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Conservative Semi-Lagrangian Finite Difference WENO Formulations with Applications to the Vlasov Equation

Published online by Cambridge University Press:  20 August 2015

Jing-Mei Qiu  and
Chi-Wang Shu
Jing-Mei Qiu*
Affiliation:
Department of Mathematical and Computer Science, Colorado School of Mines, Golden, CO 80401, USA
Chi-Wang Shu*
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
*
Corresponding author.Email:jingqiu@mines.edu
Email:shu@dam.brown.edu
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Abstract

In this paper, we propose a new conservative semi-Lagrangian (SL) finite difference (FD) WENO scheme for linear advection equations, which can serve as a base scheme for the Vlasov equation by Strang splitting [4]. The reconstruction procedure in the proposed SL FD scheme is the same as the one used in the SL finite volume (FV) WENO scheme [3]. However, instead of inputting cell averages and approximate the integral form of the equation in a FV scheme, we input point values and approximate the differential form of equation in a FD spirit, yet retaining very high order (fifth order in our experiment) spatial accuracy. The advantage of using point values, rather than cell averages, is to avoid the second order spatial error, due to the shearing in velocity (v) and electrical field (E) over a cell when performing the Strang splitting to the Vlasov equation. As a result, the proposed scheme has very high spatial accuracy, compared with second order spatial accuracy for Strang split SL FV scheme for solving the Vlasov-Poisson (VP) system. We perform numerical experiments on linear advection, rigid body rotation problem; and on the Landau damping and two-stream instabilities by solving the VP system. For comparison, we also apply (1) the conservative SL FD WENO scheme, proposed in [22] for incompressible advection problem, (2) the conservative SL FD WENO scheme proposed in [21] and (3) the non-conservative version of the SL FD WENO scheme in [3] to the same test problems. The performances of different schemes are compared by the error table, solution resolution of sharp interface, and by tracking the conservation of physical norms, energies and entropies, which should be physically preserved.

