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Asymptotic-preserving schemes for multiscale physical problems

Published online by Cambridge University Press:  09 June 2022

Shi Jin
Shi Jin*
Affiliation:
School of Mathematical Sciences, Institute of Natural Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai200240, China E-mail: shijin-m@sjtu.edu.cn
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Abstract

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We present the asymptotic transitions from microscopic to macroscopic physics, their computational challenges and the asymptotic-preserving (AP) strategies to compute multiscale physical problems efficiently. Specifically, we will first study the asymptotic transition from quantum to classical mechanics, from classical mechanics to kinetic theory, and then from kinetic theory to hydrodynamics. We then review some representative AP schemes that mimic these asymptotic transitions at the discrete level, and hence can be used crossing scales and, in particular, capture the macroscopic behaviour without resolving the microscopic physical scale numerically.

Type
Research Article
Information
Acta Numerica , Volume 31 , May 2022 , pp. 415 - 489
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© Shanghai Jiao Tong University, 2022. Published by Cambridge University Press

Footnotes

*

S. Jin was partially supported by National Key R&D Program of China Project no. 2020YFA0712000, NSFC grant no. 12031013, Strategic Priority Research Program of Chinese Academy of Sciences XDA25010401, and Shanghai Municipal Science and Technology Major Project 2021SHZDZX0102.

References

Abdulle, A., W., E, Engquist, B. and Vanden-Eijnden, E. (2012), The heterogeneous multiscale method, in Acta Numerica, Vol. 21, Cambridge University Press, pp. 1–87.Google Scholar
Agarwal, R., Yun, K.-Y. and Balakrishnan, R. (1999), Beyond Navier Stokes–Burnett equations for flow simulations in continuum-transition regime, in 30th Fluid Dynamics Conference. Available at doi:10.2514/6.1999-3580.CrossRefGoogle Scholar
Albi, G. and Pareschi, L. (2013), Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Model. Simul. 11, 1–29.CrossRefGoogle Scholar
Albi, G., Bellomo, N., Fermo, L., Ha, S.-Y., Kim, J., Pareschi, L., Poyato, D. and Soler, J. (2019), Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci. 29, 1901–2005.CrossRefGoogle Scholar
Bao, W. and Shen, J. (2005), A fourth-order time-splitting Laguerre–Hermite pseudospectral method for Bose–Einstein condensates, SIAM J. Sci. Comput. 26, 2010–2028.CrossRefGoogle Scholar
Bao, W., Jin, S. and Markowich, P. A. (2002), On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys. 175, 487–524.CrossRefGoogle Scholar
Bao, W., Jin, S. and Markowich, P. A. (2003), Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput. 25, 27–64.CrossRefGoogle Scholar
Bardos, C., Golse, F. and Levermore, D. (1991), Fluid dynamic limits of kinetic equations, I: Formal derivations, J. Statist. Phys. 63, 323–344.CrossRefGoogle Scholar
Bardos, C., Santos, R. and Sentis, R. (1984), Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc. 284, 617–649.CrossRefGoogle Scholar
Barnes, J. and Hut, P. (1986), A hierarchical $O(NlogN)$ force-calculation algorithm, Nature 324, 446–449.CrossRefGoogle Scholar
Barsukow, W., Edelmann, P. V. F., Klingenberg, C., Miczek, F. and Röpke, F. K. (2017), A numerical scheme for the compressible low-Mach number regime of ideal fluid dynamics, J. Sci. Comput. 72, 623–646.CrossRefGoogle Scholar
Bennoune, M., Lemou, M. and Mieussens, L. (2008), Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics, J. Comput. Phys. 227, 3781–3803.CrossRefGoogle Scholar
Berman, P. R., Haverkort, J. E. M. and Woerdman, J. P. (1986), Collision kernels and transport coefficients, Phys. Rev. A 34, 4647–4656.CrossRefGoogle ScholarPubMed
Bird, G. A. (1994), Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford.Google Scholar
Bisseling, R. H., Kosloff, R., Gerber, R. B., Ratner, M. A., Gibson, L. and Cerjan, C. (1987), Exact time-dependent quantum mechanical dissociation dynamics of I2He: Comparison of exact time-dependent quantum calculation with the quantum time-dependent self-consistent field (TDSCF) approximation, J. Chem. Phys. 87, 2760–2765.CrossRefGoogle Scholar
Boscarino, S., Pareschi, L. and Russo, G. (2013), Implicit–explicit Runge–Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, SIAM J. Sci. Comput. 35, A22–A51.CrossRefGoogle Scholar
Bouchut, F., Golse, F. and Pulvirenti, M. (2000), Kinetic Equations and Asymptotic Theory, Vol. 4 of Series in Applied Mathematics (Paris), Elsevier.Google Scholar
