Discretizing advection equations with rough velocity fields on non-Cartesian grids
Authors:
Pierre-Emmanuel Jabin and Datong Zhou
Journal:
Quart. Appl. Math. 82 (2024), 229-303
MSC (2020):
Primary 35F25, 65M12
DOI:
https://doi.org/10.1090/qam/1649
Published electronically:
March 30, 2023
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References |
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Abstract: We investigate the properties of discretizations of advection equations on non-Cartesian grids and graphs in general. Advection equations discretized on non-Cartesian grids have remained a long-standing challenge as the structure of the grid can lead to strong oscillations in the solution, even for otherwise constant velocity fields. We introduce a new method to track oscillations of the solution for rough velocity fields on any graph. The method in particular highlights some inherent structural conditions on the mesh for propagating regularity on solutions.
References
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- R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511–547. MR 1022305, DOI 10.1007/BF01393835
- R. Eymard, T. Gallouët, R. Herbin, and J. C. Latché, A convergent finite element-finite volume scheme for the compressible Stokes problem. II. The isentropic case, Math. Comp. 79 (2010), no. 270, 649–675. MR 2600538, DOI 10.1090/S0025-5718-09-02310-2
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- Shizan Fang, Dejun Luo, and Anton Thalmaier, Stochastic differential equations with coefficients in Sobolev spaces, J. Funct. Anal. 259 (2010), no. 5, 1129–1168. MR 2652184, DOI 10.1016/j.jfa.2010.02.014
- T. Gallouët, R. Herbin, and J.-C. Latché, Kinetic energy control in explicit finite volume discretizations of the incompressible and compressible Navier-Stokes equations, Int. J. Finite Vol. 7 (2010), no. 2, 6. MR 2753585
- Thierry Gallouët, Laura Gastaldo, Raphaele Herbin, and Jean-Claude Latché, An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations, M2AN Math. Model. Numer. Anal. 42 (2008), no. 2, 303–331. MR 2405150, DOI 10.1051/m2an:2008005
- T. Gallouët, R. Herbin, and J.-C. Latché, A convergent finite element-finite volume scheme for the compressible Stokes problem. I. The isothermal case, Math. Comp. 78 (2009), no. 267, 1333–1352. MR 2501053, DOI 10.1090/S0025-5718-09-02216-9
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- Mahir Hadžić, Andreas Seeger, Charles K. Smart, and Brian Street, Singular integrals and a problem on mixing flows, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 4, 921–943. MR 3795021, DOI 10.1016/j.anihpc.2017.09.001
- Gautam Iyer, Alexander Kiselev, and Xiaoqian Xu, Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows, Nonlinearity 27 (2014), no. 5, 973–985. MR 3207161, DOI 10.1088/0951-7715/27/5/973
- P.-E. Jabin and N. Masmoudi, DiPerna-Lions flow for relativistic particles in an electromagnetic field, Arch. Ration. Mech. Anal. 217 (2015), no. 3, 1029–1067. MR 3356994, DOI 10.1007/s00205-015-0850-5
- Pierre-Emmanuel Jabin, Critical non-Sobolev regularity for continuity equations with rough velocity fields, J. Differential Equations 260 (2016), no. 5, 4739–4757. MR 3437603, DOI 10.1016/j.jde.2015.11.028
- Flavien Léger, A new approach to bounds on mixing, Math. Models Methods Appl. Sci. 28 (2018), no. 5, 829–849. MR 3799259, DOI 10.1142/S0218202518500215
- Benoit Merlet, $L^\infty$- and $L^2$-error estimates for a finite volume approximation of linear advection, SIAM J. Numer. Anal. 46 (2007/08), no. 1, 124–150. MR 2377258, DOI 10.1137/060664057
