Stochastic discontinuous Galerkin methods (SDGM) based on fluctuation-dissipation balance. (English) Zbl 1452.65290

An approach to discretize stochastic partial differential equations based on fluctuation-dissipation balance (instead of a direct approach based on random fluxes) is proposed. The authors develop robust discontinuous Galerkin methods and apply them to various initial-boundary value problems.


65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35R60 PDEs with randomness, stochastic partial differential equations
65C30 Numerical solutions to stochastic differential and integral equations


Full Text: DOI arXiv


[1] Liu, W.; Röckner, M., Stochastic partial differential equations: an introduction (2015), Springer International Publishing · Zbl 1361.60002
[2] Prato, G. D.; Zabczyk, J., Stochastic equations in infinite dimensions (2009), Cambridge University Press
[3] Cialenco, I., Statistical inference for SPDEs: an overview, Stat Inference Stoch Process (Feb. 2018)
[4] Koski, T.; Loges, W., Asymptotic statistical inference for a stochastic heat flow problem, Stat Probab Lett, 3, 4, 185-189 (July 1985), 10. 1016/0167- 7152(85)90015-x
[5] Lototsky, S. V., Statistical inference for stochastic parabolic equations: a spectral approach, Publications Matematiques, 53, 3-45 (Jan. 2009)
[6] Fox, R. F.; Uhlenbeck, G. E., Contributions to nonequilibrium thermodynamics. II. Fluctuation theory for the Boltzmann equation, Phys Fluids, 13, 12, 2881 (1970) · Zbl 0213.54202
[7] Jing Li, P. S., Mori-Zwanzig reduced models for uncertainty quantification, arXiv (2018), arxiv preprint number 1803.02826 · Zbl 07100319
[8] Mori, H., Transport, collective motion, and Brownian motion, Prog Theor Phys, 33, 3, 423-455 (Mar. 1965)
[9] Tabak, G.; Atzberger, P. J., Stochastic reductions for inertial fluid-structure interactions subject to thermal fluctuations, SIAM J Appl Math, 75, 4, 1884-1914 (Jan. 2015)
[10] Zwanzig, R., Memory effects in irreversible thermodynamics, Phys Rev, 124, 4, 983-992 (Nov. 1961)
[11] Atzberger, P. J., Stochastic Eulerian Lagrangian methods for fluid-structure interactions with thermal fluctuations, J Comput Phys, 230, 8, 2821-2837 (Apr. 2011)
[12] Atzberger, P. J.; Kramer, P. R.; Peskin, C. S., A stochastic immersed boundary method for fluid-structure dynamics at microscopic length scales, J Comput Phys, 224, 2, 1255-1292 (June 2007)
[13] Donev, A.; Vanden-Eijnden, E.; Garcia, A.; Bell, J., On the accuracy of finite-volume schemes for fluctuating hydrodynamics, Commun Appl Math Comput Sci, 5, 2, 149-197 (June 2010)
[14] Landau, L. D.; Lifshitz, E. M., Fluid mechanics (1986), Butterworth Heineman: Butterworth Heineman Oxford, UK
[15] Uma, B.; Swaminathan, TN; Radhakrishnan, R.; Eckmann, DM; Ayyaswamy, PS, Nanoparticle Brownian motion and hydrodynamic interactions in the presence of flow fields, Phys Fluids, 23, 7 (July 2011)
[16] Bertini, L.; Brassesco, S.; Buttà, P.; Presutti, E., Stochastic phase field equations: existence and uniqueness, Ann Henri Poincaré, 3, 1, 87-98 (Mar. 2002)
[17] Da Prato, G.; Debussche, A., Stochastic Cahn-Hilliard equation, Nonlinear Anal Theory Methods Appl, 26, 2, 241-263 (Jan. 1996)
[18] Lima, E. A.B. F.; Almeida, R. C.; Oden, J. T., Analysis and numerical solution of stochastic phase-field models of tumor growth, Numer Methods Partial Differ Equ, 31, 2, 552-574 (Oct. 2014)
[19] Sancho, J.; García-Ojalvo, J.; Guo, H., Non-equilibrium Ginzburg-Landau model driven by colored noise, Phys D Nonlinear Phenom, 113, 2-4, 331-337 (Mar. 1998)
