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Entropy Production by Explicit Runge–Kutta Schemes

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Abstract

We propose and test a formula for the separate computation of the temporal and spatial entropy production of fully discrete, finite-volume, explicit Runge–Kutta discretizations of systems of conservation laws.

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Notes

  1. Physical entropy is a concave function of the state variables, but it is customary to set \(\eta \) equal to a negative function of the entropy, which causes the inequality to go in the opposite sense.

  2. SSP \(\equiv \) Strong Stability Preserving schemes [29] are RK or multistep schemes that can be written as convex combinations of Forward-Euler schemes.

  3. we are tacitly assuming here that we have a steady problem for which a terminal (steady) state is eventually attained, at which point \(U^{n+1}=U^{n}\) and, thus, \(\eta _i^{n+1} -\eta _i^n =0\) for all i.

  4. we assume that \(a_{21} a_{32} b_3 \ne 0\), which is automatically verified for \(3^{\mathrm{rd}}\)-order RK-3 schemes.

  5. For n real numbers, Jensen’s inequality yields

    $$\begin{aligned} (x_1 +\cdots +x_n )^{2}=\left( \frac{1}{n}nx_1 +\cdots +\frac{1}{n}nx_n \right) ^{2}\le \frac{1}{n}(nx_1 )^{2}+\cdots +\frac{1}{n}(nx_n )^{2}=n\left( {x_1 ^{2}+\cdots +x_n ^{2}} \right) \end{aligned}$$

    Hence, for the squared norm of the sum of \(n\,p\)-dimensional vectors we have

    $$\begin{aligned} \left( {\sum _{i=1}^n {\vec {v}_i } } \right) ^{2}= & {} \left( v_1^1 +\cdots +v_n^1\right) ^{2}+\cdots +(v_1^p +\cdots +v_n^p )^{2}\le n\left( {(v_1^1 )^{2}+\cdots +(v_n^1 )^{2}} \right) \\&+\cdots +n\left( {(v_1^p )^{2}+\cdots +(v_n^p )^{2}} \right) \\= & {} n\left( {(v_1^1 )^{2}+\cdots +(v_1^p )^{2}} \right) +\cdots +n\left( {(v_n^1 )^{2}+\cdots +(v_n^p )^{2}} \right) =n\sum _{i=1}^n {\vec {v}_i ^{2}}\end{aligned}$$

  6. An entropy conservative scheme is one for which the numerical flux verifies \(\Delta \hbox {v}_{i+\frac{1}{2}}^T \tilde{F}_{i+\frac{1}{2}} =\Delta \Theta _{i+\frac{1}{2}}\) and thus the entropy production \(\Pi _{i+\frac{1}{2}} =0\). In the scalar case, the entropy conservative flux is unique for each choice of entropy function and can be computed as \(\tilde{F}_{i+\frac{1}{2}} =\Delta \Theta _{i+\frac{1}{2}} /\Delta \hbox {v}_{i+\frac{1}{2}}\).

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Acknowledgements

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

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  1. Fluid Dynamics Group, INTA, Carretera de Ajalvir, km4, Torrejón de Ardoz (Madrid), 28850, Spain

    Carlos Lozano

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Appendix: Tadmor’s Identities

Appendix: Tadmor’s Identities

The following identities are used to obtain the discrete entropy equation for fully discrete schemes:

$$\begin{aligned}&\hbox {v}_i^T \left( F_{i+\frac{1}{2}} -F_{i-\frac{1}{2}}\right) \nonumber \\&\quad =\underbrace{\left( {\bar{\hbox {v}}}_{i+\frac{1}{2}}^T F_{i+\frac{1}{2}} -\bar{{\Theta }}_{i+\frac{1}{2}}\right) }_{\Phi _{i+\frac{1}{2}} }-\underbrace{\left( {\bar{\hbox {v}}}_{i-\frac{1}{2}}^T F_{i-\frac{1}{2}} -\bar{{\Theta }}_{i-\frac{1}{2}}\right) }_{\Phi _{i-\frac{1}{2}}}\nonumber \\&\qquad -\,{\underbrace{\frac{1}{2}\left[ \underbrace{\left( \Delta _{i+\frac{1}{2}} \hbox {v}^T F_{i+\frac{1}{2}} -\Delta _{i+\frac{1}{2}} \Theta \right) }_{\Pi _{i+\frac{1}{2}}}+\underbrace{\left( \Delta _{i-\frac{1}{2}} \hbox {v}^T F_{i-\frac{1}{2}} -\Delta _{i-\frac{1}{2}} \Theta \right) }_{\Pi _{i-\frac{1}{2}}}\right] }_{\Pi _i}}\nonumber \\&\quad (\hbox {v}_i^n )^{T}\left( U_i^{n+1} -U_i^n\right) =\eta _i^{n+1} -\eta _i^n -\underbrace{\int _{-1/2}^{1/2} {\left( {\xi +\frac{1}{2}} \right) \left( \Delta ^{n+\frac{1}{2}}\hbox {v}_i\right) ^{T}H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } }_{{E}_i^{n+\frac{1}{2}} }\nonumber \\&\quad \left( \hbox {v}_i^{n+1}\right) ^{T}\left( U_i^{n+1} -U_i^n\right) =\eta _i^{n+1} -\eta _i^n +\underbrace{\int _{-1/2}^{1/2} {\left( {\frac{1}{2}-\xi } \right) \left( \Delta ^{n+\frac{1}{2}}\hbox {v}_i \right) ^{T}H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } }_{\hbox {B}_i^{n+\frac{1}{2}} }\nonumber \\ \end{aligned}$$
(72)

