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Abstract. Discontinuous Galerkin methods are developed for solving the Vlasov-Maxwell system, methods that are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy. Such properties in general are hard to achieve within other numerical method frameworks for simulating the Vlasov-Maxwell system. The proposed scheme employs discontinuous Galerkin discretizations for both the Vlasov and the Maxwell equations, resulting in a consistent description of the distribution function and electromagnetic fields. It is proven, up to some boundary effects, that charge is conserved and the total energy can be preserved with suitable choices of the numerical flux for the Maxwell equations and the underlying approximation spaces. Error estimates are established for several flux choices. The scheme is tested on the streaming Weibel instability: the order of accuracy and conservation properties of the proposed method are verified.

In this paper, we generalize the idea in our previous work for the Vlasov-Ampère (VA) system [Y. Cheng, A. J. Christlieb, and X. Zhong. Energy conserving schemes for Vlasov-Ampère systems. J. Comput. Phys., 256:630-655, 2014] and develop energyconserving discontinuous Galerkin (DG) methods for the Vlasov-Maxwell (VM) system. The VM system is a fundamental model in the simulation of collisionless magnetized plasmas. Compared to [Y. Cheng, A. J. Christlieb, and X. Zhong. Energy conserving schemes for Vlasov-Ampère systems. J. Comput. Phys., 256:630-655, 2014], additional care needs to be taken for both the temporal and spatial discretizations to achieve similar type of conservation when the magnetic field is no longer negligible. Our proposed schemes conserve the total particle number and the total energy at the same time, therefore can obtain accurate and physically relevant numerical solutions. The main components of our methods include second order and above, explicit or implicit energy-conserving temporal discretizations, and DG methods for Vlasov and Maxwell's equations with carefully chosen numerical fluxes. Benchmark numerical tests such as the streaming Weibel instability are provided to validate the accuracy and conservation of the schemes.

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SUMMARYThe discontinuous Galerkin (DG) transport scheme is becoming increasingly popular in the atmospheric modeling due to its distinguished features, such as high-order accuracy and high-parallel efficiency. Despite the great advantages, DG schemes may produce unphysical oscillations in approximating transport equations with discontinuous solution structures including strong shocks or sharp gradients. Nonlinear limiters need to be applied to suppress the undesirable oscillations and enhance the numerical stability. It is usually very difficult to design limiters to achieve both high-order accuracy and non-oscillatory properties and even more challenging for the cubed-sphere geometry. In this paper, a simple and efficient limiter based on the weighted essentially non-oscillatory (WENO) methodology is incorporated in the DG transport framework on the cubed sphere. The uniform high-order accuracy of the resulting scheme is maintained because of the high-order nature of WENO procedures. Unlike the classic WENO limiter, for which the wide halo region may significantly impede parallel efficiency, the simple limiter requires only the information from the nearest neighboring elements without degrading the inherent high-parallel efficiency of the DG scheme. A bound-preserving filter can be further coupled in the scheme that guarantees the highly desirable positivity-preserving property for the numerical solution. The resulting scheme is high-order accurate, non-oscillatory, and positivity preserving for solving transport equations based on the cubed-sphere geometry. Extensive numerical results for several benchmark spherical transport problems are provided to demonstrate good results, both in accuracy and in non-oscillatory performance.

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