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In this paper, we develop sparse grid discontinuous Galerkin (DG) schemes for the Vlasov-Maxwell (VM) equations. The VM system is a fundamental kinetic model in plasma physics, and its numerical computations are quite demanding, due to its intrinsic high-dimensionality and the need to retain many properties of the physical solutions. To break the curse of dimensionality, we consider the sparse grid DG methods that were recently developed in [20,21] for transport equations. Such methods are based on multiwavelets on tensorized nested grids and can significantly reduce the numbers of degrees of freedom. We formulate two versions of the schemes: sparse grid DG and adaptive sparse grid DG methods for the VM system. Their key properties and implementation details are discussed. Accuracy and robustness are demonstrated by numerical tests, with emphasis on comparison of the performance of the two methods, as well as with their full grid counterparts.

This paper constructs adaptive sparse grid collocation method onto arbitrary order piecewise polynomial space. The sparse grid method is a popular technique for high dimensional problems, and the associated collocation method has been well studied in the literature. The contribution of this work is the introduction of a systematic framework for collocation onto high-order piecewise polynomial space that is allowed to be discontinuous. We consider both Lagrange and Hermite interpolation methods on nested collocation points. Our construction includes a wide range of function space, including those used in sparse grid continuous finite element method. Error estimates are provided, and the numerical results in function interpolation, integration and some benchmark problems in uncertainty quantification are used to compare different collocation schemes.

In this paper, a class of high-order central finite volume schemes is proposed for solving one-and two-dimensional hyperbolic conservation laws. Formulated on staggered meshes, the methods involve Hermite WENO (HWENO) spatial reconstructions, and Lax-Wendroff type discretizations or the natural continuous extension of Runge-Kutta methods in time. Different from the central Hermite WENO methods we developed previously in [J. Comput. Phys. 281:148-176, 2015], the spatial reconstructions, a core ingredient of the methods, are based on the zeroth-order and the first-order moments of the solution, and are implemented through a dimension-by-dimension strategy when the spatial dimension is higher than one. This leads to much simpler implementation of the methods in higher dimension and better cost efficiency. Meanwhile, the proposed methods have the attractive features of the general central Hermite WENO methods such as being compact in reconstruction and requiring neither flux splitting nor numerical fluxes, while being accurate and essentially non-oscillatory. A collection of oneand two-dimensional numerical examples is presented to demonstrate high resolution and robustness of the methods in capturing smooth and non-smooth solutions.