Bouchut, F., Ounaissa, H. and Perthame, B. (2007), Upwinding of the source term at interfaces for Euler equations with high friction, Comput. Math. Appl. 53, 361–375.CrossRefGoogle Scholar
Bourgat, J. F., Le Tallec, P., Perthame, B. and Qiu, Y. (1994), Coupling Boltzmann and Euler equations without overlapping, Contemp. Math. 157, 377–377.CrossRefGoogle Scholar
Brenier, Y. (2000), Convergence of the Vlasov–Poisson system to the incompressible Euler equations, Commun. Partial Differ. Equations 25, 737–754.CrossRefGoogle Scholar
Caflisch, R. E., Jin, S. and Russo, G. (1997), Uniformly accurate schemes for hyperbolic systems with relaxation, SIAM J. Numer. Anal. 34, 246–281.CrossRefGoogle Scholar
Carrillo, J. A., Pareschi, L. and Zanella, M. (2019), Particle based gPC methods for mean-field models of swarming with uncertainty, Commun. Comput. Phys. 25, 508–531.CrossRefGoogle Scholar
Carrillo, J. A., Wang, L., Xu, W. and Yan, M. (2021), Variational asymptotic preserving scheme for the Vlasov–Poisson–Fokker–Planck system, Multiscale Model . Simul. 19, 478–505.Google Scholar
Cercignani, C. (1988), The Boltzmann Equation and its Applications, Vol. 67 of Applied Mathematical Sciences, Springer.CrossRefGoogle Scholar
Cercignani, C., Gamba, I. M. and Levermore, C. D. (1997), High field approximations to a Boltzmann–Poisson system and boundary conditions in a semiconductor, Appl. Math. Lett. 10, 111–117.CrossRefGoogle Scholar
Chalons, C., Coquel, F., Godlewski, E., Raviart, P.-A. and Seguin, N. (2010), Godunov-type schemes for hyperbolic systems with parameter-dependent source: The case of Euler system with friction, Math. Models Methods Appl. Sci. 20, 2109–2166.CrossRefGoogle Scholar
Chalons, C., Girardin, M. and Kokh, S. (2016), An all-regime Lagrange-projection like scheme for the gas dynamics equations on unstructured meshes, Commun . Comput. Phys. 20, 188–233.CrossRefGoogle Scholar
Chen, G.-Q., Levermore, C. D. and Liu, T.-P. (1994), Hyperbolic conservation laws with stiff relaxation terms and entropy, Commun. Pure Appl. Math. 47, 787–830.CrossRefGoogle Scholar
Chen, H., Chen, S. and Matthaeus, W. H. (1992), Recovery of the Navier–Stokes equations using a lattice-gas Boltzmann method, Phys. Rev. A 45, R5339.CrossRefGoogle ScholarPubMed
Chen, S. and Doolen, G. D. (1998), Lattice Boltzmann method for fluid flows, Annu . Rev. Fluid Mech. 30, 329–364.CrossRefGoogle Scholar
Chen, Z., Liu, L. and Mu, L. (2017), DG-IMEX stochastic Galerkin schemes for linear transport equation with random inputs and diffusive scalings, J. Sci. Comput. 73, 566–592.CrossRefGoogle Scholar
Chorin, A. J. (1968), Numerical solution of the Navier–Stokes equations, Math. Comp. 22, 745–762.CrossRefGoogle Scholar
Ciccotti, G., Frenkel, D. and McDonald, I. R. (1987), Simulation of Liquids and Solids: Molecular Dynamics and Monte Carlo Methods in Statistical Mechanics, North-Holland.Google Scholar
Cordier, F., Degond, P. and Kumbaro, A. (2012), An asymptotic-preserving all-speed scheme for the Euler and Navier–Stokes equations, J. Comput. Phys. 231, 5685–5704.CrossRefGoogle Scholar
Coron, F. and Perthame, B. (1991), Numerical passage from kinetic to fluid equations, SIAM J. Numer. Anal. 28, 26–42.CrossRefGoogle Scholar
Corry, L. (2004), David Hilbert and the Axiomatization of Physics (1898–1918): From Grundlagen der Geometrie to Grundlagen der Physik, Vol. 10 of Archimedes, Springer.CrossRefGoogle Scholar
Crandall, M. G. and Lions, P.-L. (1983), Viscosity solutions of Hamilton–Jacobi equations, Trans. Amer. Math. Soc. 277, 1–42.CrossRefGoogle Scholar
Crispel, P., Degond, P. and Vignal, M.-H. (2005), An asymptotically stable discretization for the Euler–Poisson system in the quasi-neutral limit, C. R. Math. Acad. Sci. Paris 341, 323–328.CrossRefGoogle Scholar
Crispel, P., Degond, P. and Vignal, M.-H. (2007), An asymptotic preserving scheme for the two-fluid Euler–Poisson model in the quasineutral limit, J. Comput. Phys. 223, 208–234.CrossRefGoogle Scholar
Crouseilles, N. and Lemou, M. (2011), An asymptotic preserving scheme based on a micro–macro decomposition for collisional Vlasov equations: Diffusion and high-field scaling limits, Kinet. Relat. Models 4, 441.CrossRefGoogle Scholar
Crouseilles, N., Frénod, E., Hirstoaga, S. A. and Mouton, A. (2013a), Two-scale macro–micro decomposition of the Vlasov equation with a strong magnetic field, Math. Models Methods Appl. Sci. 23, 1527–1559.CrossRefGoogle Scholar
Crouseilles, N., Lemou, M. and Méhats, F. (2013b), Asymptotic preserving schemes for highly oscillatory Vlasov–Poisson equations, J. Comput. Phys. 248, 287–308.CrossRefGoogle Scholar
Cucker, F. and Smale, S. (2007), Emergent behavior in flocks, IEEE Trans. Automat. Control 52, 852–862.CrossRefGoogle Scholar
