Error estimates of Runge–Kutta discontinuous galerkin methods for the Vlasov–Maxwell system

Yang, He; Li, Fengyan

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 69-99.

In this paper, error analysis is established for Runge–Kutta discontinuous Galerkin (RKDG) methods to solve the Vlasov–Maxwell system. This nonlinear hyperbolic system describes the time evolution of collisionless plasma particles of a single species under the self-consistent electromagnetic field, and it models many phenomena in both laboratory and astrophysical plasmas. The methods involve a third order TVD Runge–Kutta discretization in time and upwind discontinuous Galerkin discretizations of arbitrary order in phase domain. With the assumption that the exact solutions have sufficient regularity, the $L^2$ errors of the particle number density function as well as electric and magnetic fields at any given time $T$ are bounded by $C{h}^{k+1/2}+C\tau {}^3$ under a CFL condition $\tau /h\le \gamma$. Here $k$ is the polynomial degree used in phase space discretization, satisfying $k>\frac{d_x+1}{2}$ (with $d_x$ being the dimension of spatial domain), $\tau$ is the time step, and $h$ is the maximum mesh size in phase space. Both $C$ and $\gamma$ are positive constants independent of $h$ and $\tau$, and they may depend on the polynomial degree $k$, time $T$, the size of the phase domain, certain mesh parameters, and some Sobolev norms of the exact solution. The analysis can be extended to RKDG methods with other numerical fluxes and to RKDG methods solving relativistic Vlasov–Maxwell equations.

B. Ayuso, J.A. Carrillo and C.-W. Shu, Discontinuous Galerkin methods for the one-dimensional Vlasov–Poisson system. Kinetic and Related Models 4 (2011) 955–989. | DOI | MR | Zbl

B. Ayuso, J.A. Carrillo and C.-W. Shu, Discontinuous Galerkin methods for the multi-dimensional Vlasov–Poisson problem. Math. Models Methods Appl. Sci. 22 (2012) 1250042. | DOI | MR | Zbl

C.K. Birdsall and A.B. Langdon, Plasma Physics Via Computer Simulation. McGraw-Hill, New York (1985).

Y. Cheng and I.M. Gamba, Numerical study of one-dimensional Vlasov–Poisson equations for infinite homogeneous stellar systems. Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 2052–2061. | DOI | MR | Zbl

Y. Cheng, I.M. Gamba, F. Li and P.J. Morrison, Discontinuous Galerkin methods for Vlasov–Maxwell equations. SIAM J. Numer. Anal. 52 (2014) 1017–1049. | DOI | MR | Zbl

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

P.G. Ciarlet, Finite element method for elliptic problems. Noth-Holland, Amsterdam (1978). | MR | Zbl

B. Cockburn and C.-W. Shu, TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52 (1989) 411–435. | MR | Zbl

B. Cockburn and C.-W. Shu, The Runge–Kutta local projection p1-discontinuous Galerkin finite element method for scalar conservation laws. ESAIM: M2AN 25 (1991) 337–361. | DOI | Numdam | MR | Zbl

G.H. Cottet and P.A. Raviart, On Particle in Cell methods for the Vlasov–Poisson equations. Transport Theory Stat. Phys. 15 (1986) 1–31. | DOI | MR

M. Crouzeix and V. Thomée, The stability in $L_p$ and $W_p^1$ of the $L_2$-projection onto finite element function spaces. Math. Comput. 178 (1987) 521–532. | MR | Zbl

B. Eliasson, Numerical modelling of the two-dimensional Fourier transformed Vlasov–Maxwell system, J. Comput. Phys. 190 (2003) 501–522. | DOI | MR | Zbl

B. Eliasson, Numerical simulations of the Fourier-transformed Vlasov–Maxwell system in higher dimensions-theory and applications. Transport Theory Stat. Phys. 39 (2011) 387–465. | DOI | MR | Zbl

R.E. Heath, I.M. Gamba, P.J. Morrison and C. Michler, A discontinuous Galerkin method for the Vlasov–Poisson system. J. Comput. Phys. 231 (2012) 1140–1174. | DOI | MR | Zbl

G. Jacobs and J.S. Hesthaven, Implicit-explicit time integration of a high-order particle-in-cell method with hyperbolic divergence cleaning. Comput. Phys. Commun. 180 (2009) 1760–1767. | DOI | MR | Zbl

A.J. Klimas and W.M. Farrell, A splitting algorithm for Vlasov simulation with filamentation filtration. J. Comput. Phys. 110 (1994) 150–163. | DOI | MR | Zbl

M.C. Pinto and M. Mehrenberger, Convergence of an adaptive semi-Lagrangian scheme for the Vlasov–Poisson system. Numer. Math. 108 (2008) 407–444. | DOI | MR | Zbl

J.-M. Qiu and C.-W. Shu, Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow. J. Comput. Phys. 230 (2011) 863–889. | DOI | MR | Zbl

C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77 (1988) 439–471. | DOI | MR | Zbl

E. Sonnendrücker, J. Roche, P. Bertrand and A. Ghizzo, The semi-Lagrangian method for the numerical resolution of the Vlasov equation. J. Comput. Phys. 149 (1999) 201–220. | DOI | MR | Zbl

A. Suzuki and T. Shigeyama, A conservative scheme for the relativistic Vlasov–Maxwell system. J. Comput. Phys. 229 (2010) 1643–1660. | DOI | MR | Zbl

H. Yang and F. Li, Discontinuous Galerkin methods for relativitistic Vlasov–Maxwell equations, in preparation.

H. Yang, F. Li and J. Qiu, Dispersion and dissipation errors of two fully discrete discontinuous Galerkin methods. J. Sci. Comput. 55 (2013) 552–574. | DOI | MR | Zbl

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

S.I. Zaki, T.J.M. Boyd and L.R.T. Gardner, A finite element code for the simulation of one-dimensional Vlasov plasmas. II. Applications. J. Comput. Phys. 79 (1988) 200–208. | DOI | Zbl

Q. Zhang and C-W Shu, Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal. 42 (2004) 641–666. | DOI | MR | Zbl

Q. Zhang and C-W Shu, Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for symmetrizable systems of conservation laws. SIAM J. Numer. Anal. 44 (2006) 1703–1720. | DOI | MR | Zbl

Q. Zhang and C-W. Shu, Stability analysis and a priori error estimates to the third order explicit Runge–Kutta discontinuous Galerkin method for scalar conservation laws. SIAM J. Numer. Anal. 48 (2010) 1038–1063. | DOI | MR | Zbl

X. Zhong and C.-W. Shu, Numerical resolution of discontinuous Galerkin methods for time dependent wave equations. Comput. Methods Appl. Mech. Eng. 200 (2011) 2814–2827. | DOI | MR | Zbl