A High-Order Semi-Lagrangian Finite Difference Method for Nonlinear Vlasov and BGK Models

doi:10.1007/s42967-021-00156-z

Abstract

Abstract: In this paper, we propose a new conservative high-order semi-Lagrangian finite difference (SLFD) method to solve linear advection equation and the nonlinear Vlasov and BGK models. The finite difference scheme has better computational flexibility by working with point values, especially when working with high-dimensional problems in an operator splitting setting. The reconstruction procedure in the proposed SLFD scheme is motivated from the SL finite volume scheme. In particular, we define a new sliding average function, whose cell averages agree with point values of the underlying function. By developing the SL finite volume scheme for the sliding average function, we derive the proposed SLFD scheme, which is high-order accurate, mass conservative and unconditionally stable for linear problems. The performance of the scheme is showcased by linear transport applications, as well as the nonlinear Vlasov-Poisson and BGK models. Furthermore, we apply the Fourier stability analysis to a fully discrete SLFD scheme coupled with diagonally implicit Runge-Kutta (DIRK) method when applied to a stiff two-velocity hyperbolic relaxation system. Numerical stability and asymptotic accuracy properties of DIRK methods are discussed in theoretical and computational aspects.

Key words: Semi-Lagrangian, WENO, Finite difference, Vlasov-Poisson, BGK equation, Linear stability

CLC Number:

65M25

Linjin Li, Jingmei Qiu, Giovanni Russo. A High-Order Semi-Lagrangian Finite Difference Method for Nonlinear Vlasov and BGK Models[J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 170-198.

TrendMD

References

1. Boscarino, S., Cho, S.-Y., Russo, G., Yun, S.-B.: High order conservative semi-Lagrangian scheme for the BGK model of the Boltzmann equation. arXiv: 1905. 03660 (2019)
2. Carrillo, J., Vecil, F.: Nonoscillatory interpolation methods applied to Vlasov-based models. SIAM J. Sci. Comput. 29(3), 1179–1206 (2007)
3. Cheng, C.-Z., Knorr, G.: The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22(3), 330–351 (1976)
4. Crouseilles, N., Mehrenberger, M., Sonnendrücker, E.: Conservative semi-Lagrangian schemes for Vlasov equations. J. Comput. Phys. 229(6), 1927–1953 (2010)
5. Ding, M., Cai, X., Guo, W., Qiu, J.-M.: A semi-Lagrangian discontinuous Galerkin (DG)-local DG method for solving convection-diffusion equations. J. Comput. Phys. 409, 109295 (2020)
6. Ding, M., Qiu, J.-M., Shu, R.: Accuracy and stability analysis of the semi-Lagrangian method for stiff hyperbolic relaxation systems and kinetic BGK model (2021) (in preparation)
7. Filbet, F., Sonnendrücker, E.: Comparison of Eulerian Vlasov solvers. Comput. Phys. Commun. 150(3), 247–266 (2003)
8. Filbet, F., Sonnendrücker, E., Bertrand, P.: Conservative numerical schemes for the Vlasov equation. J. Comput. Phys. 172(1), 166–187 (2001)
9. Gottlieb, S., Ketcheson, D., Shu, C.-W.: High order strong stability preserving time discretizations. J. Sci. Comput. 38(3), 251–289 (2009)
10. Groppi, M., Russo, G., Stracquadanio, G.: High order semi-Lagrangian methods for the BGK equation. arXiv: 1411. 7929 (2014)
11. Guo, W., Nair, R.D., Qiu, J.-M.: A conservative semi-Lagrangian discontinuous Galerkin scheme on the cubed sphere. Mon. Weather Rev. 142(1), 457–475 (2014)
12. Harris, L.M., Lauritzen, P.H., Mittal, R.: A flux-form version of the conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed sphere grid. J. Comput. Phys. 230(4), 1215-1237 (2011)
13. Hu, J., Shu, R., Zhang, X.: Asymptotic-preserving and positivity-preserving implicit-explicit schemes for the stiff BGK equation. SIAM J. Numer. Anal. 56(2), 942–973 (2018)
14. LeVeque, R.: High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Numer. Anal. 33(2), 627–665 (1996)
15. Lin, S.-J., Rood, R.B.: Multidimensional flux-form semi-Lagrangian transport schemes. Mon. Weather Rev. 124(9), 2046–2070 (1996)
16. Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)
17. Liu, Y., Shu, C.-W., Zhang, M.P.: On the positivity of linear weights in WENO approximations. Acta Math. Appl. Sin. Engl. Ser. 25, 503–538 (2009)
18. Pieraccini, S., Puppo, G.: Implicit-explicit schemes for BGK kinetic equations. J. Sci. Comput. 32(1), 1–28 (2007)
19. Qiu, J.-M., Christlieb, A.: A conservative high order semi-Lagrangian WENO method for the Vlasov equation. J. Comput. Phys. 229(4), 1130–1149 (2010)
20. Qiu, J.-M., Shu, C.-W.: Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow. J. Comput. Phys. 230(4), 863–889 (2011)
21. Russo, G., Filbet, F.: Semilagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics. Kinet. Relat. Models 2(1), 231–250 (2009)
22. Russo, G., Santagati, P., Yun, S.-B.: Convergence of a semi-Lagrangian scheme for the BGK model of the Boltzmann equation. SIAM J. Numer. Anal. 50(3), 1111–1135 (2012)
23. Russo, G., Yun, S.-B.: Convergence of a semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation. SIAM J. Numer. Anal. 56(6), 3580–3610 (2018)
24. Shu, C.-W.: High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Rev. 51, 82–126 (2009)
25. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)
26. Umeda, T., Ashour-Abdalla, M., Schriver, D.: Comparison of numerical interpolation schemes for onedimensional electrostatic Vlasov code. J. Plasma Phys. 72(06), 1057–1060 (2006)
27. Wanner, G., Hairer, E.: Solving Ordinary Differential Equations II. Springer, Berlin (1996)

