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An h-Adaptive RKDG Method for the Vlasov–Poisson System

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Abstract

In this paper, we propose a new h-adaptive indicator for the Runge–Kutta discontinuous Galerkin (RKDG) scheme in simulations of the Vlasov–Poisson (VP) system. This adaptive indicator, tailored for the VP system, is based on the principle that each cell assumes solution variations as equally as possible. Under the framework of the RKDG method, such adaptive indicator is particularly simple and cheap for the computation. Its effectiveness is demonstrated by extensive numerical tests. The detailed adaptive algorithm as well as some important implementation issues, including the grid and data structure, adaptive criteria, data prolongation/projection and mesh projection, is presented.

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References

  1. Ayuso, B., Carrillo, J.A., Shu, C.-W.: Discontinuous Galerkin methods for the one-dimensional Vlasov–Poisson system. Kinet. Relat. Model. 4, 955–989 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berger, M.J., Oliger, J.: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53(3), 484–512 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Birdsall, C.K., Langdon, A.B.: Plasma Physics Via Computer Simulation. CRC Press, Boca Raton (2004)

    Google Scholar 

  4. Biswas, R., Devine, K., Flaherty, J.: Parallel, adaptive finite element methods for conservation laws. Appl. Numer. Math. 14, 255–283 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carrillo, J.A., Vecil, F.: Nonoscillatory interpolation methods applied to Vlasov-based models. SIAM J. Sci. Comput. 29, 1179–1206 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38(5), 1676–1706 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheng, C.-Z., Knorr, G.: The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22, 330–351 (1976)

    Article  Google Scholar 

  8. Cheng, Y., Gamba, I.M., Morrison, P.J.: Study of conservation and recurrence of Runge–Kutta discontinuous Galerkin schemes for Vlasov–Poisson systems. J. Sci. Comput. 56(2), 319–349 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cockburn, B., Kanschat, G., Perugia, I., Schötzau, D.: Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids. SIAM J. Numer. Anal. 39, 264–285 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cockburn, B., Shu, C.-W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Crouseilles, N., Mehrenberger, M., Sonnendrücker, E.: Conservative semi-Lagrangian schemes for Vlasov equations. J. Comput. Phys. 229(6), 1927–1953 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dedner, A., Makridakis, C., Ohlberger, M.: Error control for a class of Runge–Kutta discontinuous Galerkin methods for nonlinear conservation laws. SIAM J. Numer. Anal. 45, 514–538 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Devine, K., Flaherty, J.: Parallel adaptive \(hp\)-refinement techniques for conservation laws. Appl. Numer. Math. 20, 367–386 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eastwood, J.W.: Particle simulation methods in plasma physics. Comput. Phys. Commun. 43, 89–106 (1986)

    Article  Google Scholar 

  15. Fijalkow, E.: A numerical solution to the Vlasov equation. Comput. Phys. Commun. 116(2–3), 319–328 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Filbet, F., Sonnendrücker, E.: Comparison of Eulerian Vlasov solvers. Comput. Phys. Commun. 150(3), 247–266 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Filbet, F., Sonnendrucker, E., Bertrand, P.: Conservative numerical schemes for the Vlasov equation. J. Comput. Phys. 172(1), 166–187 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Flaherty, J., Loy, R., Shephard, M., Szymanski, B., Teresco, J., Ziantz, L.: Adaptive local refinement with octree load-balancing for the parallel solution of three-dimensional conservation laws. J. Parallel Distrib. Comput. 47, 139–152 (1997)

    Article  MATH  Google Scholar 

  19. Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hartmann, R., Houston, P.: Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws. SIAM J. Sci. Comput. 24, 979–1004 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Heath, R., Gamba, I., Morrison, P., Michler, C.: A discontinuous Galerkin method for the Vlasov–Poisson system. J. Comput. Phys. 231, 1140–1174 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hockney, R.W., Eastwood, J.W.: Computer Simulation Using Particles. CRC Press, Boca Raton (2010)

    MATH  Google Scholar 

  23. Klimas, A., Farrell, W.: A splitting algorithm for Vlasov simulation with filamentation filtration. J. Comput. Phys. 110, 150–163 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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)

    Article  MathSciNet  MATH  Google Scholar 

  25. Qiu, J.-M., Shu, C.-W.: Conservative semi-Lagrangian finite difference WENO formulations with applications to the Vlasov equation. Commun. Comput. Phys. 10(4), 979–1000 (2011)

    Article  MathSciNet  Google Scholar 

  26. Qiu, J.-M., Shu, C.-W.: Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: theoretical analysis and application to the Vlasov–Poisson system. J. Comput. Phys. 230(23), 8386–8409 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Remacle, J.-F., Flaherty, J., Shephard, M.: An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems. SIAM Rev. 45, 53–72 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Rossmanith, J.A., Seal, D.C.: A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations. J. Comput. Phys. 230, 6203–6232 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Schaeffer, J.: Global existence of smooth solutions to the Vlasov Poisson system in three dimensions. Commun. Part. Differ. Equ. 16(8–9), 1313–1335 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shen, C., Qiu, J.-M., Christlieb, A.: Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations. J. Comput. Phys. 230(10), 3780–3802 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  32. Sonnendrücker, E., Roche, J., Bertrand, P., Ghizzo, A.: The semi-Lagrangian method for the numerical resolution of the Vlasov equation. J. Comput. Phys. 149(2), 201–220 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  33. 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. 273, 618–639 (2014)

    Article  MathSciNet  Google Scholar 

  34. Zaki, S., Boyd, T., Gardner, L.: A finite element code for the simulation of one-dimensional Vlasov plasmas. II. Applications. J. Comput. Phys. 79, 200–208 (1988)

    Article  MATH  Google Scholar 

  35. Zaki, S., Gardner, L., Boyd, T.: A finite element code for the simulation of one-dimensional Vlasov plasmas. I. Theory. J. Comput. Phys. 79, 184–199 (1988)

    Article  MATH  Google Scholar 

  36. Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229, 3091–3120 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhou, T., Guo, Y., Shu, C.-W.: Numerical study on Landau damping. Phys. D 157(4), 322–333 (2001)

    Article  MATH  Google Scholar 

  38. Zhu, H., Qiu, J.: Adaptive Runge–Kutta discontinuous Galerkin methods using different indicators: one-dimensional case. J. Comput. Phys. 228, 6957–6976 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhu, H., Qiu, J.: An \(h\)-adaptive RKDG method with troubled-cell indicator for two-dimensional hyperbolic conservation laws. Adv. Comput. Math. 39, 445–463 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Natural Sciences, Nanjing University of Posts and Telecommunications, Nanjing, 210023, Jiangsu, People’s Republic of China

    Hongqiang Zhu

  2. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Jianxian Qiu

  3. Department of Mathematics, University of Houston, Houston, TX, 77204-3008, USA

    Jing-Mei Qiu

Authors
  1. Hongqiang Zhu
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  2. Jianxian Qiu
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  3. Jing-Mei Qiu
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Correspondence to Jing-Mei Qiu.

Additional information

The research is partially supported by NSFC Grants 11201242, 91530107 and 11571290, NSF Grants NSF-DMS-1217008 and NSF-DMS-1522777, Jiangsu Government Scholarship for Overseas Studies, and Air Force Office of Scientific Computing FA9550-12-0318.

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Zhu, H., Qiu, J. & Qiu, JM. An h-Adaptive RKDG Method for the Vlasov–Poisson System. J Sci Comput 69, 1346–1365 (2016). https://doi.org/10.1007/s10915-016-0238-1

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  • DOI: https://doi.org/10.1007/s10915-016-0238-1

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