[20] Atzberger, P. J., Spatially adaptive stochastic numerical methods for intrinsic fluctuations in reaction-diffusion systems, J Comput Phys, 229, 9, 34743501 (May 2010)
[21] Kim, C.; Nonaka, A.; Bell, JB; Garcia, AL; Donev, A., Stochastic simulation of reaction-diffusion systems: a fluctuating-hydrodynamics approach, J Chem Phys, 146, 12, 124110 (Mar. 2017)
[22] Atzberger, P. J., Velocity correlations of a thermally fluctuating Brownian particle: a novel model of the hydrodynamic coupling, Phys Lett A, 351, 4-5, 225-230 (Mar. 2006)
[23] Balboa Usabiaga, F.; Bell, JB; Delgado-Buscalioni, R.; Donev, A.; Fai, TG; Griffith, BE, Staggered schemes for fluctuating hydrodynamics, Multiscale Model Simul, 10, 4, 1369-1408 (Jan. 2012)
[24] Delong, S.; Sun, Y.; Griffith, B. E.; Vanden-Eijnden, E.; Donev, A., Multiscale temporal integrators for fluctuating hydrodynamics, Phys Rev E, 90, 6 (Dec. 2014)
[25] Lin, G.; Wan, X.; Su, C. H.; Karniadakis, G. E., Stochastic computational fluid mechanics, Comput Sci Eng, 9, 2, 21-29 (Mar. 2007)
[26] Kramer, P. R.; Peskin, C. S.; Atzberger, P. J., On the foundations of the stochastic immersed boundary method, Comput Methods Appl Mech Eng, 197, 25-28, 2232-2249 (Apr. 2008)
[27] Wang, Y.; Sigurdsson, J. K.; Atzberger, P. J., Fluctuating hydrodynamics methods for dynamic coarse-grained implicit-solvent simulations in LAMMPS, SIAM J Sci Comput, 38, 5, S62-S77 (Jan. 2016)
[28] Hairer, M., An introduction to stochastic PDEs, arXiv (2009), arxiv preprint number 0907.4178
[29] Lieb, E.; Loss, M., Analysis (2001), American Mathematical Society
[30] Zhang, Z.; Karniadakis, G. E., Numerical methods for stochastic partial differential equations with white noise (2017), Springer · Zbl 1380.65021
[31] Caruana, M.; Friz, P., Partial differential equations driven by rough paths, J Differ Equ, 247, 1, 140-173 (2009) · Zbl 1167.35386
[32] Holden, H.; Øksendal, B.; Ubøe, J.; Zhang, T., Stochastic partial differential equations driven by levy processes, (Stochastic partial differential equations: a modeling, white noise functional approach (2010), Springer New York: Springer New York New York, NY), 213-256
[33] Johnson, J. B., Thermal agitation of electricity in conductors, Phys Rev, 32, 1, 97-109 (July 1928)
[34] Nyquist, H., Thermal agitation of electric charge in conductors, Phys Rev, 32, 1, 110-113 (July 1928)
[35] Callen, H. B.; Welton, T. A., Irreversibility and generalized noise, Phys Rev, 83, 1, 34-40 (July 1951)
[36] Onsager, L., Reciprocal relations in irreversible processes. I, Phys Rev, 37, 4, 405-426 (Feb. 1931)
[37] Reichl, L. E., A modern course in statistical physics (Apr. 2016), Wiley-VCH Verlag GmbH & Co. KGaA
[38] Atzberger, P. J., Incorporating shear into stochastic Eulerian-Lagrangian methods for rheo- logical studies of complex fluids and soft materials, Phys D Nonlinear Phenom, 265, 57-70 (Dec. 2013)
[39] Plunkett, P.; Hu, J.; Siefert, C.; Atzberger, P. J., Spatially adaptive stochastic methods for fluid-structure interactions subject to thermal fluctuations in domains with complex geometries, J Comput Phys, 277, 121-137 (Nov. 2014)
[40] Wang, Y.; Lei, H.; Atzberger, P. J., Fluctuating hydrodynamic methods for fluid-structure interactions in confined channel geometries, Appl Math Mech, 39, 1, 125-152 (Dec. 2017)
[41] Arnold, D. N.; Falk, R. S.; Winther, R., Finite element exterior calculus: from Hodge theory to numerical stability, Bull Am Math Soc, 47, 2, 281-354 (Jan. 2010)
