We have developed a deterministic conservative solver for the inhomogeneous Fokker-Planck-Landau equation coupled with the Poisson equation, which is a classical mean-field primary model for collisional plasmas. Two subproblems, i.e. the Vlasov-Poisson problem and homogeneous Landau problem, are obtained through time-splitting methods, and treated separately by the Runge-Kutta Discontinuous Galerkin method and a conservative spectral method, respectively. To ensure conservation when projecting between the two different computing grids, a special conservation routine is designed to link the solutions of these two subproblems. This conservation routine accurately enforces conservation of moments in Fourier space. The entire numerical scheme is implemented with parallelization with hybrid MPI and OpenMP. Numerical experiments are provided to study linear and nonlinear Landau Damping problems and two-stream flow problem as well.

A Runge-Kutta discontinuous Galerkin solver for 2D Boltzmann model equations: Verification and analysis of computational performance AIP Conf. Proc. 1501, 381 (2012) Abstract. In the present work, we propose a deterministic numerical solver for the homogeneous Boltzmann equation based on Discontinuous Galerkin (DG) methods. The weak form of the collision operator is approximated by a quadratic form in linear algebra setting. We employ the property of "shifting symmetry" in the weight matrix to reduce the computing complexity from theoretical O(N 3 ) down to O(N 2 ), with N the total number of freedom for d-dimensional velocity space. In addition, the sparsity is also explored to further reduce the storage complexity. To apply lower order polynomials and resolve loss of conserved quantities, we invoke the conservation routine at every time step to enforce the conservation of desired moments (mass, momentum and/or energy), with only linear complexity. Due to the locality of the DG schemes, the whole computing process is well parallelized using hybrid OpenMP and MPI. The current work only considers integrable angular cross-sections under elastic and/or inelastic interaction laws. Numerical results on 2-D and 3-D problems are shown.

In the present work, we propose a deterministic numerical solver for the space homogeneous Boltzmann equation based on discontinuous Galerkin (DG) methods. Such an application has been rarely studied. The main goal of this manuscript is to generate a conservative solver for the collisional operator. As the key part, the weak form of the collision operator is approximated within subspaces of piecewise polynomials. In order to save the computational cost and to resolve loss of conservation laws due to numerical approximations, we propose the following combined procedures. First, the collision operator is projected onto a subspace of basis polynomials up to first order. Then, at every time step, a conservation routine is employed to enforce the preservation of desired moments (mass, momentum, and/or energy) by solving a constrained minimization problem with only linear complexity. The asymptotic error analysis shows the validity and guarantees the accuracy of these two procedures. The approximated collision integral is finally written as a quadratic form in a linear algebra setting. The theoretical number of operations for evaluating the complete set of the approximated collision matrix would be of order O(N 3) with N being the total number of freedom for d-dimensional velocity space. However, we have found and applied a "shifting symmetries" property in the collision weight matrix that consists of finding a minimal set of basis matrices that can exactly reconstruct the complete family of such a matrix. This procedure reduces the computation and storage of the collision matrix down to O(N 2). In addition, the matrix is highly sparse, yielding actual complexity O(N 2−1/d), with d being number of dimensions. Due to the locality of the DG schemes, the whole computing process is well performed with parallelization using hybrid OpenMP and message passing interface. The current work only considers the homogeneous Boltzmann equation with integrable angular cross sections under elastic and/or inelastic interaction laws. No transport is included in this manuscript. We only focus on the approximation of time dynamics for conservative binary collisions. The numerical results on two-dimensional and three-dimensional problems are provided.

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