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A hybrid HWENO-based method of lines transpose approach for Vlasov simulations

  • School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.52396/JUST-2021-0007
  • Received Date: 04 January 2021
  • Rev Recd Date: 20 March 2021
  • Publish Date: 31 March 2021
    Abstract

  • Abstract

    A new type hybrid Hermite weighted essentially non-oscillatory (HWENO) schemes in the implicit method of lines transpose (MOLT) framework is designed for solving one-dimensional linear transport equations and further applied to the Vlasov-Poisson (VP) simulations via dimensional splitting. Compared with the WENO-based MOLT method given in J. Comput. Phys. [2016, 327: 337-367], the new proposed hybrid HWENO-based MOLT scheme has two advantages. The first is the HWENO schemes using the stencils narrower than those of the WENO schemes with the same order of accuracy. The second is that the schemes can adapt between the linear scheme and the HWENO scheme automatically. In summary, the hybrid HWENO scheme keeps the simplicity and robustness of the simple WENO scheme, while it has higher efficiency with less numerical errors in smooth regions and less computational costs as well. Benchmark examples are given to demonstrate the robustness and good performance of the proposed scheme.

    Abstract

    A new type hybrid Hermite weighted essentially non-oscillatory (HWENO) schemes in the implicit method of lines transpose (MOLT) framework is designed for solving one-dimensional linear transport equations and further applied to the Vlasov-Poisson (VP) simulations via dimensional splitting. Compared with the WENO-based MOLT method given in J. Comput. Phys. [2016, 327: 337-367], the new proposed hybrid HWENO-based MOLT scheme has two advantages. The first is the HWENO schemes using the stencils narrower than those of the WENO schemes with the same order of accuracy. The second is that the schemes can adapt between the linear scheme and the HWENO scheme automatically. In summary, the hybrid HWENO scheme keeps the simplicity and robustness of the simple WENO scheme, while it has higher efficiency with less numerical errors in smooth regions and less computational costs as well. Benchmark examples are given to demonstrate the robustness and good performance of the proposed scheme.
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    • References(36)
  • [1]
    Zhou T, Guo Y, Shu C W. Numerical study on Landau damping. Physica D: Nonlinear Phenomena, 2001, 157(4): 322-333.
    [2]
    Filbet F, Sonnendrücker E. Comparison of Eulerian Vlasov solvers. Computer Physics Communications, 2003, 150(3): 247-266.
    [3]
    Cheng Y, Gamba I M, Morrison P J. Study of conservation and recurrence of Runge-Kutta discontinuous Galerkin schemes for Vlasov-Poisson systems. Journal of Scientific Computing, 2013, 56, 319-349.
    [4]
    Cheng Y, Christlieb A J, Zhong X. Numerical study of the two-species Vlasov-Ampère system: Energy-conserving schemes and the current-driven ion-acoustic instability. Journal of Computational Physics, 2015, 288: 66-85.
    [5]
    Cai X, Guo W, Qiu J M. A high order semi-Lagrangian discontinuous Galerkin method for Vlasov-Poisson simulations without operator splitting. Journal of Computational Physics, 2018, 354: 529-551.
    [6]
    Begue M, Ghizzo A, Bertrand P. Two-dimensional Vlasov simulation of Raman scattering and plasma beatwave acceleration on parallel computers. Journal of Computational Physics,1999, 151(2): 458-478.
    [7]
    Besse N, Segré J, Sonnendrücker E. Semi-Lagrangian schemes for the two-dimensional Vlasov-Poisson system on unstructured meshes. Transport Theory and Statistical Physics, 2005, 34(3-5): 311-332.
    [8]
    Cheng C, Knorr G. The integration of the Vlasov equation in configuration space. Journal of Computational Physics, 1976, 22(3): 330-351.
    [9]
    Christlieb A, Guo W, Morton M, et al. A high order time splitting method based on integral deferred correction for semi-Lagrangian Vlasov simulations. Journal of Computational Physics, 2014, 267: 7-27.
    [10]
    Crouseilles N, Mehrenberger M, Sonnendrücker E. Conservative semi-Lagrangian schemes for Vlasov equations. Journal of Computational Physics, 2010, 229(6): 1927-1953.
    [11]
    Crouseilles N, Respaud T, Sonnendrücker E. A forward semi-Lagrangian method for the numerical solution of the Vlasov equation. Computer Physics Communications, 2009, 180(10): 1730-1745.
    [12]
    Filbet F, Sonnendrücker E, Bertrand P. Conservative numerical schemes for the Vlasov equation. Journal of Computational Physics, 2001, 172(1): 166-187.
    [13]
    Guo W, Qiu J M. Hybrid semi-Lagrangian finite element-finite difference methods for the Vlasov equation. Journal of Computational Physics, 2013, 234: 108-132.
    [14]