The first one is straightforward. To prove the second one use

$$\begin{aligned} H(\hbox {v})= & {} \frac{\partial U}{\partial \hbox {v}}=(\eta _{UU} )^{-1} \nonumber \\ \hbox {v}(\xi )= & {} {\bar{\hbox {v}}}_i^{n+\frac{1}{2}} +\xi \Delta ^{n+\frac{1}{2}}\hbox {v}_i \nonumber \\ {\bar{\hbox {v}}}_i^{n+\frac{1}{2}}= & {} \frac{1}{2}\left( \hbox {v}_i^{n+1} +\hbox {v}_i^n\right) \nonumber \\ \Delta ^{n+\frac{1}{2}}\hbox {v}_i= & {} \hbox {v}_i^{n+1} -\hbox {v}_i^n \end{aligned}$$
(73)

to rewrite

$$\begin{aligned}&(\hbox {v}_i^n )^{T}\left( U_i^{n+1} -U_i^n\right) =(\hbox {v}_i^n )^{T}\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {U_\mathrm{v} d\hbox {v}} =(\hbox {v}_i^n )^{T}\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {H(\hbox {v})d\hbox {v}} \nonumber \\&\quad =(\hbox {v}_i^n )^{T}\int _{-1/2}^{1/2} {H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } \nonumber \\&\quad \hbox {Likewise}, \nonumber \\&\quad \eta _i^{n+1} -\eta _i^n =\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {(\eta _\mathrm{v} )^{T}d\hbox {v}} =\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {(\eta _U )^{T}U_\mathrm{v} d\hbox {v}} =\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {\hbox {v}^{T}H(\hbox {v})d\hbox {v}} \nonumber \\&\quad =\int _{-1/2}^{1/2} {\left( {\bar{\hbox {v}}}_i^{n+\frac{1}{2}} +\xi \Delta ^{n+\frac{1}{2}}\hbox {v}_i\right) ^{T}H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } =(\hbox {v}_i^n )^{T}\int _{-1/2}^{1/2} {H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } \nonumber \\&\qquad +\int _{-1/2}^{1/2} {\left( {\xi +\frac{1}{2}} \right) \left( \Delta ^{n+\frac{1}{2}}\hbox {v}_i\right) ^{T}H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } =(\hbox {v}_i^n )^{T}\left( U_i^{n+1} -U_i^n\right) +{E}_i^{n+\frac{1}{2}}\nonumber \\ \end{aligned}$$
(74)

In a similar fashion,

$$\begin{aligned}&\left( \hbox {v}_i^{n+1}\right) ^{T}\left( U_i^{n+1} -U_i^n\right) =\left( \hbox {v}_i^{n+1}\right) ^{T}\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {U_\mathrm{v} d\hbox {v}} =(\hbox {v}_i^n )^{T}\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {H(\hbox {v})d\hbox {v}} \nonumber \\&\quad =\left( \hbox {v}_i^{n+1}\right) ^{T}\int _{-1/2}^{1/2} {H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } \nonumber \\&\quad \hbox {Likewise},\nonumber \\&\quad \eta _i^{n+1} -\eta _i^n =\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {(\eta _\mathrm{v} )^{T}d\hbox {v}} =\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {(\eta _U )^{T}U_\mathrm{v} d\hbox {v}} =\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {\hbox {v}^{T}H(\hbox {v})d\hbox {v}} \nonumber \\&\quad =\int _{-1/2}^{1/2} {\left( {\bar{\hbox {v}}}_i^{n+\frac{1}{2}} +\xi \Delta ^{n+\frac{1}{2}}\hbox {v}_i\right) ^{T}H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } =\left( \hbox {v}_i^{n+1} \right) ^{T}\int _{-1/2}^{1/2} {H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } \nonumber \\&\qquad -\int _{-1/2}^{1/2} {\left( {\frac{1}{2}-\xi } \right) \left( \Delta ^{n+\frac{1}{2}}\hbox {v}_i\right) ^{T}H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } =\left( \hbox {v}_i^{n+1}\right) ^{T}\left( U_i^{n+1} -U_i^n\right) -B_i^{n+\frac{1}{2}}\nonumber \\ \end{aligned}$$
(75)

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Lozano, C. Entropy Production by Explicit Runge–Kutta Schemes. J Sci Comput 76, 521–564 (2018). https://doi.org/10.1007/s10915-017-0627-0

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