Daus, E. S., Jin, S. and Liu, L. (2019), Spectral convergence of the stochastic Galerkin approximation to the Boltzmann equation with multiple scales and large random perturbation in the collision kernel, Kinet. Relat. Models 12, 909–922.CrossRefGoogle Scholar
Degond, P. and Deluzet, F. (2017), Asymptotic-preserving methods and multiscale models for plasma physics, J. Comput. Phys. 336, 429–457.CrossRefGoogle Scholar
Degond, P. and Tang, M. (2011), All speed scheme for the low Mach number limit of the isentropic Euler equations, Commun. Comput. Phys. 10, 1–31.CrossRefGoogle Scholar
Degond, P., Deluzet, F. and Doyen, D. (2017), Asymptotic-preserving particle-in-cell methods for the Vlasov–Maxwell system in the quasi-neutral limit, J. Comput. Phys. 330, 467–492.CrossRefGoogle Scholar
Degond, P., Deluzet, F. and Negulescu, C. (2010a), An asymptotic preserving scheme for strongly anisotropic elliptic problems, Multiscale Model. Simul. 8, 645–666.CrossRefGoogle Scholar
Degond, P., Deluzet, F. and Savelief, D. (2012a), Numerical approximation of the Euler–Maxwell model in the quasineutral limit, J. Comput. Phys. 231, 1917–1946.CrossRefGoogle Scholar
Degond, P., Deluzet, F., Lozinski, A., Narski, J. and Negulescu, C. (2012b), Duality-based asymptotic-preserving method for highly anisotropic diffusion equations, Commun. Math. Sci. 10, 1–31.CrossRefGoogle Scholar
Degond, P., Deluzet, F., Navoret, L., Sun, A.-B. and Vignal, M.-H. (2010b), Asymptotic-preserving particle-in-cell method for the Vlasov–Poisson system near quasineutrality, J. Comput. Phys. 229, 5630–5652.CrossRefGoogle Scholar
Degond, P., Jin, S. and Mieussens, L. (2005), A smooth transition model between kinetic and hydrodynamic equations, J. Comput. Phys. 209, 665–694.CrossRefGoogle Scholar
Degond, P., Lozinski, A., Narski, J. and Negulescu, C. (2012c), An asymptotic-preserving method for highly anisotropic elliptic equations based on a micro–macro decomposition, J. Comput. Phys. 231, 2724–2740.CrossRefGoogle Scholar
Dellacherie, S. (2010), Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number, J. Comput. Phys. 229, 978–1016.CrossRefGoogle Scholar
Deng, J. (2012), Asymptotic preserving schemes for semiconductor Boltzmann equation in the diffusive regime, Numer. Math. Theory Methods Appl. 5, 278–296.CrossRefGoogle Scholar
Deserno, M. and Holm, C. (1998), How to mesh up Ewald sums, II: An accurate error estimate for the particle–particle particle–mesh algorithm, J. Chem. Phys. 109, 7694–7701.CrossRefGoogle Scholar
Deshpande, S. (1986), Kinetic theory based new upwind methods for inviscid compressible flows, in 24th Aerospace Sciences Meeting. Available at doi:10.2514/6.1986-275.CrossRefGoogle Scholar
Dimarco, G. and Pareschi, L. (2011), Exponential Runge–Kutta methods for stiff kinetic equations, SIAM J. Numer. Anal. 49, 2057–2077.CrossRefGoogle Scholar
Dimarco, G., Loubère, R. and Vignal, M.-H. (2017), Study of a new asymptotic preserving scheme for the Euler system in the low Mach number limit, SIAM J. Sci. Comput. 39, A2099–A2128.CrossRefGoogle Scholar
Drukker, K. (1999), Basics of surface hopping in mixed quantum/classical simulations, J. Comput. Phys. 153, 225–272.CrossRefGoogle Scholar
Duan, Z. H. and Krasny, R. (2000), An Ewald summation based multipole method, J. Chem. Phys. 113, 3492–3495.CrossRefGoogle Scholar
Dumbser, M., Enaux, C. and Toro, E. F. (2008), Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J. Comput. Phys. 227, 3971–4001.CrossRefGoogle Scholar
Durstenfeld, R. (1964), Algorithm 235: Random permutation, Commun. Assoc. Comput. Mach. 7, 420.Google Scholar
E, W. and Engquist, B. (2003), The heterogeneous multiscale methods, Commun. Math. Sci. 1, 87–132.CrossRefGoogle Scholar
E, W. and Yu, B. (2018), The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems, Commun. Math. Statist. 6, 1–12.CrossRefGoogle Scholar
Fang, D., Jin, S. and Sparber, C. (2018), An efficient time-splitting method for the Ehrenfest dynamics, Multiscale Model. Simul. 16, 900–921.CrossRefGoogle Scholar
Feireisl, E., Lukác̆ová-Medvidová, M., Nec̆asová, S., Novotný, A. and She, B. (2018), Asymptotic preserving error estimates for numerical solutions of compressible Navier–Stokes equations in the low Mach number regime, Multiscale Model. Simul. 16, 150–183.CrossRefGoogle Scholar
Filbet, F. and Jin, S. (2010), A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys. 229, 7625–7648.CrossRefGoogle Scholar
Filbet, F. and Rodrigues, L. M. (2016), Asymptotically stable particle-in-cell methods for the Vlasov–Poisson system with a strong external magnetic field, SIAM J. Numer. Anal. 54, 1120–1146.CrossRefGoogle Scholar