    Parker G J, Hitchon W N G. Convected scheme simulations of the electron distribution function in a positive column plasma. Japanese Journal of Applied Physics, 1997, 36(7S): 4799.
    [15]
    Qiu J M, Shu C W. Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov-Poisson system. Journal of Computational Physics, 2011, 230(23): 8386-8409.
    [16]
    Rossmanith J, Seal D. A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations. Journal of Computational Physics, 2011, 230(16): 6203-6232.
    [17]
    Sonnendrücker E, Roche J, Bertrand P, et al. The semi-Lagrangian method for the numerical resolution of the Vlasov equation. Journal of Computational Physics, 1999, 149(2): 201-220.
    [18]
    Birdsall C K, Langdon A B. Plasma Physics Via Computer Simulaition. Boca Raton, FL: CRC Press, 2005.
    [19]
    Verboncoeur J P. Particle simulation of plasmas: Review and advances. Plasma Physics and Controlled Fusion, 2005, 47(5A): A231-A260.
    [20]
    Hockney R W, Eastwood J W. Computer Simulation Using Particles. Boca Raton, FL: CRC Press, 2010.
    [21]
    Barnes J, Hut P. A hierarchical O(N log N) force-calculation algorithm. Nature,1986,324: 446-449.
    [22]
    Evstatiev E G, Shadwick B A. Variational formulation of particle algorithms for kinetic plasma simulations. Journal of Computational Physics, 2013, 245: 376-398.
    [23]
    Christlieb A, Guo W, Jiang Y. A WENO-based method of lines transpose approach for Vlasov simulations. Journal of Computational Physics, 2016, 327: 337-367.
    [24]
    Schemann M, Bornemann F. An adaptive Rothe method for the wave equation. Computing and Visualization in Science, 1998, 1(3): 137-144.
    [25]
    Salazar A J, Raydan M, Campo A. Theoretical analysis of the exponential transversal method of lines for the diffusion equation. Numerical Methods for Partial Differential Equations, 2000, 16(1): 30-41.
    [26]
    Causley M F, Cho H, Christlieb A J, et al. Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution. SIAM Journal on Numerical Analysis, 2016, 54(3): 1635-1652.
    [27]
    Cheng Y, Christlieb A J, Guo W, et al. An asymptotic preserving Maxwell solver resulting in the Darwin limit of electrodynamics. Journal of Scientific Computing, 2017, 71(3): 959-993.
    [28]
    Liu H, Qiu J. Finite difference Hermite WENO schemes for hyperbolic conservation laws. Journal of Scientific Computing, 2015, 63(2): 548-572.
    [29]
    Liu H, Qiu J. Finite difference Hermite WENO schemes for conservation laws, II: An alternative approach. Journal of Scientific Computing, 2016, 66(2): 598-624.
    [30]
    Zhao Z, Zhu J, Chen Y, et al. A new hybrid WENO scheme for hyperbolic conservation laws. Computers & Fluids, 2019, 179: 422-436.
    [31]
    Qiu J, Shu C W. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: One-dimensional case. Journal of Computational Physics, 2004, 193(1): 115-135.
    [32]
    Zhao Z, Chen Y, Qiu J. A hybrid Hermite WENO scheme for hyperbolic conservation laws. Journal of Computational Physics, 2020, 405: 109175.
    [33]
    Zhu J, Qiu J. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method, III: Unstructured meshes. Journal of Scientific Computing, 2009, 39(2): 293-321.
    [34]
    Qiu J, Shu C W. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case. Computers & Fluids, 2005, 34(6): 642-663.
    [35]
    Zhu J, Qiu J. A new type of modified WENO schemes for solving hyperbolic conservation laws. SIAM Journal on Scientific Computing,2017, 39(3): A1089-A1113.
    [36]
    Zhu J, Shu C W. A new type of multi-resolution WENO schemes with increasingly higher order of accuracy. Journal of Computational Physics, 2018, 375: 659-683.
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    [1]
    Zhou T, Guo Y, Shu C W. Numerical study on Landau damping. Physica D: Nonlinear Phenomena, 2001, 157(4): 322-333.
    [2]
    Filbet F, Sonnendrücker E. Comparison of Eulerian Vlasov solvers. Computer Physics Communications, 2003, 150(3): 247-266.
    [3]
    Cheng Y, Gamba I M, Morrison P J. Study of conservation and recurrence of Runge-Kutta discontinuous Galerkin schemes for Vlasov-Poisson systems. Journal of Scientific Computing, 2013, 56, 319-349.
    [4]
    Cheng Y, Christlieb A J, Zhong X. Numerical study of the two-species Vlasov-Ampère system: Energy-conserving schemes and the current-driven ion-acoustic instability. Journal of Computational Physics, 2015, 288: 66-85.
    [5]
    Cai X, Guo W, Qiu J M. A high order semi-Lagrangian discontinuous Galerkin method for Vlasov-Poisson simulations without operator splitting. Journal of Computational Physics, 2018, 354: 529-551.
    [6]
    Begue M, Ghizzo A, Bertrand P. Two-dimensional Vlasov simulation of Raman scattering and plasma beatwave acceleration on parallel computers. Journal of Computational Physics,1999, 151(2): 458-478.
    [7]
    Besse N, Segré J, Sonnendrücker E. Semi-Lagrangian schemes for the two-dimensional Vlasov-Poisson system on unstructured meshes. Transport Theory and Statistical Physics, 2005, 34(3-5): 311-332.
