含混合导数的二维对流扩散方程的保正格式
Positivity-preserving scheme for two-dimensional advection-diffusion equations including mixed derivatives
摘要：
在这项工作中，我们提出了一个保正格式求解二维对流扩散方程包括混合导数项，以提高精度的低阶方法。在没有混合导数的情况下，这些方程的解已经得到了详细的研究，而混合导数项的保正解却很少受到关注。一个二维扩散方程，其中的解析解是已知的，数值求解显示该计划的适用性。在局部热平衡的假设下，将其进一步应用于二维柱坐标系中的Fokker-Planck碰撞算符。对于一个热平衡问题，它表明，该计划的粒子数和能量守恒，而保持的积极性是确保和稳态解是麦克斯韦分布。
英文摘要：
In this work, we propose a positivity-preserving scheme for solving two-dimensional advection-diffusion equations including mixed derivative terms, in order to improve the accuracy of lower-order methods. The solution to these equations, in the absence of mixed derivatives, has been studied in detail, while positivity-preserving solutions to mixed derivative terms have received much less attention. A two-dimensional diffusion equation, for which the analytical solution is known, is solved numerically to show the applicability of the scheme. It is further applied to the Fokker-Planck collision operator in two-dimensional cylindrical coordinates under the assumption of local thermal equilibrium. For a thermal equilibration problem, it is shown that the scheme conserves particle number and energy, while the preservation of positivity is ensured and the steady-state solution is the Maxwellian distribution.
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