Filbet, F. and Rodrigues, L. M. (2017), Asymptotically preserving particle-in-cell methods for inhomogeneous strongly magnetized plasmas, SIAM J. Numer. Anal. 55, 2416–2443.CrossRefGoogle Scholar
Filbet, F. and Rodrigues, L. M. (2020), Asymptotics of the three-dimensional Vlasov equation in the large magnetic field limit, J. Éc. Polytech. Math. 7, 1009–1067.CrossRefGoogle Scholar
Filbet, F., Hu, J. and Jin, S. (2012), A numerical scheme for the quantum Boltzmann equation with stiff collision terms, ESAIM Math. Model. Numer. Anal. 46, 443–463.CrossRefGoogle Scholar
Filbet, F., Rodrigues, L. M. and Zakerzadeh, H. (2021), Convergence analysis of asymptotic preserving schemes for strongly magnetized plasmas, Numer. Math. 149, 549–593.CrossRefGoogle Scholar
Foch, J. D. (1973), On higher order hydrodynamic theories of shock structure, in The Boltzmann Equation (Cohen, E. G. D. and Thirring, W., eds), Springer, pp. 123–140.CrossRefGoogle Scholar
Fornberg, B. (1996), A Practical Guide to Pseudospectral Methods, Vol. 1 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.CrossRefGoogle Scholar
Frenkel, D. and Smit, B. (2001), Understanding Molecular Simulation: From Algorithms to Applications, Vol. 1 of Computational Science Series, Elsevier.Google Scholar
Frénod, E., Hirstoaga, S. A., Lutz, M. and Sonnendrücker, E. (2015), Long time behaviour of an exponential integrator for a Vlasov–Poisson system with strong magnetic field, Commun. Comput. Phys. 18, 263–296.CrossRefGoogle Scholar
Frénod, E., Salvarani, F. and Sonnendrücker, E. (2009), Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method, Math. Models Methods Appl. Sci. 19, 175–197.CrossRefGoogle Scholar
Gallagher, I., Saint-Raymond, L. and Texier, B. (2013), From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich.Google Scholar
Gamba, I. M., Jin, S. and Liu, L. (2019), Micro–macro decomposition based asymptotic-preserving numerical schemes and numerical moments conservation for collisional nonlinear kinetic equations, J. Comput. Phys. 382, 264–290.CrossRefGoogle Scholar
Georges, A., Kotliar, G., Krauth, W. and Rozenberg, M. J. (1996), Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Modern Phys. 68, 13.CrossRefGoogle Scholar
Gérard, P., Markowich, P. A., Mauser, N. J. and Poupaud, F. (1997), Homogenization limits and Wigner transforms, Commun . Pure Appl. Math. 50, 323–379.3.0.CO;2-C>CrossRefGoogle Scholar
Golse, F. (2003), The mean-field limit for the dynamics of large particle systems, Journées Équations aux Dérivées Partielles 9, 1–47.CrossRefGoogle Scholar
Golse, F. and Paul, T. (2017), The Schrödinger equation in the mean-field and semiclassical regime, Arch. Ration. Mech. Anal. 223, 57–94.CrossRefGoogle Scholar
Golse, F., Jin, S. and Levermore, C. D. (1999), The convergence of numerical transfer schemes in diffusive regimes, I: Discrete-ordinate method, SIAM J. Numer. Anal. 36, 1333–1369.CrossRefGoogle Scholar
Golse, F., Jin, S. and Paul, T. (2021), On the convergence of time splitting methods for quantum dynamics in the semiclassical regime, Found. Comput. Math. 21, 613–647.CrossRefGoogle Scholar
Gosse, L. and Toscani, G. (2002), An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math. Acad. Sci. Paris 334, 337–342.CrossRefGoogle Scholar
Goudon, T., Nieto, J., Poupaud, F. and Soler, J. (2005), Multidimensional high-field limit of the electrostatic Vlasov–Poisson–Fokker–Planck system, J. Differ. Equations 213, 418–442.CrossRefGoogle Scholar
Greengard, L. and Rokhlin, V. (1987), A fast algorithm for particle simulations, J. Comput. Phys. 73, 325–348.CrossRefGoogle Scholar
Guillard, H. and Viozat, C. (1999), On the behaviour of upwind schemes in the low Mach number limit, Comput. Fluids 28, 63–86.CrossRefGoogle Scholar
Gunzburger, M. D., Webster, C. G. and Zhang, G. (2014), Stochastic finite element methods for partial differential equations with random input data, in Acta Numerica, Vol. 23, Cambridge University Press, pp. 521–650.Google Scholar
Haack, J., Jin, S. and Liu, J.-G. (2012), An all-speed asymptotic-preserving method for the isentropic Euler and Navier–Stokes equations, Commun. Comput. Phys. 12, 955–980.CrossRefGoogle Scholar
Hairer, E., Lubich, C. and Wang, B. (2020), A filtered Boris algorithm for charged-particle dynamics in a strong magnetic field, Numer. Math. 144, 787–809.CrossRefGoogle Scholar
He, X. and Luo, L.-S. (1997), Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation, Phys. Rev. E 56, 6811.CrossRefGoogle Scholar
Heller, E. J. (2006), Guided Gaussian wave packets, Acc. Chem. Res. 39, 127–134.CrossRefGoogle ScholarPubMed
Hetenyi, B., Bernacki, K. and Berne, B. J. (2002), Multiple ‘time step’ Monte Carlo, .J. Chem. Phys 117, 8203–8207.CrossRefGoogle Scholar