[1]	Kunlei Zhao, Yulong Du, Li Yuan. A New Sixth-Order WENO Scheme for Solving Hyperbolic Conservation Laws [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 3-30.
[2]	Jie Du, Yang Yang. High-Order Bound-Preserving Finite Difference Methods for Multispecies and Multireaction Detonations [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 31-63.
[3]	Jun Zhu, Jianxian Qiu. New Finite Difference Mapped WENO Schemes with Increasingly High Order of Accuracy [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 64-96.
[4]	Scott E. Field, Sigal Gottlieb, Zachary J. Grant, Leah F. Isherwood, Gaurav Khanna. A GPU-Accelerated Mixed-Precision WENO Method for Extremal Black Hole and Gravitational Wave Physics Computations [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 97-115.
[5]	Andrew Christlieb, Matthew Link, Hyoseon Yang, Ruimeng Chang. High-Order Semi-Lagrangian WENO Schemes Based on Non-polynomial Space for the Vlasov Equation [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 116-142.
[6]	M. Semplice, E. Travaglia, G. Puppo. One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 143-169.
[7]	Yifei Wan, Yinhua Xia. A New Hybrid WENO Scheme with the High-Frequency Region for Hyperbolic Conservation Laws [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 199-234.
[8]	Bao-Shan Wang, Wai Sun Don, Alexander Kurganov, Yongle Liu. Fifth-Order A-WENO Schemes Based on the Adaptive Diffusion Central-Upwind Rankine-Hugoniot Fluxes [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 295-314.
[9]	Xucheng Meng, Yaguang Gu, Guanghui Hu. A Fourth-Order Unstructured NURBS-Enhanced Finite Volume WENO Scheme for Steady Euler Equations in Curved Geometries [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 315-342.
[10]	G. Puppo, M. Semplice, G. Visconti. Quinpi: Integrating Conservation Laws with CWENO Implicit Methods [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 343-369.
[11]	Liang Li, Jun Zhu, Chi-Wang Shu, Yong-Tao Zhang. A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 403-427.
[12]	Zepeng Liu, Yan Jiang, Mengping Zhang, Qingyuan Liu. High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 485-528.
[13]	Xiaoyi Liu, Tingchun Wang, Shilong Jin, Qiaoqiao Xu. Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation [J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1509-1530.
[14]	Hendrik Ranocha, Gregor J. Gassner. Preventing Pressure Oscillations Does Not Fix Local Linear Stability Issues of Entropy-Based Split-Form High-Order Schemes [J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 880-903.
[15]	Xiaofeng Cai, Wei Guo, Jing-Mei Qiu. Comparison of Semi-Lagrangian Discontinuous Galerkin Schemes for Linear and Nonlinear Transport Simulations [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 3-33.