Energy conserving discontinuous Galerkin spectral element method for the Vlasov–Poisson system
Introduction
The Vlasov–Poisson system describes the behaviour of a collisionless plasma subject to electrostatic effects in terms of the corresponding distribution function in phase space, thus representing one of the basic models in plasma physics. Due to the high dimensionality of the problem, efficient numerical techniques for the computation of approximate solutions are of paramount importance. Moreover, such techniques should ideally reproduce the main features of the system, namely: the absence of dissipation, the stability of the solution and the conservation of physical invariants such as the total number of particles, the total energy and the total momentum. In the literature, various approaches have been proposed, corresponding to different trade-offs concerning the fulfillment of these requirements. Semi-Lagrangian schemes emphasize computational efficiency, obtained using large time steps [13], [19], [27], [29], [48], [52]. Conservative finite difference formulations, constructed in terms of the Hamiltonian formulation of the problem, result in conservation equations for the physical invariants of the discrete system which parallel those of the continuous one [36]. Discontinuous Galerkin methods, finally, have been considered because of their locality, which allows optimal scaling in massively parallel computations, as well as their accuracy and flexibility in terms of the computational grid [3], [4], [5], [9], [10], [12], [29], [32], [44].
In this paper, we consider a discontinuous Galerkin (DG) spectral element method for the Vlasov–Poisson system with the introduction of a local reconstruction of the electric field which ensures exact discrete energy conservation up to errors introduced by the time discretization. This choice is motivated by the following considerations. The DG method provides the highest locality of the numerical discretization, representing an optimal choice for massively parallel computations, it combines the use of upwind numerical fluxes with high order accuracy, so that the numerical solution tends to be stable but not excessively dissipative, and it permits the use of nonuniform and even nonconforming grids [7], [40]. Moreover, the DG discretization ensures discrete conservation of the total number of particles in a very natural way. The spectral element version of the DG method is obtained by using a quadrature formula for the computation of the integrals appearing in the finite element formulation and by collocating the discrete degrees of freedom at the quadrature nodes [6], [35] and, for high polynomial orders, results in a more efficient method compared to the standard DG one. The spectral element DG method has been used extensively for fluid dynamics computations where, as is the case for the Vlasov–Poisson system, the flow regime is strongly advection dominated but yet the solution does not develop shocks (for instance, see [46] for the case of atmospheric flows). The use of a special reconstruction of the electric field computed in the solution of the Poisson problem is introduced to compensate the spurious energy sources resulting from the upwind numerical flux employed in the DG method, as we discuss in details in the sequel of the paper; such a reconstruction is local (i.e. defined at the element level), in order to preserve the locality of the formulation, and consistent with the accuracy of the scheme, so that it does not degrade the overall convergence rate.
The resulting formulation can be regarded as a modification of the method proposed and analyzed in [3], [4], [5], the main modifications consisting in the facts that a) the method is of spectral element type and b) energy conservation is achieved using a local reconstruction of the electric field, instead of solving multiple Poisson problems. We notice that these two aspects are independent from each other, and in particular the energy conserving reconstruction of the electric field is also applicable in the framework of a standard DG method. We also observe that an alternative approach to obtain energy conserving formulations for the Vlasov–Poisson system can be found in [9] (these ideas are then extended in [10] to the Maxwell system), where the starting point for the numerical discretization is provided by Ampère's law and where energy conserving schemes are presented also for the completely discretized problem in space and time. The counterpart to this is that, by discretizing Ampère's law, charge conservation is not preserved for the numerical solution and most of the time marching schemes discussed in [9] are implicit; in this respect, the pros and the cons of the method of [9] are dual compared with those of the scheme discussed in the present paper.
After discretizing the Vlasov–Poisson system in space with the spectral element DG formulation, we obtain an Ordinary Differential Equation (ODE) which is then integrated in time following the classical method of lines. Typically, this is done with explicit Runge–Kutta (RK) methods, along the lines of the RKDG method introduced in the nowadays classical series of papers [16], [18], [21], [22], [24]. This leads to an accurate and stable formulation, which computational cost however tends to be adversely affected by a somewhat stringent stability condition on the time-step, especially for high order polynomial spaces. A possible alternative to this strategy is represented by the use of exponential time integrators [37], [53], and we present some preliminary comparisons between RK and exponential time marching schemes in the present paper. An alternative approach, which we postpone for further investigation, is the use of semi-Lagrangian DG methods [44], [45].
