A High-Order Semi-Lagrangian Finite Difference Method for Nonlinear Vlasov and BGK Models
Tóm tắt
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.
Tài liệu tham khảo
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)
Carrillo, J., Vecil, F.: Nonoscillatory interpolation methods applied to Vlasov-based models. SIAM J. Sci. Comput. 29(3), 1179–1206 (2007)
Cheng, C.-Z., Knorr, G.: The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22(3), 330–351 (1976)
Crouseilles, N., Mehrenberger, M., Sonnendrücker, E.: Conservative semi-Lagrangian schemes for Vlasov equations. J. Comput. Phys. 229(6), 1927–1953 (2010)
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)
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)
Filbet, F., Sonnendrücker, E.: Comparison of Eulerian Vlasov solvers. Comput. Phys. Commun. 150(3), 247–266 (2003)
Filbet, F., Sonnendrücker, E., Bertrand, P.: Conservative numerical schemes for the Vlasov equation. J. Comput. Phys. 172(1), 166–187 (2001)
Gottlieb, S., Ketcheson, D., Shu, C.-W.: High order strong stability preserving time discretizations. J. Sci. Comput. 38(3), 251–289 (2009)
Groppi, M., Russo, G., Stracquadanio, G.: High order semi-Lagrangian methods for the BGK equation. arXiv:1411.7929 (2014)
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)
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)
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)
LeVeque, R.: High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Numer. Anal. 33(2), 627–665 (1996)
Lin, S.-J., Rood, R.B.: Multidimensional flux-form semi-Lagrangian transport schemes. Mon. Weather Rev. 124(9), 2046–2070 (1996)
Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)
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)
Pieraccini, S., Puppo, G.: Implicit-explicit schemes for BGK kinetic equations. J. Sci. Comput. 32(1), 1–28 (2007)
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)
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)
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)
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)
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)
Shu, C.-W.: High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Rev. 51, 82–126 (2009)
Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)
Umeda, T., Ashour-Abdalla, M., Schriver, D.: Comparison of numerical interpolation schemes for one-dimensional electrostatic Vlasov code. J. Plasma Phys. 72(06), 1057–1060 (2006)
Wanner, G., Hairer, E.: Solving Ordinary Differential Equations II. Springer, Berlin (1996)