We present a novel discontinuous Galerkin algorithm for the solution of a class of Fokker–Planck collision operators. These operators arise in many fields of physics, and our particular application is for kinetic plasma simulations. In particular, we focus on an operator often known as the ‘Lenard–Bernstein’ or ‘Dougherty’ operator. Several novel algorithmic innovations, based on the concept of weak equality, are reported. These weak equalities are used to define weak operators that compute primitive moments, and are also used to determine a reconstruction procedure that allows an efficient and accurate discretization of the diffusion term. We show that when two integrations by parts are used to construct the discrete weak form, and finite velocity-space extents are accounted for, a scheme that conserves density, momentum and energy exactly is obtained. One novel feature is that the requirements of momentum and energy conservation lead to unique formulas to compute primitive moments. Careful definition of discretized moments also ensure that energy is conserved in the piecewise linear case, even though the kinetic-energy term, $v^{2}$ is not included in the basis set used in the discretization. A series of benchmark problems is presented and shows that the scheme conserves momentum and energy to machine precision. Empirical evidence also indicates that entropy is a non-decreasing function. The collision terms are combined with the Vlasov equation to study collisional Landau damping and plasma heating via magnetic pumping.

中文翻译：

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非线性 Dougherty–Fokker–Planck 碰撞算子的保守不连续 Galerkin 格式

我们提出了一种新颖的不连续 Galerkin 算法，用于求解一类 Fokker-Planck 碰撞算子。这些算子出现在许多物理领域，我们的特殊应用是动力学等离子体模拟。特别是，我们关注一个通常被称为“Lenard-Bernstein”或“Dougherty”算子的算子。报告了一些基于弱平等概念的新算法创新。这些弱等式用于定义计算原始矩的弱算子，也用于确定允许对扩散项进行有效和准确离散化的重建过程。我们表明，当使用两个部分积分来构造离散弱形式，并考虑有限速度空间范围时，一种守恒密度的方案，正好得到动量和能量。一个新颖的特点是动量和能量守恒的要求导致了计算原始矩的独特公式。离散矩的仔细定义也确保能量在分段线性情况下是守恒的，即使动能项， $v^{2}$ 不包括在离散化中使用的基组中。提出了一系列基准问题，并表明该方案对机器精度保持动量和能量。经验证据还表明熵是一个非递减函数。碰撞项与 Vlasov 方程相结合，通过磁泵研究碰撞朗道阻尼和等离子体加热。