- Benoît Merlet and Julien Vovelle, Error estimate for finite volume scheme, Numer. Math. 106 (2007), no. 1, 129–155. MR 2286009, DOI 10.1007/s00211-006-0053-y
- Benoît Perthame and Anne-Laure Dalibard, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2319–2335. MR 2471920, DOI 10.1090/S0002-9947-08-04656-4
- André Schlichting and Christian Seis, Convergence rates for upwind schemes with rough coefficients, SIAM J. Numer. Anal. 55 (2017), no. 2, 812–840. MR 3631392, DOI 10.1137/16M1084882
- Christian Seis, Maximal mixing by incompressible fluid flows, Nonlinearity 26 (2013), no. 12, 3279–3289. MR 3141856, DOI 10.1088/0951-7715/26/12/3279
- Christian Seis, A quantitative theory for the continuity equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 34 (2017), no. 7, 1837–1850. MR 3724758, DOI 10.1016/j.anihpc.2017.01.001
- Christian Seis, Optimal stability estimates for continuity equations, Proc. Roy. Soc. Edinburgh Sect. A 148 (2018), no. 6, 1279–1296. MR 3869178, DOI 10.1017/S0308210518000173
- Chad M. Topaz and Andrea L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math. 65 (2004), no. 1, 152–174. MR 2111591, DOI 10.1137/S0036139903437424
- Xicheng Zhang, Stochastic flows of SDEs with irregular coefficients and stochastic transport equations, Bull. Sci. Math. 134 (2010), no. 4, 340–378. MR 2651896, DOI 10.1016/j.bulsci.2009.12.004
References
- Giovanni Alberti, Gianluca Crippa, and Anna L. Mazzucato, Exponential self-similar mixing and loss of regularity for continuity equations, C. R. Math. Acad. Sci. Paris 352 (2014), no. 11, 901–906 (English, with English and French summaries). MR 3268760, DOI 10.1016/j.crma.2014.08.021
- Luigi Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math. 158 (2004), no. 2, 227–260. MR 2096794, DOI 10.1007/s00222-004-0367-2
- Luigi Ambrosio, Myriam Lecumberry, and Stefania Maniglia, Lipschitz regularity and approximate differentiability of the DiPerna-Lions flow, Rend. Sem. Mat. Univ. Padova 114 (2005), 29–50 (2006). MR 2207860
- F. Ben Belgacem and P.-E. Jabin, Convergence of numerical approximations to non-linear continuity equations with rough force fields, Arch. Ration. Mech. Anal. 234 (2019), no. 2, 509–547. MR 3995045, DOI 10.1007/s00205-019-01396-3
- Fethi Ben Belgacem and Pierre-Emmanuel Jabin, Compactness for nonlinear continuity equations, J. Funct. Anal. 264 (2013), no. 1, 139–168. MR 2995703, DOI 10.1016/j.jfa.2012.10.005
- Anna Bohun, François Bouchut, and Gianluca Crippa, Lagrangian flows for vector fields with anisotropic regularity, Ann. Inst. H. Poincaré C Anal. Non Linéaire 33 (2016), no. 6, 1409–1429. MR 3569235, DOI 10.1016/j.anihpc.2015.05.005
- François Bouchut, Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Ration. Mech. Anal. 157 (2001), no. 1, 75–90. MR 1822415, DOI 10.1007/PL00004237
- François Bouchut and Gianluca Crippa, Lagrangian flows for vector fields with gradient given by a singular integral, J. Hyperbolic Differ. Equ. 10 (2013), no. 2, 235–282. MR 3078074, DOI 10.1142/S0219891613500100
- Jean Bourgain, Haim Brezis, and Petru Mironescu, Another look at Sobolev spaces, Optimal control and partial differential equations, IOS, Amsterdam, 2001, pp. 439–455. MR 3586796
- Franck Boyer, Analysis of the upwind finite volume method for general initial- and boundary-value transport problems, IMA J. Numer. Anal. 32 (2012), no. 4, 1404–1439. MR 2991833, DOI 10.1093/imanum/drr031
- Yann Brenier, Felix Otto, and Christian Seis, Upper bounds on coarsening rates in demixing binary viscous liquids, SIAM J. Math. Anal. 43 (2011), no. 1, 114–134. MR 2765685, DOI 10.1137/090775142
- Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
