SMAI logo

Journal of Computational Mathematics

Maximum-principle-satisfying second-order Intrusive Polynomial Moment scheme
Jonas Kusch1; Graham W. Alldredge2; Martin Frank1
1 Karlsruhe Institute of Technology, Karlsruhe, Germany
2 FU Berlin, Berlin, Germany
The SMAI Journal of computational mathematics, Volume 5 (2019), pp. 23-51.

Using standard intrusive techniques when solving hyperbolic conservation laws with uncertainties can lead to oscillatory solutions as well as nonhyperbolic moment systems. The Intrusive Polynomial Moment (IPM) method ensures hyperbolicity of the moment system while restricting oscillatory over- and undershoots to specified bounds. In this contribution, we derive a second-order discretization of the IPM moment system which fulfills the maximum principle. This task is carried out by investigating violations of the specified bounds due to the errors from the numerical optimization required by the scheme. This analysis gives weaker conditions on the entropy that is used, allowing the choice of an entropy which enables choosing the exact minimal and maximal value of the initial condition as bounds. Solutions calculated with the derived scheme are nonoscillatory while fulfilling the maximum principle. The second-order accuracy of our scheme leads to significantly reduced numerical costs.

Published online:
DOI: 10.5802/smai-jcm.42
Classification: 35L65, 35R60, 65M08
Keywords: uncertainty quantification, conservation laws, maximum principle, moment system, hyperbolic, oscillations
Jonas Kusch 1; Graham W. Alldredge 2; Martin Frank 1

1 Karlsruhe Institute of Technology, Karlsruhe, Germany
2 FU Berlin, Berlin, Germany
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{SMAI-JCM_2019__5__23_0,
     author = {Jonas Kusch and Graham W. Alldredge and Martin Frank},
     title = {Maximum-principle-satisfying second-order {Intrusive} {Polynomial} {Moment} scheme},
     journal = {The SMAI Journal of computational mathematics},
     pages = {23--51},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {5},
     year = {2019},
     doi = {10.5802/smai-jcm.42},
     zbl = {07090178},
     mrnumber = {3928534},
     language = {en},
     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.42/}
}
TY  - JOUR
AU  - Jonas Kusch
AU  - Graham W. Alldredge
AU  - Martin Frank
TI  - Maximum-principle-satisfying second-order Intrusive Polynomial Moment scheme
JO  - The SMAI Journal of computational mathematics
PY  - 2019
SP  - 23
EP  - 51
VL  - 5
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.42/
DO  - 10.5802/smai-jcm.42
LA  - en
ID  - SMAI-JCM_2019__5__23_0
ER  - 
%0 Journal Article
%A Jonas Kusch
%A Graham W. Alldredge
%A Martin Frank
%T Maximum-principle-satisfying second-order Intrusive Polynomial Moment scheme
%J The SMAI Journal of computational mathematics
%D 2019
%P 23-51
%V 5
%I Société de Mathématiques Appliquées et Industrielles
%U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.42/
%R 10.5802/smai-jcm.42
%G en
%F SMAI-JCM_2019__5__23_0
Jonas Kusch; Graham W. Alldredge; Martin Frank. Maximum-principle-satisfying second-order Intrusive Polynomial Moment scheme. The SMAI Journal of computational mathematics, Volume 5 (2019), pp. 23-51. doi : 10.5802/smai-jcm.42. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.42/

[1] G. Alldredge; C. D Hauck; A. L. Tits High-order entropy-based closures for linear transport in slab geometry II: A computational study of the optimization problem, SIAM Journal on Scientific Computing, Volume 34 (2012) no. 4, p. B361-B391 | MR | Zbl

[2] G. W. Alldredge; C. D. Hauck; D. P. OĹeary; A. L. Tits Adaptive change of basis in entropy-based moment closures for linear kinetic equations, Journal of Computational Physics, Volume 258 (2014), pp. 489-508 | DOI | MR | Zbl

[3] J. B. Bell; C. N. Dawson; G. R. Shubin An unsplit, higher order Godunov method for scalar conservation laws in multiple dimensions, Journal of Computational Physics, Volume 74 (1988) no. 1, pp. 1-24 | Zbl

[4] C. Canuto; A. Quarteroni Approximation results for orthogonal polynomials in Sobolev spaces, Mathematics of Computation, Volume 38 (1982) no. 157, pp. 67-86 | DOI | MR | Zbl

[5] K. M Case; P. F. Zweifel Linear transport theory, Addison-Wesley Pub. Co., 1967 | Zbl

[6] S. Chandrasekhar Stochastic problems in physics and astronomy, Reviews of modern physics, Volume 15 (1943) no. 1, pp. 1-89 | DOI | MR | Zbl

[7] P. Colella Multidimensional upwind methods for hyperbolic conservation laws, Journal of Computational Physics, Volume 87 (1990) no. 1, pp. 171-200 | MR | Zbl