[42] Arnold, D. N.; Bochev, P. B.; Lehoucq, R. B.; Nicolaides, R. A.; Shashkov, M., Compatible spatial discretizations, 142 (2007), Springer Science & Business Media
[43] Hyman, J.; Morel, J.; Shashkov, M.; Steinberg, S., Mimetic finite difference methods for diffusion equations, Comput Geosci, 6, 3/4, 333-352 (2002) · Zbl 1023.76033
[44] Desbrun, M.; Hirani, A.; Marsden, J., Discrete exterior calculus for variational problems in computer vision and graphics, (42nd IEEE International conference on decision and control (IEEE cat. No.03CH37475) (2003), IEEE)
[45] Gillette, A.; Holst, M.; Zhu, Y., Finite element exterior calculus for evolution problems, J Comput Math, 35, 2, 187-212 (Mar. 2017)
[46] 0ksendal, B., Stochastic differential equations: an introduction with applications (2003), Springer Berlin Heidelberg
[47] Hairer, M.; Bréhier, C-E; Stuart, A. M., Weak error estimates for trajectories of SPDEs for Spectral Galerkin discretization, J Computat Math, 36, 2, 159-182 (2018) · Zbl 1413.65008
[48] Allen, E. J.; Novosel, S. J.; Zhang, Z., Finite element and difference approximation of some linear stochastic partial differential equations, Stoch Stoch Rep, 64, 1-2, 117-142 (May 1998)
[49] Du, Q.; Zhang, T., Numerical approximation of some linear stochastic partial differential equations driven by special additive noises, SIAM J Numer Anal, 40, 4, 1421-1445 (Jan. 2002)
[50] Kloeden, P. E.; Platen, E., Numerical solution of stochastic differential equations (1992), Springer Berlin Heidelberg · Zbl 0925.65261
[51] Reed, W. H.; Hill, T. R., Triangular mesh methods for the neutron transport equation, (Los Alamos report LA-UR (1973)), 73-479
[52] Cockburn, B.; Hou, S.; Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Math Comput, 54, 190 (1990), 545-545 · Zbl 0695.65066
[53] Cockburn, B.; Lin, S.-Y.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J Comput Phys, 84, 1, 90-113 (1989) · Zbl 0677.65093
[54] Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J Comput Phys, 141, 2, 199-224 (Apr. 1998)
[55] Cockburn, B.; Shu, C.-W., The Runge-Kutta local projection Pˆdiscontinuous-Galerkin finite element method for scalar conservation laws, ESAIM Math Model Numer Anal, 25, 3, 337-361 (1991) · Zbl 0732.65094
[56] Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math Comput, 52, 186 (1989), 411-411 · Zbl 0662.65083
[57] Arnold, D. N., An interior penalty finite element method with discontinuous elements, SIAM J Numer Anal, 19, 4, 742-760 (Aug. 1982)
[58] Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J Comput Phys, 131, 2, 267-279 (Mar. 1997)
[59] Brezzi, F.; Manzini, G.; Marini, D.; Pietra, P.; Russo, A., Discontinuous Galerkin approximations for elliptic problems, Numer Methods Partial Differ Equ, 16, 4, 365-378 (2000) · Zbl 0957.65099
[60] Cockburn, B.; Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J Numer Anal, 35, 6, 2440-2463 (Dec. 1998)
[61] Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J Numer Anal, 39, 5, 1749-1779 (2002) · Zbl 1008.65080
[62] Hesthaven, J. S.; Warburton, T., Nodal discontinuous Galerkin methods (2008), Springer: Springer New York · Zbl 1134.65068
[63] Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection- dominated problems, J Sci Comput, 16, 3, 173-261 (Sept. 2001)
[64] Cockburn, B.; Dong, B., An analysis of the minimal dissipation local discontinuous Galerkin method for convection-diffusion problems, J Sci Comput, 32, 2, 233-262 (Mar. 2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.