On the convergence of the regularized entropy-based moment method for kinetic equations
Alldredge, Graham W.1 ; Frank, Martin2 ; Giesselmann, Jan3
1 Berlin, Germany
2 Department of Mathematics, Karlsruhe Institute of Technology, Karlsruhe, Germany
3 Department of Mathematics, Technical University of Darmstadt, Darmstadt, Germany
The SMAI Journal of computational mathematics, Tome 9 (2023), pp. 1-29

The entropy-based moment method is a well-known discretization for the velocity variable in kinetic equations which has many desirable theoretical properties but is difficult to implement with high-order numerical methods. The regularized entropy-based moment method was recently introduced to remove one of the main challenges in the implementation of the entropy-based moment method, namely the requirement of the realizability of the numerical solution. In this work we use the method of relative entropy to prove the convergence of the regularized method to the original method as the regularization parameter goes to zero and give convergence rates. Our main assumptions are the boundedness of the velocity domain and that the original moment solution is Lipschitz continuous in space and bounded away from the boundary of realizability. We provide results from numerical simulations showing that the convergence rates we prove are optimal.

Publié le :
DOI : 10.5802/smai-jcm.93

Alldredge, Graham W. 1 ; Frank, Martin 2 ; Giesselmann, Jan 3

1 Berlin, Germany
2 Department of Mathematics, Karlsruhe Institute of Technology, Karlsruhe, Germany
3 Department of Mathematics, Technical University of Darmstadt, Darmstadt, Germany 
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Alldredge, Graham W.; Frank, Martin; Giesselmann, Jan. On the convergence of the regularized entropy-based moment method for kinetic equations. The SMAI Journal of computational mathematics, Tome 9 (2023), pp. 1-29. doi: 10.5802/smai-jcm.93

[1] Alldredge, Graham W.; Frank, Martin; Hauck, Cory D. A regularized entropy-based moment method for kinetic equations, SIAM J. Appl. Math., Volume 79 (2019) no. 5, pp. 1627-1653 | DOI | MR | Zbl

[2] Alldredge, Graham W.; Schneider, Florian A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension, J. Comput. Phys., Volume 295 (2015), pp. 665-684 | DOI | MR | Zbl

[3] Berthelin, Florent; Tzavaras, Athanasios E.; Vasseur, Alexis From discrete velocity Boltzmann equations to gas dynamics before shocks, J. Stat. Phys., Volume 135 (2009) no. 1, pp. 153-173 | DOI | MR | Zbl

[4] Berthelin, Florent; Vasseur, Alexis From kinetic equations to multidimensional isentropic gas dynamics before shocks, SIAM J. Math. Anal., Volume 36 (2005) no. 6, pp. 1807-1835 | DOI | MR | Zbl

[5] Borwein, Jonathan M.; Lewis, Adrian S. Duality Relationships for Entropy-Like Minimization Problems, SIAM J. Control Optim., Volume 29 (1991) no. 2, pp. 325-338 | MR | Zbl | DOI

[6] Boyd, Stephen; Vandenberghe, Lieven Convex optimization, Cambridge University Press, 2004 | DOI

[7] Cercignani, Carlo The Boltzmann Equation and Its Applications, Springer, 1988 | DOI

[8] Coulombel, Jean-François; Goudon, Thierry Entropy-Based Moment Closure For Kinetic Equations: Riemann Problem And Invariant Regions, J. Hyperbolic Differ. Equ., Volume 6 (2006), pp. 649-671 | MR | DOI | Zbl

[9] Dafermos, Constantine M. Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften, 325, Springer, 2016, xxxviii+826 pages | DOI | MR

[10] Evans, Lawrence C. Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 2010

[11] Giesselmann, Jan; Lattanzio, Corrado; Tzavaras, Athanasios E. Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics, Arch. Ration. Mech. Anal., Volume 223 (2017) no. 3, pp. 1427-1484 | DOI | MR | Zbl

[12] Giesselmann, Jan; Tzavaras, Athanasios E. Stability properties of the Euler-Korteweg system with nonmonotone pressures, Appl. Anal., Volume 96 (2017) no. 9, pp. 1528-1546 | DOI | MR | Zbl

[13] Hauck, Cory D. High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., Volume 9 (2011), pp. 187-205 | MR | DOI | Zbl

[14] Hauck, Cory D.; Levermore, C. David; Tits, André L. Convex Duality and Entropy-Based Moment Closures: Characterizing Degenerate Densities, SIAM J. Control Optim., Volume 47 (2008) no. 4, pp. 1977-2015 | DOI | MR | Zbl

[15] Junk, Michael Domain of Definition of Levermore’s Five Moment System, J. Stat. Phys., Volume 93 (1998) no. 5-6, pp. 1143-1167 | MR | DOI | Zbl

[16] Junk, Michael Maximum entropy for reduced moment problems, Math. Models Methods Appl. Sci., Volume 10 (2000), pp. 1001-1025 | MR | DOI | Zbl

[17] Krupa, Sam G. Criteria for the a-contraction and stability for the piecewise-smooth solutions to hyperbolic balance laws, Commun. Math. Sci., Volume 18 (2020), pp. 1493-1537 | Zbl | MR | DOI

[18] Levermore, C. David Moment Closure Hierarchies for Kinetic Theories, J. Stat. Phys., Volume 83 (1996), pp. 1021-1065 | MR | DOI | Zbl

[19] Lewis, E. E.; Miller, W. F. Jr. Computational Methods in Neutron Transport, John Wiley & Sons, 1984

[20] Markowich, Peter A.; Ringhofer, Christian A.; Schmeiser, Christian Semiconductor Equations, Springer, 1990 | DOI

[21] Mihalas, Dimitri; Weibel-Mihalas, Barbara Foundations of Radiation Hydrodynamics, Courier Corporation, 1999

[22] Serre, Denis; Vasseur, Alexis About the relative entropy method for hyperbolic systems of conservation laws, A panorama of mathematics: pure and applied (Contemporary Mathematics), Volume 658, American Mathematical Society, 2016, pp. 237-248 | DOI | MR | Zbl

[23] Tzavaras, Athanasios E. Relative entropy in hyperbolic relaxation, Commun. Math. Sci., Volume 3 (2005) no. 2, pp. 119-132 | MR | Zbl | DOI

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