Hill, N. R. (1990), Gaussian beam migration, Geophysics 55, 1416–1428.CrossRefGoogle Scholar
Hu, J. and Jin, S. (2016), A stochastic Galerkin method for the Boltzmann equation with uncertainty, J. Comput. Phys. 315, 150–168.CrossRefGoogle Scholar
Hu, J. and Shu, R. (2019), A second-order asymptotic-preserving and positivity-preserving exponential Runge–Kutta method for a class of stiff kinetic equations, Multiscale Model. Simul. 17, 1123–1146.CrossRefGoogle Scholar
Hu, J. and Shu, R. (2021), On the uniform accuracy of implicit–explicit backward differentiation formulas (IMEX-BDF) for stiff hyperbolic relaxation systems and kinetic equations, Math. Comp. 90, 641–670.CrossRefGoogle Scholar
Hu, J., Jin, S. and Li, Q. (2017), Asymptotic-preserving schemes for multiscale hyperbolic and kinetic equations, in Handbook of Numerical Analysis, Vol. XVIII, Elsevier, pp. 103–129.Google Scholar
Hu, J., Jin, S. and Yan, B. (2012), A numerical scheme for the quantum Fokker–Planck–Landau equation efficient in the fluid regime, Commun. Comput. Phys. 12, 1541–1561.CrossRefGoogle Scholar
Jabin, P.-E. and Wang, Z. (2017), Mean field limit for stochastic particle systems, in Active Particles, Vol. 1, Advances in Theory, Models, and Applications (Bellomo, N. et al., eds), Springer, pp. 379–402.Google Scholar
Jin, S. (1995), Runge–Kutta methods for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys. 122, 51–67.CrossRefGoogle Scholar
Jin, S. (1999), Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput. 21, 441–454.CrossRefGoogle Scholar
Jin, S. (2010), Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review, in Lecture Notes for Summer School on Methods and Models of Kinetic Theory (M&MKT), pp. 177–216.Google Scholar
Jin, S. and Levermore, C. D. (1996), Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys. 126, 449–467.CrossRefGoogle Scholar
Jin, S. and Li, L. (2022a), On the mean field limit of the random batch method for interacting particle systems, Sci. China Math. 65, 169–202.Google Scholar
Jin, S. and Li, L. (2022b), Random batch methods for classical and quantum interacting particle systems and statistical samplings. To appear in Active Particles II (Bellomo, N. et al., eds).CrossRefGoogle Scholar
Jin, S. and Li, Q. (2013), A BGK-penalization-based asymptotic-preserving scheme for the multispecies Boltzmann equation, Numer. Methods Partial Differ. Equations 29, 1056–1080.CrossRefGoogle Scholar
Jin, S. and Li, X. (2020), Random batch algorithms for quantum Monte Carlo simulations, Commun. Comput. Phys. 28, 1907–1936.Google Scholar
Jin, S. and Liu, L. (2017), An asymptotic-preserving stochastic Galerkin method for the semiconductor Boltzmann equation with random inputs and diffusive scalings, Multiscale Model. Simul. 15, 157–183.CrossRefGoogle Scholar
Jin, S. and Pareschi, L. (2001), Asymptotic-preserving (AP) schemes for multiscale kinetic equations: A unified approach, in Hyperbolic Problems: Theory, Numerics, Applications (Freistühler, H. and Warnecke, G., eds), Vol. 141 of International Series of Numerical Mathematics, Birkhäuser, pp. 573–582.CrossRefGoogle Scholar
Jin, S. and Shu, R. (2017), A stochastic asymptotic-preserving scheme for a kinetic-fluid model for disperse two-phase flows with uncertainty, J. Comput. Phys. 335, 905–924.CrossRefGoogle Scholar
Jin, S. and Wang, L. (2011), An asymptotic preserving scheme for the Vlasov–Poisson–Fokker–Planck system in the high field regime, Acta Math. Sci. 31, 2219–2232.CrossRefGoogle Scholar
Jin, S. and Wang, L. (2013), Asymptotic-preserving numerical schemes for the semiconductor Boltzmann equation efficient in the high field regime, SIAM J. Sci. Comput. 35, B799–B819.CrossRefGoogle Scholar
Jin, S. and Xin, Z. (1995), The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Commun. Pure Appl. Math. 48, 235–276.CrossRefGoogle Scholar
Jin, S. and Yan, B. (2011), A class of asymptotic-preserving schemes for the Fokker–Planck–Landau equation, J. Comput. Phys. 230, 6420–6437.CrossRefGoogle Scholar
Jin, S. and Zhou, Z. (2013), A semi-Lagrangian time splitting method for the Schrödinger equation with vector potentials, Commun. Inform. Syst. 13, 247–289.CrossRefGoogle Scholar
Jin, S. and Zhu, Y. (2018), Hypocoercivity and uniform regularity for the Vlasov–Poisson–Fokker–Planck system with uncertainty and multiple scales, SIAM J. Math. Anal. 50, 1790–1816.CrossRefGoogle Scholar
Jin, S., Levermore, C. D. and McLaughlin, D. W. (1999), The semiclassical limit of the defocusing NLS hierarchy, Commun. Pure Appl. Math. 52, 613–654.3.0.CO;2-L>CrossRefGoogle Scholar
Jin, S., Li, L. and Liu, J.-G. (2020a), Random batch methods (RBM) for interacting particle systems, J. Comput. Phys. 400, 108877.CrossRefGoogle Scholar