Keywords

65Semi-Lagrangian methodsfinite difference/finite volume schemeconservative schemeWENO reconstructionVlasov equationLandau dampingtwo-stream instability
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 4 , October 2011 , pp. 979 - 1000
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Boyd, J. P., Chebyshev and Fourier Spectral Methods, Courier Dover Publications, 2001.Google Scholar
[2]Carlini, E., Ferretti, R., and Russo, G., A weighted essentially nonoscillatory, large time-step scheme for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 27(3) (2006), 1071–1091.Google Scholar
[3]Carrillo, J. A. and Vecil, F., Nonoscillatory interpolation methods applied to Vlasov-Based models, SIAM J. Sci. Comput., 29 (2007), 1179–1206.Google Scholar
[4]Cheng, C. Z. and Knorr, G., The integration of the Vlasov equation in configuration space, J. Comput. Phys., 22(3) (1976), 330–351.Google Scholar
[5]Childs, P. N. and Morton, K. W., Characteristic Galerkin methods for scalar conservation laws in one dimension, SIAM J. Numer. Anal., 27(3) (1990), 553–594.Google Scholar
[6]Cockburn, B., Johnson, C., Shu, C.-W., and Tadmor, E., Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Springer, New York, 1998.Google Scholar
[7]Colella, P. and Woodward, P. R., The piecewise parabolic method (PPM) for gas-dynamical simulations, J. Comput. Phys., 54(1) (1984), 174–201.Google Scholar
[8]Crouseilles, N., Latu, G., and Sonnendrucker, E., Hermite spline interpolation on patches for parallelly solving the Vlasov-Poisson equation, Int. J. Appl. Math. Comput. Sci., 17(3) (2007), 335–349.Google Scholar
[9]Crouseilles, N., Mehrenberger, M., and Sonnendrucker, E., Conservative semi-Lagrangian schemes for Vlasov equations, J. Comput. Phys., 229(6) (2010), 1927–1953.Google Scholar
[10]Filbet, F. and Sonnendrucker, E., Comparison of eulerian Vlasov solvers, Comput. Phys. Commun., 150(3) (2003), 247–266.CrossRefGoogle Scholar
[11]Filbet, F. and Sonnendrücker, E., Comparison of Eulerian Vlasov solvers, Comput. Phys. Commun., 150(3) (2003), 247–266.Google Scholar
[12]Filbet, F., Sonnendrücker, E., and Bertrand, P., Conservative numerical schemes for the Vlasov equation, J. Comput. Phys., 172(1) (2001), 166–187.Google Scholar
[13]Filbet, F., Sonnendrucker, E., and Bertrand, P., Conservative numerical schemes for the Vlasov equation. J. Comput. Phys., 172(1) (2001), 166–187.Google Scholar
[14]Gottlieb, S., Ketcheson, D. I., and Shu, C.-W., High order strong stability preserving time discretizations, J. Sci. Comput., 38(3) (2009), 251–289.Google Scholar
[15]Huot, F., Ghizzo, A., Bertrand, P., Sonnendrucker, E., and Coulaud, O., Instability of the time splitting scheme for the one-dimensional and relativistic Vlasov-Maxwell system, J. Comput. Phys., 185(2) (2003), 512–531.Google Scholar
[16]Jiang, G.-S. and Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126(1) (1996), 202–228.CrossRefGoogle Scholar
[17]Lee, T. and Lin, C. L., A characteristic Galerkin method for discrete Boltzmann equation, J. Comput. Phys., 171(1) (2001), 336–356.Google Scholar
[18]LeVeque, R. J., High-resolution conservative algorithms for advection in incompressible flow, SIAM J. Numer. Anal., (1996), 627–665.Google Scholar
[19]Lin, S. J. and Rood, R. B., An explicit flux-form semi-Lagrangian shallow-water model on the sphere, Quart. J. Royal. Meteorologic. Soc., 123(544) (1997), 2477–2498.Google Scholar
[20]Liu, Y., Shu, C.-W., and M, Zhang, On the positivity of linear weights in WENO approximations, Acta. Math. Appl. Sin., 25 (2009), 503–538.Google Scholar
[21]Qiu, J.-M. and Christlieb, A., A Conservative high order semi-Lagrangian WENO method for the Vlasov Equation, J. Comput. Phys., 229(4) (2010), 1130–1149.Google Scholar
[22]Qiu, J.-M. and Shu, C.-W., Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow, J. Comput. Phys., 230 (2011), 863–889.Google Scholar
[23]Qiu, J.-M. and Shu, C.-W., Convergence of Godunov-type schemes for scalar conservation laws under large time steps, SIAM J. Numer. Anal., 46 (2008), 2211–2237.Google Scholar
[24]Qiu, J.-X. and Shu, C.-W., Finite difference WENO schemes with Lax-Wendroff-type time discretizations, SIAM J. Sci. Comput., 24(6) (2003), 2185–2200.CrossRefGoogle Scholar
[25]Sebastian, K. and Shu, C.-W., Multidomain WENO finite difference method with interpolation at subdomain interfaces, J. Sci. Comput., 19(1) (2003), 405–438.CrossRefGoogle Scholar
[26]Shu, C.-W., High order weighted essentially non-oscillatory schemes for convection dominated problems, SIAM Rev., 51 (2009), 82–126.Google Scholar
[27]Sonnendruecker, E., Roche, J., Bertrand, P., and Ghizzo, A., The semi-Lagrangian method for the numerical resolution of the Vlasov equation, J. Comput. Phys., 149(2) (1999), 201–220.Google Scholar
[28]Takewaki, H., Nishiguchi, A., and Yabe, T., Cubic interpolated pseudo-particle method (CIP) for solving hyperbolic-type equations, J. Comput. Phys., 61(2) (1985), 261–268.Google Scholar
[29]Umeda, T., Ashour-Abdalla, M., and Schriver, D., Comparison of numerical interpolation schemes for one-dimensional electrostatic Vlasov code, J. Plasma. Phys., 72(06) (2006), 1057–1060.Google Scholar
[30]Xiu, D. and Karniadakis, G. E., A semi-Lagrangian high-order method for Navier-Stokes equations, J. Comput. Phys., 172(2) (2001), 658–684.Google Scholar
[31]Yabe, T., Xiao, F., and Utsumi, T., The constrained interpolation profile method for multiphase analysis, J. Comput. Phys., 169(2) (2001), 556–593.Google Scholar