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Entropy Production by Implicit Runge–Kutta Schemes

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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

  1. 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].

  2. 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 \( (\tilde{A}^{T} \tilde{A})^{ - 1} \tilde{A}^{T} . \).

  3. 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.

  4. 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.

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Acknowledgements

This work has been supported by the Spanish Ministry of Defence and INTA under the research program “Termofluidodinámica” (IGB99001).

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  1. Computational Aerodynamics Group, National Institute for Aerospace Technology (INTA), Torrejón de Ardoz, Spain

    Carlos Lozano

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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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