Jin, S., Li, L. and Sun, Y. (2020b), On the Random Batch Method for second order interacting particle systems. Available at arXiv:2011.10778 (to appear in Multiscale Model. Simul.).Google Scholar
Jin, S., Li, L., Xu, Z. and Zhao, Y. (2021a), A random batch Ewald method for particle systems with Coulomb interactions, SIAM J. Sci. Comput. 43, B937–B960.CrossRefGoogle Scholar
Jin, S., Liu, J.-G. and Ma, Z. (2017a), Uniform spectral convergence of the stochastic Galerkin method for the linear transport equations with random inputs in diffusive regime and a micro–macro decomposition-based asymptotic-preserving method, Res. Math. Sci. 4, 1–25.CrossRefGoogle Scholar
Jin, S., Ma, Z. and Wu, K. (2021b), Asymptotic-preserving neural networks for multiscale time-dependent linear transport equations. Available at arXiv:2111.02541.Google Scholar
Jin, S., Markowich, P. and Sparber, C. (2011), Mathematical and computational methods for semiclassical Schrödinger equations, in Acta Numerica, Vol. 20, Cambridge University Press, pp. 121–209.Google Scholar
Jin, S., Pareschi, L. and Toscani, G. (2000), Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal. 38, 913–936.CrossRefGoogle Scholar
Jin, S., Sparber, C. and Zhou, Z. (2017b), On the classical limit of a time-dependent self-consistent field system: Analysis and computation, Kinet. Relat. Models 10, 263–298.CrossRefGoogle Scholar
Jin, S., Wu, H. and Yang, X. (2008), Gaussian beam methods for the Schrödinger equation in the semi-classical regime: Lagrangian and Eulerian formulations, Commun. Math. Sci. 6, 995–1020.CrossRefGoogle Scholar
Jin, S., Xiu, D. and Zhu, X. (2015), Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings, J. Comput. Phys. 289, 35–52.CrossRefGoogle Scholar
Kevrekidis, I. G., Gear, C. W. and Hummer, G. (2004), Equation-free: The computer-aided analysis of complex multiscale systems, AIChE J. 50, 1346–1355.CrossRefGoogle Scholar
Klainerman, S. and Majda, A. (1981), Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Commun. Pure Appl. Math. 34, 481–524.CrossRefGoogle Scholar
Klar, A. (1998), An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM J. Numer. Anal. 35, 1073–1094.CrossRefGoogle Scholar
Klar, A. and Schmeiser, C. (2001), Numerical passage from radiative heat transfer to nonlinear diffusion models, Math. Models Methods Appl. Sci. 11, 749–767.CrossRefGoogle Scholar
Klar, A., Neunzert, H. and Struckmeier, J. (2000), Transition from kinetic theory to macroscopic fluid equations: A problem for domain decomposition and a source for new algorithms, Transp. Theory Statist. Phys. 29, 93–106.CrossRefGoogle Scholar
Klein, R. (1995), Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics, I: One-dimensional flow, J. Comput. Phys. 121, 213–237.CrossRefGoogle Scholar
Koura, K. and Matsumoto, H. (1991), Variable soft sphere molecular model for inverse-power-law or Lennard-Jones potential, Phys. Fluids A 3, 2459–2465.CrossRefGoogle Scholar
Küpper, K., Frank, M. and Jin, S. (2016), An asymptotic preserving two-dimensional staggered grid method for multiscale transport equations, SIAM J. Numer. Anal. 54, 440–461.CrossRefGoogle Scholar
Lanford, O. E. III (1975), Time evolution of large classical systems, in Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, WA, 1974), Vol. 38 of Lecture Notes in Physics, Springer, pp. 1–111.CrossRefGoogle Scholar
Larsen, E. W. and Morel, J. E. (1989), Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes, II, J. Comput. Phys. 83, 212–236.CrossRefGoogle Scholar
Larsen, E. W., Morel, J. E. and Miller, W. F. Jr (1987), Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes, J. Comput. Phys. 69, 283–324.CrossRefGoogle Scholar
Lasser, C. and Lubich, C. (2020), Computing quantum dynamics in the semiclassical regime, in Acta Numerica, Vol. 29, Cambridge University Press, pp. 229–401.Google Scholar
Lax, P. D. and Levermore, D. (1983), The small dispersion limit of the Korteweg–de Vries equation, I, Commun. Pure Appl. Math. 36, 253–290.CrossRefGoogle Scholar
Lelièvre, T. and Stoltz, G. (2016), Partial differential equations and stochastic methods in molecular dynamics, in Acta Numerica, Vol. 25, Cambridge University Press, pp. 681–880.Google Scholar
Lemou, M. and Mieussens, L. (2008), A new asymptotic preserving scheme based on micro–macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput. 31, 334–368.CrossRefGoogle Scholar
Leung, S. and Qian, J. (2009), Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime, J. Comput. Phys. 228, 2951–2977.CrossRefGoogle Scholar
Lewis, E. E. and Miller, W. F. (1984), Computational Methods of Neutron Transport, Wiley.Google Scholar
Li, L. and Yang, C. (2021), Asymptotic preserving scheme for anisotropic elliptic equations with deep neural network. Available at arXiv:2104.05337.Google Scholar
Li, L., Li, Y., Liu, J.-G., Liu, Z. and Lu, J. (2020a), A stochastic version of Stein variational gradient descent for efficient sampling, Commun. Appl. Math. Comput. Sci. 15, 37–63.CrossRefGoogle Scholar
Li, L., Xu, Z. and Zhao, Y. (2020b), A random-batch Monte Carlo method for many-body systems with singular kernels, SIAM J. Sci. Comput. 42, A1486–A1509.CrossRefGoogle Scholar
Li, Q. and Pareschi, L. (2014), Exponential Runge–Kutta for the inhomogeneous Boltzmann equations with high order of accuracy, J. Comput. Phys. 259, 402–420.CrossRefGoogle Scholar
Li, Q. and Wang, L. (2017), Uniform regularity for linear kinetic equations with random input based on hypocoercivity, SIAM/ASA J. Uncertainty Quantif. 5, 1193–1219.CrossRefGoogle Scholar
Li, Z., Kovachki, N. B., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A. and Anandkumar, A. (2021), Fourier neural operator for parametric partial differential equations, in 9th International Conference on Learning Representations (ICLR 2021). Available at open-review.net.Google Scholar
Liang, J., Tan, P., Zhao, Y., Li, L., Jin, S., Hong, L. and Xu, Z. (2022), Super-scalable molecular dynamics algorithm, J. Chem. Phys. 156, 014114.CrossRefGoogle Scholar
Lions, P.-L. and Paul, T. (1993), Sur les mesures de Wigner, Rev. Mat. Iberoamer. 9, 553–618.CrossRefGoogle Scholar
Liu, C., Xu, K., Sun, Q. and Cai, Q. (2016), A unified gas-kinetic scheme for continuum and rarefied flows, IV: Full Boltzmann and model equations, J. Comput. Phys. 314, 305–340.CrossRefGoogle Scholar
Liu, J.-G. and Mieussens, L. (2010), Analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit, SIAM J. Numer. Anal. 48, 1474–1491.CrossRefGoogle Scholar
Liu, L. and Jin, S. (2018), Hypocoercivity based sensitivity analysis and spectral convergence of the stochastic Galerkin approximation to collisional kinetic equations with multiple scales and random inputs, Multiscale Model. Simul. 16, 1085–1114.CrossRefGoogle Scholar
Liu, T.-P. and Yu, S.-H. (2004), Boltzmann equation: Micro–macro decompositions and positivity of shock profiles, Commun. Math. Phys. 246, 133–179.CrossRefGoogle Scholar
Loève, M. (1977), Probability Theory, fourth edition, Springer.Google Scholar
Lu, L., Jin, P., Pang, G., Zhang, Z. and Karniadakis, G. E. (2021a), Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators, Nature Mach. Intell. 3, 218–229.CrossRefGoogle Scholar
Lu, Y., Wang, L. and Xu, W. (2021b), Solving multiscale steady radiative transfer equation using neural networks with uniform stability. Available at arXiv:2110.07037.Google Scholar
Luty, B. A., Davis, M. E., Tironi, I. G. and Van Gunsteren, W. F. (1994), A comparison of particle–particle, particle–mesh and Ewald methods for calculating electrostatic interactions in periodic molecular systems, Mol. Simul. 14, 11–20.CrossRefGoogle Scholar
Makri, N. and Miller, W. H. (1987), Time-dependent self-consistent field (TDSCF) approximation for a reaction coordinate coupled to a harmonic bath: Single and multiple configuration treatments, J. Chem. Phys. 87, 5781–5787.CrossRefGoogle Scholar
Markowich, P. A., Pietra, P. and Pohl, C. (1999), Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit, Numer. Math. 81, 595–630.CrossRefGoogle Scholar
Markowich, P. A., Ringhofer, C. A. and Schmeiser, C. (1990), Semiconductor Equations, Springer.CrossRefGoogle Scholar
Martin, M. G., Chen, B. and Siepmann, J. I. (1998), A novel Monte Carlo algorithm for polarizable force fields: Application to a fluctuating charge model for water, J. Chem. Phys. 108, 3383–3385.CrossRefGoogle Scholar
Marx, D. and Hutter, J. (2009), Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods, Cambridge University Press.CrossRefGoogle Scholar
McKean, H. P. (1967), Propagation of chaos for a class of non-linear parabolic equations, in Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), pp. 41–57.Google Scholar
Miczek, F., Röpke, F. K. and Edelmann, P. V. (2015), New numerical solver for flows at various Mach numbers, Astronom. Astrophys. 576, A50.CrossRefGoogle Scholar
Motsch, S. and Tadmor, E. (2014), Heterophilious dynamics enhances consensus, SIAM Rev. 56, 577–621.CrossRefGoogle Scholar
Narski, J. and Ottaviani, M. (2014), Asymptotic preserving scheme for strongly anisotropic parabolic equations for arbitrary anisotropy direction, Comput. Phys. Commun. 185, 3189–3203.CrossRefGoogle Scholar
Nieto, J., Poupaud, F. and Soler, J. (2001), High-field limit for the Vlasov–Poisson–Fokker–Planck system, Arch. Ration. Mech. Anal. 158, 29–59.CrossRefGoogle Scholar