- Didier Bresch and Pierre-Emmanuel Jabin, Global weak solutions of PDEs for compressible media: a compactness criterion to cover new physical situations, Shocks, singularities and oscillations in nonlinear optics and fluid mechanics, Springer INdAM Ser., vol. 17, Springer, Cham, 2017, pp. 33–54. MR 3675552
- Didier Bresch and Pierre-Emmanuel Jabin, Global existence of weak solutions for compressible Navier-Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor, Ann. of Math. (2) 188 (2018), no. 2, 577–684. MR 3862947, DOI 10.4007/annals.2018.188.2.4
- Elia Bruè and Quoc-Hung Nguyen, Sharp regularity estimates for solutions of the continuity equation drifted by Sobolev vector fields, Anal. PDE 14 (2021), no. 8, 2539–2559. MR 4377866, DOI 10.2140/apde.2021.14.2539
- Martin Burger, Yasmin Dolak-Struss, and Christian Schmeiser, Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions, Commun. Math. Sci. 6 (2008), no. 1, 1–28. MR 2397995
- Nicolas Champagnat and Pierre-Emmanuel Jabin, Well posedness in any dimension for Hamiltonian flows with non BV force terms, Comm. Partial Differential Equations 35 (2010), no. 5, 786–816. MR 2753620, DOI 10.1080/03605301003646705
- Nicolas Champagnat and Pierre-Emmanuel Jabin, Strong solutions to stochastic differential equations with rough coefficients, Ann. Probab. 46 (2018), no. 3, 1498–1541. MR 3785594, DOI 10.1214/17-AOP1208
- Maria Colombo, Gianluca Crippa, and Stefano Spirito, Renormalized solutions to the continuity equation with an integrable damping term, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 1831–1845. MR 3396434, DOI 10.1007/s00526-015-0845-y
- Gianluca Crippa and Camillo De Lellis, Estimates and regularity results for the DiPerna-Lions flow, J. Reine Angew. Math. 616 (2008), 15–46. MR 2369485, DOI 10.1515/CRELLE.2008.016
- François Delarue and Frédéric Lagoutière, Probabilistic analysis of the upwind scheme for transport equations, Arch. Ration. Mech. Anal. 199 (2011), no. 1, 229–268. MR 2754342, DOI 10.1007/s00205-010-0322-x
- François Delarue, Frédéric Lagoutière, and Nicolas Vauchelet, Convergence order of upwind type schemes for transport equations with discontinuous coefficients, J. Math. Pures Appl. (9) 108 (2017), no. 6, 918–951. MR 3723162, DOI 10.1016/j.matpur.2017.05.012
- R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511–547. MR 1022305, DOI 10.1007/BF01393835
- R. Eymard, T. Gallouët, R. Herbin, and J. C. Latché, A convergent finite element-finite volume scheme for the compressible Stokes problem. II. The isentropic case, Math. Comp. 79 (2010), no. 270, 649–675. MR 2600538, DOI 10.1090/S0025-5718-09-02310-2
- R. Eymard, T. Gallouët, R. Herbin, and J.-C. Latché, Convergence of the MAC scheme for the compressible Stokes equations, SIAM J. Numer. Anal. 48 (2010), no. 6, 2218–2246. MR 2763662, DOI 10.1137/090779863
- Shizan Fang, Dejun Luo, and Anton Thalmaier, Stochastic differential equations with coefficients in Sobolev spaces, J. Funct. Anal. 259 (2010), no. 5, 1129–1168. MR 2652184, DOI 10.1016/j.jfa.2010.02.014
- T. Gallouët, R. Herbin, and J.-C. Latché, Kinetic energy control in explicit finite volume discretizations of the incompressible and compressible Navier-Stokes equations, Int. J. Finite Vol. 7 (2010), no. 2, 6. MR 2753585
- Thierry Gallouët, Laura Gastaldo, Raphaele Herbin, and Jean-Claude Latché, An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations, M2AN Math. Model. Numer. Anal. 42 (2008), no. 2, 303–331. MR 2405150, DOI 10.1051/m2an:2008005
- T. Gallouët, R. Herbin, and J.-C. Latché, A convergent finite element-finite volume scheme for the compressible Stokes problem. I. The isothermal case, Math. Comp. 78 (2009), no. 267, 1333–1352. MR 2501053, DOI 10.1090/S0025-5718-09-02216-9
