computational Mathematics, London Math. Soc. Lecture Notes Ser. 36393-139, Hong Kong (2009)
Merriam, M.L.: An entropy-based approach to nonlinear stability. NASA TM-101086, March 1989
Harten, A.: On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49(1), 151–164 (1983)
Tadmor, E.: Entropy functions for symmetric systems of conservation laws. J. Math. Anal. Appl. 122(2), 355–359 (1987)
Burrage, K.: Stability and efficiency properties of implicit Runge–Kutta methods. PhD Thesis, University of Auckland, 1978
Lambert, J.: Numerical methods for ordinary differential systems. The initial value problem. Wiley, ISBN 0-471-92990-5, 1991
Burrage, K., Butcher, J.: Stability criteria for implicit Runge–Kutta methods. SIAM J. Numer. Anal. 16, 46–57 (1979)
Kraaijevanger, J., Spijker, M.N.: Algebraic stability and error propagation in Runge–Kutta methods. Appl. Numer. Math. 5(1–2), 71–87 (1989). https://doi.org/10.1016/0168-9274(89)90025-1
Zakerzadeh, H., Fjordholm, U.: High-order accurate, fully discrete entropy stable schemes for scalar conservation laws. IMA J. Numer. Anal. 36(2), 633–654 (2016)
Tadmor, E., Zhong, W.: Novel entropy stable schemes for 1D and 2D fluid equations. In: Benzoni-Gavage, S., Serre, D. (eds.) Hyperbolic problems: theory, numerics, applications, pp. 1111–1119. Springer, Berlin (2008)
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This work has been supported by the Spanish Ministry of Defence and INTA under the research program “Termofluidodinámica” (IGB99001).
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Lozano, C. Entropy Production by Implicit Runge–Kutta Schemes. J Sci Comput 79, 1832–1853 (2019). https://doi.org/10.1007/s10915-019-00914-5
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DOI: https://doi.org/10.1007/s10915-019-00914-5