[8] B. Després; G. Poëtte; D. Lucor Robust Uncertainty Propagation in Systems of Conservation Laws with the Entropy Closure Method, Springer International Publishing (2013), pp. 105-149

[9] B. Dubroca; A. Klar Half-moment closure for radiative transfer equations, Journal of Computational Physics, Volume 180 (2002) no. 2, pp. 584-596 | Zbl

[10] C. K. Garrett; C. Hauck; J. Hill Optimization and large scale computation of an entropy-based moment closure, Journal of Computational Physics, Volume 302 (2015), pp. 573-590 | DOI | MR | Zbl

[11] R. G Ghanem; P. D. Spanos Stochastic Finite Elements: A Spectral Approach, Dover, 2003 | Zbl

[12] D. Gottlieb; D. Xiu Galerkin method for wave equations with uncertain coefficients, Commun. Comput. Phys, Volume 3 (2008) no. 2, pp. 505-518 | MR | Zbl

[13] S. Gottlieb; C.-W. Shu; E. Tadmor Strong stability-preserving high-order time discretization methods, SIAM review, Volume 43 (2001) no. 1, pp. 89-112 | DOI | MR | Zbl

[14] J.-L. Guermond; M. Nazarov; B. Popov; Y. Yang A second-order maximum principle preserving Lagrange finite element technique for nonlinear scalar conservation equations, SIAM Journal on Numerical Analysis, Volume 52 (2014) no. 4, pp. 2163-2182 | DOI | MR | Zbl

[15] C. Hauck; R. McClarren Positive P N Closures, SIAM Journal on Scientific Computing, Volume 32 (2010) no. 5, pp. 2603-2626 | MR | Zbl

[16] C. D. Hauck High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci, Volume 9 (2011) no. 1, pp. 187-205 | DOI | MR | Zbl

[17] H. Holden; N. H. Risebro Front tracking for hyperbolic conservation laws, 152, Springer, 2015 | MR | Zbl

[18] J. Kusch Uncertainty Quantification for Hyperbolic Equations, RWTH Aachen University (2015), pp. 1-23

[19] R. J. LeVeque Numerical Methods for Conservation Laws, Birkhäuser Verlag Basel, 1992 | Zbl

[20] R. J. LeVeque Nonlinear conservation laws and finite volume methods, Computational methods for astrophysical fluid flow, Springer, 1998, pp. 1-159 | Zbl

[21] C. D. Levermore Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, Volume 83 (1996) no. 5-6, pp. 1021-1065 | DOI | MR | Zbl

[22] E. E. Lewis; W. F. Miller Computational Methods of Neutron Transport, John Wiley and Sons, Inc., New York, NY, 1984 | Zbl

[23] X.-D. Liu A maximum principle satisfying modification of triangle based adapative stencils for the solution of scalar hyperbolic conservation laws, SIAM journal on numerical analysis, Volume 30 (1993) no. 3, pp. 701-716 | DOI | MR

[24] X.-D. Liu; S. Osher Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes I, SIAM Journal on Numerical Analysis, Volume 33 (1996) no. 2, pp. 760-779 | DOI | MR | Zbl

[25] G. Poëtte; B. Després; D. Lucor Uncertainty quantification for systems of conservation laws, Journal of Computational Physics, Volume 228 (2009) no. 7, pp. 2443-2467 | DOI | MR

[26] G. Poëtte; B. Després; D. Lucor Treatment of uncertain material interfaces in compressible flows, Computer Methods in Applied Mechanics and Engineering, Volume 200 (2011) no. 1, pp. 284-308 | MR | Zbl

[27] G. Poëtte; B. Després; D. Lucor Uncertainty propagation for systems of conservation laws, high order stochastic spectral methods, Spectral and High Order Methods for Partial Differential Equations, Springer, 2011, pp. 293-305 | DOI | Zbl

[28] G. C. Pomraning The Equations of Radiation Hydrodynamics, Oxford, 1973

[29] C.-W. Shu Total-variation-diminishing time discretizations, SIAM Journal on Scientific and Statistical Computing, Volume 9 (1988) no. 6, pp. 1073-1084 | DOI | MR | Zbl

[30] N. Wiener The homogeneous chaos, American Journal of Mathematics, Volume 60 (1938) no. 4, pp. 897-936 | MR | Zbl

[31] D. Xiu; G. Em Karniadakis Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, Computer Methods in Applied Mechanics and Engineering, Volume 191 (2002) no. 43, pp. 4927-4948 | DOI | MR | Zbl

[32] X. Zhang; C.-W. Shu On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, Journal of Computational Physics, Volume 229 (2010) no. 23, pp. 8918-8934 | DOI | MR | Zbl

[33] X. Zhang; C.-W. Shu Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (2011) | Zbl

Cited by Sources: