Skip to main content
Top
Published in: Journal of Scientific Computing 1/2019

27-06-2018

A Class of Low Dissipative Schemes for Solving Kinetic Equations

Authors: Giacomo Dimarco, Cory Hauck, Raphaël Loubère

Published in: Journal of Scientific Computing | Issue 1/2019

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We introduce an extension of the fast semi-Lagrangian scheme developed in J Comput Phys 255:680–698 (2013) in an effort to increase the spatial accuracy of the method. The basic idea of this extension is to modify the first-order accurate transport step of the original semi-Lagrangian scheme to allow for a general piecewise polynomial reconstruction of the distribution function. For each discrete velocity, we update the solution not at cell centers, but rather at the extreme points of the spatial reconstruction, the locations of which are different for each discrete velocity and change with time. Several approaches are discussed for evaluating the collision operator at these extreme points using only cell center values by making special assumption on the spatial variation of the collision operator. The result is a class of schemes that preserves the structure of the solution over very long times when compared to existing schemes in the literature. As a proof of concept, the new method is implemented in a one-dimensional setting, using piecewise linear reconstructions of the distribution function together with a related reconstruction of the collision operator. The method is derived both for the relatively simple Bhatnagar–Gross–Krook (BGK) operator as well as for the classical Boltzmann operator. Several numerical tests are used to assess the performance of the implementation, including comparisons with the original method in J Comput Phys 255:680–698 (2013) and with classical semi-Lagrangian methods of first and second order. In convergence tests, we observe uniform second-order accuracy across all flow regimes for the BGK operator and nearly second-order accuracy for the Boltzmann operator. In addition, we observe that the method outperforms the classical semi-Lagrangian approach, in particular when resolving fine solution structures in space.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Footnotes
1
We consider different locations to highlight areas where we observe the largest differences in the methods.
 
2
A mesh convergence test, not reported, confirms that the wave locations for all schemes converge to the same location when the number of points increases.
 
Literature
1.
go back to reference Bird, G.A.: Molecular Gas Dynamics and Direct Simulation of Gas Flows. Clarendon Press, Oxford (1994) Bird, G.A.: Molecular Gas Dynamics and Direct Simulation of Gas Flows. Clarendon Press, Oxford (1994)
2.
go back to reference Birsdall, C.K., Langdon, A.B.: Plasma Physics Via Computer Simulation. Series in Plasma Physics. Institute of Physics (IOP), London (2004) Birsdall, C.K., Langdon, A.B.: Plasma Physics Via Computer Simulation. Series in Plasma Physics. Institute of Physics (IOP), London (2004)
3.
go back to reference Bobylev, A.V., Rjasanow, S.: Difference scheme for the Boltzmann equation based on the fast Fourier transform. Eur. J. Mech. B. Fluids 16(2), 293–306 (1997)MathSciNetMATH Bobylev, A.V., Rjasanow, S.: Difference scheme for the Boltzmann equation based on the fast Fourier transform. Eur. J. Mech. B. Fluids 16(2), 293–306 (1997)MathSciNetMATH
4.
go back to reference Bobylev, A.V., Palczewski, A., Schneider, J.: On approximation of the Boltzmann equation by discrete velocity models. C. R. Acad. Sci. Paris Ser. I. Math 320, 639–644 (1995)MathSciNetMATH Bobylev, A.V., Palczewski, A., Schneider, J.: On approximation of the Boltzmann equation by discrete velocity models. C. R. Acad. Sci. Paris Ser. I. Math 320, 639–644 (1995)MathSciNetMATH
6.
7.
go back to reference Cercignani, C., Illner, R., Pulvirenti, Mario: The Mathematical Theory of Dilute Gases, vol. 106. Springer Science & Business Media, New York (2013)MATH Cercignani, C., Illner, R., Pulvirenti, Mario: The Mathematical Theory of Dilute Gases, vol. 106. Springer Science & Business Media, New York (2013)MATH
8.
go back to reference Chacón, L., del-Castillo-Negrete, D., Hauck, C.D.: An asymptotic-preserving semi-Lagrangian algorithm for the time-dependent anisotropic heat transport equation. J. Comp. Phys. 272, 719–746 (2014)MathSciNetCrossRefMATH Chacón, L., del-Castillo-Negrete, D., Hauck, C.D.: An asymptotic-preserving semi-Lagrangian algorithm for the time-dependent anisotropic heat transport equation. J. Comp. Phys. 272, 719–746 (2014)MathSciNetCrossRefMATH
9.
go back to reference Cheng, C.Z., Knorr, G.: The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22, 330–351 (1976)CrossRef Cheng, C.Z., Knorr, G.: The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22, 330–351 (1976)CrossRef
10.
go back to reference Crouseilles, N., Respaud, T., Sonnendrucker, E.: A forward semi-Lagrangian method for the numerical solution of the Vlasov equation. Comp. Phys. Commun. 180(10), 1730–1745 (2009)MathSciNetCrossRefMATH Crouseilles, N., Respaud, T., Sonnendrucker, E.: A forward semi-Lagrangian method for the numerical solution of the Vlasov equation. Comp. Phys. Commun. 180(10), 1730–1745 (2009)MathSciNetCrossRefMATH
11.
go back to reference Crouseilles, N., Mehrenberger, M., Sonnendrucker, E.: Conservative semi-Lagrangian schemes for Vlasov equations. J. Comp. Phys. 229, 1927–1953 (2010)MathSciNetCrossRefMATH Crouseilles, N., Mehrenberger, M., Sonnendrucker, E.: Conservative semi-Lagrangian schemes for Vlasov equations. J. Comp. Phys. 229, 1927–1953 (2010)MathSciNetCrossRefMATH
12.
13.
go back to reference Desvillettes, L., Mischler, S.: About the splitting algorithm for Boltzmann and BGK equations. Math. Mod. Methods Appl. Sci. 6, 1079–1101 (1996)CrossRefMATH Desvillettes, L., Mischler, S.: About the splitting algorithm for Boltzmann and BGK equations. Math. Mod. Methods Appl. Sci. 6, 1079–1101 (1996)CrossRefMATH
14.
go back to reference Dimarco, G., Hauck, C., Loubère, R.: Multidimensional high order FK schemes for the Boltzmann equation (in progress) Dimarco, G., Hauck, C., Loubère, R.: Multidimensional high order FK schemes for the Boltzmann equation (in progress)
15.
go back to reference Dimarco, G.: The hybrid moment guided Monte Carlo method for the Boltzmann equation. Kin. Rel. Models 6, 291–315 (2013)CrossRefMATH Dimarco, G.: The hybrid moment guided Monte Carlo method for the Boltzmann equation. Kin. Rel. Models 6, 291–315 (2013)CrossRefMATH
16.
go back to reference Dimarco, G., Loubère, R.: Towards an ultra efficient kinetic scheme. Part I: basics on the BGK equation. J. Comput. Phys. 255, 680–698 (2013)MathSciNetCrossRefMATH Dimarco, G., Loubère, R.: Towards an ultra efficient kinetic scheme. Part I: basics on the BGK equation. J. Comput. Phys. 255, 680–698 (2013)MathSciNetCrossRefMATH
17.
go back to reference Dimarco, G., Loubère, R.: Towards an ultra efficient kinetic scheme. Part II: the high order case. J. Comput. Phys. 255, 699–719 (2013)MathSciNetCrossRefMATH Dimarco, G., Loubère, R.: Towards an ultra efficient kinetic scheme. Part II: the high order case. J. Comput. Phys. 255, 699–719 (2013)MathSciNetCrossRefMATH
18.
go back to reference Dimarco, G., Pareschi, L.: A fluid solver independent hybrid method for multiscale kinetic equations. SIAM J. Sci. Comput. 32, 603–634 (2010)MathSciNetCrossRefMATH Dimarco, G., Pareschi, L.: A fluid solver independent hybrid method for multiscale kinetic equations. SIAM J. Sci. Comput. 32, 603–634 (2010)MathSciNetCrossRefMATH
19.
go back to reference Dimarco, G., Pareschi, L.: Asymptotic preserving implicit-explicit Runge–Kutta methods for non linear kinetic equations. SIAM J. Numer. Anal. 49, 2057–2077 (2011)MathSciNetCrossRefMATH Dimarco, G., Pareschi, L.: Asymptotic preserving implicit-explicit Runge–Kutta methods for non linear kinetic equations. SIAM J. Numer. Anal. 49, 2057–2077 (2011)MathSciNetCrossRefMATH
20.
go back to reference Dimarco, G., Pareschi, L.: Exponential Runge–Kutta methods for stiff kinetic equations. SIAM J. Numer. Anal. 51, 1064–1087 (2013)MathSciNetCrossRefMATH Dimarco, G., Pareschi, L.: Exponential Runge–Kutta methods for stiff kinetic equations. SIAM J. Numer. Anal. 51, 1064–1087 (2013)MathSciNetCrossRefMATH
22.
go back to reference Dimarco, G., Loubère, R., Narski, J., Rey, T.: An efficient numerical method for solving the Boltzmann equation in multidimensions. J. Comp. Phys. 353, 46–81 (2018)MathSciNetCrossRefMATH Dimarco, G., Loubère, R., Narski, J., Rey, T.: An efficient numerical method for solving the Boltzmann equation in multidimensions. J. Comp. Phys. 353, 46–81 (2018)MathSciNetCrossRefMATH
23.
go back to reference Filbet, F., Russo, G.: High order numerical methods for the space non-homogeneous Boltzmann equation. J. Comput. Phys. 186, 457–480 (2003)MathSciNetCrossRefMATH Filbet, F., Russo, G.: High order numerical methods for the space non-homogeneous Boltzmann equation. J. Comput. Phys. 186, 457–480 (2003)MathSciNetCrossRefMATH
24.
go back to reference Filbet, F., Sonnendrücker, E., Bertrand, P.: Conservative numerical schemes for the Vlasov equation. J. Comput. Phys. 172, 166–187 (2001)MathSciNetCrossRefMATH Filbet, F., Sonnendrücker, E., Bertrand, P.: Conservative numerical schemes for the Vlasov equation. J. Comput. Phys. 172, 166–187 (2001)MathSciNetCrossRefMATH
25.
go back to reference Filbet, F., Mouhot, C., Pareschi, L.: Solving the Boltzmann equation in N log2 N. SIAM J. Sci. Comput. 28(3), 1029–1053 (2007)CrossRefMATH Filbet, F., Mouhot, C., Pareschi, L.: Solving the Boltzmann equation in N log2 N. SIAM J. Sci. Comput. 28(3), 1029–1053 (2007)CrossRefMATH
26.
go back to reference Gamba, I.M., Tharkabhushaman, S.H.: Spectral-Lagrangian based methods applied to computation of non-equilibrium statistical states. J. Comput. Phys. 228, 2012–2036 (2009)MathSciNetCrossRef Gamba, I.M., Tharkabhushaman, S.H.: Spectral-Lagrangian based methods applied to computation of non-equilibrium statistical states. J. Comput. Phys. 228, 2012–2036 (2009)MathSciNetCrossRef
27.
go back to reference Gamba, I.M., Haack, J.R., Hauck, C.D., Hu, J.: A fast spectral method for the Boltzmann collision operator with general collision kernels. SIAM J. Sci. Comput. 39, B658–B674 (2017)MathSciNetCrossRefMATH Gamba, I.M., Haack, J.R., Hauck, C.D., Hu, J.: A fast spectral method for the Boltzmann collision operator with general collision kernels. SIAM J. Sci. Comput. 39, B658–B674 (2017)MathSciNetCrossRefMATH
28.
go back to reference Groppi, M., Russo, G., Stracquadanio, G.: High order semi-Lagrangian methods for the BGK equation. Commun. Math. Sci. 14, 389414 (2016)MathSciNetCrossRefMATH Groppi, M., Russo, G., Stracquadanio, G.: High order semi-Lagrangian methods for the BGK equation. Commun. Math. Sci. 14, 389414 (2016)MathSciNetCrossRefMATH
29.
go back to reference Groppi, M., Russo, G., Stracquadanio, G.: Boundary conditions for semi-Lagrangian methods for the BGK model. Commun. Appl. Ind. Math 7(3), 138164 (2016)MathSciNetMATH Groppi, M., Russo, G., Stracquadanio, G.: Boundary conditions for semi-Lagrangian methods for the BGK model. Commun. Appl. Ind. Math 7(3), 138164 (2016)MathSciNetMATH
30.
go back to reference Gross, E.P., Bathnagar, P.L., Krook, M.: A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954)CrossRefMATH Gross, E.P., Bathnagar, P.L., Krook, M.: A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954)CrossRefMATH
31.
go back to reference Güçlü, Y., Hitchon, W.N.G.: A high order cell-centered semi-Lagrangian scheme for multi-dimensional kinetic simulations of neutral gas flows. J. Comput. Phys. 231, 3289–3316 (2012)MathSciNetCrossRefMATH Güçlü, Y., Hitchon, W.N.G.: A high order cell-centered semi-Lagrangian scheme for multi-dimensional kinetic simulations of neutral gas flows. J. Comput. Phys. 231, 3289–3316 (2012)MathSciNetCrossRefMATH
32.
go back to reference Güçlü, Y., Christlieb, A.J., Hitchon, W.N.G.: Arbitrarily high order convected scheme solution of the Vlasov–Poisson system. J. Comput. Phys. 270, 711–752 (2014)MathSciNetCrossRefMATH Güçlü, Y., Christlieb, A.J., Hitchon, W.N.G.: Arbitrarily high order convected scheme solution of the Vlasov–Poisson system. J. Comput. Phys. 270, 711–752 (2014)MathSciNetCrossRefMATH
33.
go back to reference Hauck, C.D., McClarren, R.G.: A collision-based hybrid method for time-dependent, linear, kinetic transport equations. Multiscale Model. Simul. 11, 1197–1227 (2013)MathSciNetCrossRefMATH Hauck, C.D., McClarren, R.G.: A collision-based hybrid method for time-dependent, linear, kinetic transport equations. Multiscale Model. Simul. 11, 1197–1227 (2013)MathSciNetCrossRefMATH
34.
go back to reference Homolle, T., Hadjiconstantinou, N.: A low-variance deviational simulation Monte Carlo for the Boltzmann equation. J. Comput. Phys. 226, 2341–2358 (2007)MathSciNetCrossRefMATH Homolle, T., Hadjiconstantinou, N.: A low-variance deviational simulation Monte Carlo for the Boltzmann equation. J. Comput. Phys. 226, 2341–2358 (2007)MathSciNetCrossRefMATH
35.
go back to reference Homolle, T., Hadjiconstantinou, N.: Low-variance deviational simulation Monte Carlo. Phys. Fluids 19, 041701 (2007)CrossRefMATH Homolle, T., Hadjiconstantinou, N.: Low-variance deviational simulation Monte Carlo. Phys. Fluids 19, 041701 (2007)CrossRefMATH
36.
go back to reference Jin, S.: Efficient asymptotic-preserving (ap) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21, 441454 (1999)MathSciNetCrossRef Jin, S.: Efficient asymptotic-preserving (ap) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21, 441454 (1999)MathSciNetCrossRef
37.
go back to reference LeVeque, R.J.: Numerical Methods for Conservation Laws, Lectures in Mathematics. Birkhauser Verlag, Basel (1992)CrossRefMATH LeVeque, R.J.: Numerical Methods for Conservation Laws, Lectures in Mathematics. Birkhauser Verlag, Basel (1992)CrossRefMATH
38.
go back to reference Mieussens, L.: Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamic. Math. Mod. Methods App. Sci. 10, 1121–1149 (2000)MathSciNetMATH Mieussens, L.: Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamic. Math. Mod. Methods App. Sci. 10, 1121–1149 (2000)MathSciNetMATH
39.
40.
go back to reference Palczewski, A., Schneider, J.: Existence, stability, and convergence of solutions of discrete velocity models to the Boltzmann equation. J. Stat. Phys. 91, 307–326 (1998)MathSciNetCrossRefMATH Palczewski, A., Schneider, J.: Existence, stability, and convergence of solutions of discrete velocity models to the Boltzmann equation. J. Stat. Phys. 91, 307–326 (1998)MathSciNetCrossRefMATH
41.
go back to reference Palczewski, A., Schneider, J., Bobylev, A.V.: A consistency result for a discrete-velocity model of the Boltzmann equation. SIAM J. Numer. Anal. 34, 1865–1883 (1997)MathSciNetCrossRefMATH Palczewski, A., Schneider, J., Bobylev, A.V.: A consistency result for a discrete-velocity model of the Boltzmann equation. SIAM J. Numer. Anal. 34, 1865–1883 (1997)MathSciNetCrossRefMATH
42.
go back to reference Pareschi, L., Russo, G.: Numerical solution of the Boltzmann equation I: spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37, 12171245 (2000)MathSciNetCrossRefMATH Pareschi, L., Russo, G.: Numerical solution of the Boltzmann equation I: spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37, 12171245 (2000)MathSciNetCrossRefMATH
43.
go back to reference Pareschi, L., Toscani, G.: Interacting Multi-agent Systems. Kinetic Equations and Monte Carlo Methods. Oxford University Press, New York (2013)MATH Pareschi, L., Toscani, G.: Interacting Multi-agent Systems. Kinetic Equations and Monte Carlo Methods. Oxford University Press, New York (2013)MATH
44.
go back to reference Pareschi, L., Russo, G., Toscani, G.: Fast spectral methods for the Fokker PlanckLandau collision operator. J. Comput. Phys. 165, 216236 (2000)CrossRefMATH Pareschi, L., Russo, G., Toscani, G.: Fast spectral methods for the Fokker PlanckLandau collision operator. J. Comput. Phys. 165, 216236 (2000)CrossRefMATH
45.
go back to reference Qiu, J.-M., Christlieb, A.: A Conservative high order semi-Lagrangian WENO method for the Vlasov equation. J. Comput. Phys. 229, 1130–1149 (2010)MathSciNetCrossRefMATH Qiu, J.-M., Christlieb, A.: A Conservative high order semi-Lagrangian WENO method for the Vlasov equation. J. Comput. Phys. 229, 1130–1149 (2010)MathSciNetCrossRefMATH
46.
go back to reference Qiu, J.-M., Shu, C.-W.: Conservative semi-Lagrangian finite difference WENO formulations with applications to the Vlasov equation. Commun. Comput. Phys. 10, 979–1000 (2011)MathSciNetCrossRefMATH Qiu, J.-M., Shu, C.-W.: Conservative semi-Lagrangian finite difference WENO formulations with applications to the Vlasov equation. Commun. Comput. Phys. 10, 979–1000 (2011)MathSciNetCrossRefMATH
48.
go back to reference Sod, G.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1–31 (1978)MathSciNetCrossRefMATH Sod, G.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1–31 (1978)MathSciNetCrossRefMATH
49.
go back to reference Sonnendrücker, E., Roche, J., Bertrand, P., Ghizzo, A.: The semi-Lagrangian method for the numerical resolution of the Vlasov equation. J. Comput. Phys. 149, 201220 (1999)MathSciNetCrossRefMATH Sonnendrücker, E., Roche, J., Bertrand, P., Ghizzo, A.: The semi-Lagrangian method for the numerical resolution of the Vlasov equation. J. Comput. Phys. 149, 201220 (1999)MathSciNetCrossRefMATH
51.
go back to reference Xiong, T., Qiu, J.-M., Xu, Z., Christlieb, A.: High order maximum principle preserving semi-Lagrangian finite difference WENO schemes for the Vlasov equation. J. Comput. Phys. 73, 618639 (2014)MathSciNetMATH Xiong, T., Qiu, J.-M., Xu, Z., Christlieb, A.: High order maximum principle preserving semi-Lagrangian finite difference WENO schemes for the Vlasov equation. J. Comput. Phys. 73, 618639 (2014)MathSciNetMATH
Metadata
Title
A Class of Low Dissipative Schemes for Solving Kinetic Equations
Authors
Giacomo Dimarco
Cory Hauck
Raphaël Loubère
Publication date
27-06-2018
Publisher
Springer US
Published in
Journal of Scientific Computing / Issue 1/2019
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-018-0776-9

Other articles of this Issue 1/2019

Journal of Scientific Computing 1/2019 Go to the issue

Premium Partner