The remaining of the paper is organized as follows. In Section 2, the Vlasov–Poisson system is introduced. The DG numerical discretization is discussed in Section 3, which mostly follows the ideas of [3], [4]. Section 4 is devoted to the illustration of the local reconstruction strategy for the electric field. Section 5 then addresses all the aspects related with the spectral element formulation. In Section 6 we summarize the time marching schemes considered for the time integration of the resulting ODE system. The resulting numerical scheme is validated numerically in Section 7, and finally some conclusions are drawn in Section 8.
Section snippets
The Vlasov–Poisson system
In d spatial dimensions, the Vlasov–Poisson system is where is the particle distribution function, is the electric field, being the electrostatic potential, and is the particle density. Eq. (2.1), supplemented with an initial condition , must be solved for times and phase space coordinates . Here we assume that and and enforce periodic boundary conditions on both and
Discontinuous Galerkin discretization of the Vlasov–Poisson system
Relevant references concerning the introduction and the analysis of the DG method are in [16], [18], [21], [22], [24], [34], [47] for hyperbolic problems and [2], [8], [11], [23] for elliptic problems. Concerning the specific application to the Vlasov–Poisson system, the DG method has then been studied in [3], [4], [5], [12], [32]; in particular, error estimates of order , where k is the polynomial degree of the local finite element spaces, are shown in [3], [4] for the particle distribution
An energy conserving formulation
In this section, we first analyze to which extent the energy balance (2.9) is reproduced by the discrete method, and then we show how it is possible to obtain an exactly energy conserving spatial discretization by taking advantage of the freedom in the choice of the finite element spaces for and and in the definition of .
The total energy appearing in (2.9) is the sum of two contributions: the kinetic energy and the electrostatic energy, and the conversion between these two energies
Spectral element DG discretization of the Vlasov–Poisson system
To obtain the spectral element formulation corresponding to (3.4), (3.10) and (3.11), one simply makes the formal substitution introducing the quadrature formulae defined in Section 3.1. The resulting scheme can be analyzed within the framework of the generalized Galerkin methods, which is discussed in [15], Chapter 5. We provide now the details for the energy conserving formulation addressed in Theorem 4.2, since all the other cases can be easily deduced from this one.
We consider the
Time discretization
Eq. (5.3) is an ordinary differential equation which can be written as where f is the array of the nodal values of the distribution function and R denotes the right-hand-side; (6.1) can be discretized in time using a time integrator method, following the classical approach of the method of lines. The standard choice is represented by Runge–Kutta type methods, and we consider here the classical RK4 method. Hence, introducing a time step Δt and time levels , for , to
Numerical validation
In this section we address the numerical validation of the spectral element DG formulation. The various test cases are selected to assess the performance of the method concerning the following aspects: convergence rates, overall ability to represent relevant physical phenomena, robustness and discrete conservation properties. References for the considered test cases can be found in [12], [17], [19], [32], [41], [43], [51]. For Sections 7.1, 7.2 and 7.3, the emphasis is on the spatial
Conclusion and future developments
In this work, a spectral element DG formulation for the Vlasov–Poisson equation has been discussed, with particular attention to its discrete conservation properties. The scheme conserves naturally the number of particle, and a suitable reconstruction of the electric field has been discussed resulting in exact energy conservation (up to errors introduced by the time discretization). The accuracy and stability of the resulting method have been investigated using some classical test cases.
Acknowledgements
Many ideas addressed in this work are the result of various discussions with Blanca Ayuso de Dios and Soheil Hajian.
We would also like to thank Luca Bonaventura, Daniel Y. Le Roux, Omar Maj, Michael Kraus and Philip J. Morrison for many interesting comments and suggestions.
The first author (Éric Madaule) worked at this paper during his Master Degree at the ENSEIRB-Matmeca engineering school in Bordeaux.
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