- Thierry Goudon, Julie Llobell, and Sebastian Minjeaud, An asymptotic preserving scheme on staggered grids for the barotropic Euler system in low Mach regimes, Numer. Methods Partial Differential Equations 36 (2020), no. 5, 1098–1128. MR 4158319, DOI 10.1002/num.22466
- Mahir Hadžić, Andreas Seeger, Charles K. Smart, and Brian Street, Singular integrals and a problem on mixing flows, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 4, 921–943. MR 3795021, DOI 10.1016/j.anihpc.2017.09.001
- Gautam Iyer, Alexander Kiselev, and Xiaoqian Xu, Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows, Nonlinearity 27 (2014), no. 5, 973–985. MR 3207161, DOI 10.1088/0951-7715/27/5/973
- P.-E. Jabin and N. Masmoudi, DiPerna-Lions flow for relativistic particles in an electromagnetic field, Arch. Ration. Mech. Anal. 217 (2015), no. 3, 1029–1067. MR 3356994, DOI 10.1007/s00205-015-0850-5
- Pierre-Emmanuel Jabin, Critical non-Sobolev regularity for continuity equations with rough velocity fields, J. Differential Equations 260 (2016), no. 5, 4739–4757. MR 3437603, DOI 10.1016/j.jde.2015.11.028
- Flavien Léger, A new approach to bounds on mixing, Math. Models Methods Appl. Sci. 28 (2018), no. 5, 829–849. MR 3799259, DOI 10.1142/S0218202518500215
- Benoit Merlet, $L^\infty$- and $L^2$-error estimates for a finite volume approximation of linear advection, SIAM J. Numer. Anal. 46 (2007/08), no. 1, 124–150. MR 2377258, DOI 10.1137/060664057
- Benoît Merlet and Julien Vovelle, Error estimate for finite volume scheme, Numer. Math. 106 (2007), no. 1, 129–155. MR 2286009, DOI 10.1007/s00211-006-0053-y
- Benoît Perthame and Anne-Laure Dalibard, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2319–2335. MR 2471920, DOI 10.1090/S0002-9947-08-04656-4
- André Schlichting and Christian Seis, Convergence rates for upwind schemes with rough coefficients, SIAM J. Numer. Anal. 55 (2017), no. 2, 812–840. MR 3631392, DOI 10.1137/16M1084882
- Christian Seis, Maximal mixing by incompressible fluid flows, Nonlinearity 26 (2013), no. 12, 3279–3289. MR 3141856, DOI 10.1088/0951-7715/26/12/3279
- Christian Seis, A quantitative theory for the continuity equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 34 (2017), no. 7, 1837–1850. MR 3724758, DOI 10.1016/j.anihpc.2017.01.001
- Christian Seis, Optimal stability estimates for continuity equations, Proc. Roy. Soc. Edinburgh Sect. A 148 (2018), no. 6, 1279–1296. MR 3869178, DOI 10.1017/S0308210518000173
- Chad M. Topaz and Andrea L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math. 65 (2004), no. 1, 152–174. MR 2111591, DOI 10.1137/S0036139903437424
- Xicheng Zhang, Stochastic flows of SDEs with irregular coefficients and stochastic transport equations, Bull. Sci. Math. 134 (2010), no. 4, 340–378. MR 2651896, DOI 10.1016/j.bulsci.2009.12.004
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Additional Information
Pierre-Emmanuel Jabin
Affiliation:
Department of Mathematics and Huck Institutes, Pennsylvania State University, State College, PA 16801
MR Author ID:
660988
Email:
pejabin@psu.edu
Datong Zhou
Affiliation:
Department of Mathematics, Pennsylvania State University, State College, PA 16801
Email:
dbz5086@psu.edu
Received by editor(s):
November 14, 2022
Received by editor(s) in revised form:
December 19, 2022
Published electronically:
March 30, 2023
Additional Notes:
The authors were partially supported by NSF DMS Grants 2049020, 2219397, and 2205694.
Dedicated:
This paper is dedicated to Bob Pego and to the many unique ideas and contributions that he has brought to PDEs and the field of applied math in general.
Article copyright:
© Copyright 2023
Brown University