    [8]
    Cheng C, Knorr G. The integration of the Vlasov equation in configuration space. Journal of Computational Physics, 1976, 22(3): 330-351.
    [9]
    Christlieb A, Guo W, Morton M, et al. A high order time splitting method based on integral deferred correction for semi-Lagrangian Vlasov simulations. Journal of Computational Physics, 2014, 267: 7-27.
    [10]
    Crouseilles N, Mehrenberger M, Sonnendrücker E. Conservative semi-Lagrangian schemes for Vlasov equations. Journal of Computational Physics, 2010, 229(6): 1927-1953.
    [11]
    Crouseilles N, Respaud T, Sonnendrücker E. A forward semi-Lagrangian method for the numerical solution of the Vlasov equation. Computer Physics Communications, 2009, 180(10): 1730-1745.
    [12]
    Filbet F, Sonnendrücker E, Bertrand P. Conservative numerical schemes for the Vlasov equation. Journal of Computational Physics, 2001, 172(1): 166-187.
    [13]
    Guo W, Qiu J M. Hybrid semi-Lagrangian finite element-finite difference methods for the Vlasov equation. Journal of Computational Physics, 2013, 234: 108-132.
    [14]
    Parker G J, Hitchon W N G. Convected scheme simulations of the electron distribution function in a positive column plasma. Japanese Journal of Applied Physics, 1997, 36(7S): 4799.
    [15]
    Qiu J M, Shu C W. Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov-Poisson system. Journal of Computational Physics, 2011, 230(23): 8386-8409.
    [16]
    Rossmanith J, Seal D. A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations. Journal of Computational Physics, 2011, 230(16): 6203-6232.
    [17]
    Sonnendrücker E, Roche J, Bertrand P, et al. The semi-Lagrangian method for the numerical resolution of the Vlasov equation. Journal of Computational Physics, 1999, 149(2): 201-220.
    [18]
    Birdsall C K, Langdon A B. Plasma Physics Via Computer Simulaition. Boca Raton, FL: CRC Press, 2005.
    [19]
    Verboncoeur J P. Particle simulation of plasmas: Review and advances. Plasma Physics and Controlled Fusion, 2005, 47(5A): A231-A260.
    [20]
    Hockney R W, Eastwood J W. Computer Simulation Using Particles. Boca Raton, FL: CRC Press, 2010.
    [21]
    Barnes J, Hut P. A hierarchical O(N log N) force-calculation algorithm. Nature,1986,324: 446-449.
    [22]
    Evstatiev E G, Shadwick B A. Variational formulation of particle algorithms for kinetic plasma simulations. Journal of Computational Physics, 2013, 245: 376-398.
    [23]
    Christlieb A, Guo W, Jiang Y. A WENO-based method of lines transpose approach for Vlasov simulations. Journal of Computational Physics, 2016, 327: 337-367.
    [24]
    Schemann M, Bornemann F. An adaptive Rothe method for the wave equation. Computing and Visualization in Science, 1998, 1(3): 137-144.
    [25]
    Salazar A J, Raydan M, Campo A. Theoretical analysis of the exponential transversal method of lines for the diffusion equation. Numerical Methods for Partial Differential Equations, 2000, 16(1): 30-41.
    [26]
    Causley M F, Cho H, Christlieb A J, et al. Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution. SIAM Journal on Numerical Analysis, 2016, 54(3): 1635-1652.
    [27]
    Cheng Y, Christlieb A J, Guo W, et al. An asymptotic preserving Maxwell solver resulting in the Darwin limit of electrodynamics. Journal of Scientific Computing, 2017, 71(3): 959-993.
    [28]
    Liu H, Qiu J. Finite difference Hermite WENO schemes for hyperbolic conservation laws. Journal of Scientific Computing, 2015, 63(2): 548-572.
    [29]
    Liu H, Qiu J. Finite difference Hermite WENO schemes for conservation laws, II: An alternative approach. Journal of Scientific Computing, 2016, 66(2): 598-624.
    [30]
    Zhao Z, Zhu J, Chen Y, et al. A new hybrid WENO scheme for hyperbolic conservation laws. Computers & Fluids, 2019, 179: 422-436.
    [31]
    Qiu J, Shu C W. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: One-dimensional case. Journal of Computational Physics, 2004, 193(1): 115-135.
    [32]
    Zhao Z, Chen Y, Qiu J. A hybrid Hermite WENO scheme for hyperbolic conservation laws. Journal of Computational Physics, 2020, 405: 109175.
    [33]
    Zhu J, Qiu J. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method, III: Unstructured meshes. Journal of Scientific Computing, 2009, 39(2): 293-321.
    [34]
    Qiu J, Shu C W. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case. Computers & Fluids, 2005, 34(6): 642-663.
    [35]
    Zhu J, Qiu J. A new type of modified WENO schemes for solving hyperbolic conservation laws. SIAM Journal on Scientific Computing,2017, 39(3): A1089-A1113.
    [36]
    Zhu J, Shu C W. A new type of multi-resolution WENO schemes with increasingly higher order of accuracy. Journal of Computational Physics, 2018, 375: 659-683.
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