Noelle, S., Bispen, G., Arun, K. R., Lukác̆ová-Medvidová, M. and Munz, C.-D. (2014), A weakly asymptotic preserving low Mach number scheme for the Euler equations of gas dynamics, SIAM J. Sci. Comput. 36, B989–B1024.CrossRefGoogle Scholar
Pareschi, L. and Russo, G. (2005), Implicit–explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput. 25, 129–155.Google Scholar
Perthame, B. (1990), Boltzmann type schemes for gas dynamics and the entropy property, SIAM J. Numer. Anal. 27, 1405–1421.CrossRefGoogle Scholar
Prendergast, K. H. and Xu, K. (1993), Numerical hydrodynamics from gas-kinetic theory, J. Comput. Phys. 109, 53–66.CrossRefGoogle Scholar
Qian, Y.-H., d’Humières, D. and Lallemand, P. (1992), Lattice BGK models for Navier–Stokes equation, EPL (Europhysics Letters) 17, 479.CrossRefGoogle Scholar
Raissi, M., Perdikaris, P. and Karniadakis, G. E. (2019), Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378, 686–707.CrossRefGoogle Scholar
Ricketson, L. F. and Chacón, L. (2020), An energy-conserving and asymptotic-preserving charged-particle orbit implicit time integrator for arbitrary electromagnetic fields, J. Comput. Phys. 418, 109639.CrossRefGoogle Scholar
Rivell, T. (2006), Notes on earth atmospheric entry for Mars sample return missions. Technical report TP-2006-213486, NASA Ames Research Center, Moffett Field, CA, USA.Google Scholar
Rokhlin, V. (1985), Rapid solution of integral equations of classical potential theory, J. Comput. Phys. 60, 187–207.CrossRefGoogle Scholar
Russo, G. and Smereka, P. (2013), The Gaussian wave packet transform: Efficient computation of the semi-classical limit of the Schrödinger equation, 1: Formulation and the one dimensional case, J. Comput. Phys. 233, 192–209.CrossRefGoogle Scholar
Schütte, C. and Bornemann, F. A. (1999), On the singular limit of the quantum-classical molecular dynamics model, SIAM J. Appl. Math. 59, 1208–1224.CrossRefGoogle Scholar
Sparber, C., Markowich, P. and Mauser, N. (2003), Wigner functions versus WKB-methods in multivalued geometrical optics, Asymptot. Anal. 33, 153–187.Google Scholar
Stanley, H. E. (1971), Phase Transitions and Critical Phenomena, Clarendon Press, Oxford.Google Scholar
Sun, W., Jiang, S. and Xu, K. (2015), An asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations, J. Comput. Phys. 285, 265–279.CrossRefGoogle Scholar
Szepessy, A. (2011), Langevin molecular dynamics derived from Ehrenfest dynamics, Math. Models Methods Appl. Sci. 21, 2289–2334.CrossRefGoogle Scholar
Tan, D. C., Zhang, T., Chang, T. and Zheng, Y. X. (1994), Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differ. Equations 112, 1–32.CrossRefGoogle Scholar
Temam, R. (1969), Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires (ii), Arch. Ration. Mech. Anal. 33, 377–385.CrossRefGoogle Scholar
Tully, J. C. (1998), Mixed quantum–classical dynamics, Faraday Discussions 110, 407–419.CrossRefGoogle Scholar
Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I. and Shochet, O. (1995), Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75, 1226.CrossRefGoogle Scholar
Villani, C. (2009), Hypocoercivity, Vol. 202 of Memoirs of the American Mathematical Society, American Mathematical Society.CrossRefGoogle Scholar
Wang, Y., Ying, W. and Tang, M. (2018), Uniformly convergent scheme for strongly anisotropic diffusion equations with closed field lines, SIAM J. Sci. Comput. 40, B1253–B1276.CrossRefGoogle Scholar
Wigner, E. P. (1932), On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749–759.CrossRefGoogle Scholar
Xiu, D. (2010), Numerical Methods for Stochastic Computations, Princeton University Press.Google Scholar
Xiu, D. and Karniadakis, G. E. (2002), The Wiener–Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput. 24, 619–644.CrossRefGoogle Scholar
Yan, B. and Jin, S. (2013), A successive penalty-based asymptotic-preserving scheme for kinetic equations, SIAM J. Sci. Comput. 35, A150–A172.CrossRefGoogle Scholar
Yang, C., Deluzet, F. and Narski, J. (2019), On the numerical resolution of anisotropic equations with high order differential operators arising in plasma physics, J. Comput. Phys. 386, 502–523.CrossRefGoogle Scholar
Ying, L., Biros, G. and Zorin, D. (2004), A kernel-independent adaptive fast multipole algorithm in two and three dimensions, J. Comput. Phys. 196, 591–626.CrossRefGoogle Scholar
Zakerzadeh, H. (2017), On the Mach-uniformity of the Lagrange-projection scheme, ESAIM Math. Model. Numer. Anal. 51, 1343–1366.Google Scholar
Zhu, Y. and Jin, S. (2017), The Vlasov–Poisson–Fokker–Planck system with uncertainty and a one-dimensional asymptotic preserving method, Multiscale Model. Simul. 15, 1502–1529.CrossRefGoogle Scholar