The fractional Klein-Kramers equation describes the process of subdiffusion in the presence of an external force field in phase space and incorporates a fractional operator in time of order α, 0 < α < 1. We present a family of finite volume schemes for the fractional Klein-Kramers equation, that includes first or second-order schemes in phase space, and implicit or explicit schemes in time with an order of accuracy that can change between α and $2-\alpha$. It is proved, for the open domain, that the schemes satisfy the positivity preserving property. The positivity preserving property for the explicit schemes imposes a strong condition in the relation between time step, space step and phase step, for small values of α, highlighting the advantage of using implicit schemes in these cases. For a bounded domain in space, two types of boundary conditions are considered, absorbing boundary conditions and reflecting boundary conditions. The inclusion of boundary conditions leads to some technical complications that require changes in the schemes near the boundary. The positivity preserving property holds for the new formulation and the overall accuracy is ensured with the use of non-uniform meshes. Numerical tests are presented in the end to show the convergence of the finite volume schemes.

分数阶Klein-Kramers方程描述了在相空间中存在外力场的情况下的子扩散过程，并以α，0 <  α  <1的时间合并了分数算子。我们给出了分数阶的有限体积方案族Klein-Kramers方程，包括相空间中的一阶或二阶方案，以及时间上的隐式或显式方案，其精确度顺序可以在α和α之间变化。$2-\alpha$。对于开放域，证明了该方案满足正保持性。对于较小的α值，显式方案的正保持性在时间步长，空间步长和相位步长之间的关系中强加了条件。，强调了在这些情况下使用隐式方案的优势。对于空间中的有界域，考虑两种类型的边界条件：吸收边界条件和反映边界条件。包含边界条件会导致一些技术复杂性，需要在边界附近更改方案。保持阳性的特性适用于新配方，并且使用非均匀网格可确保整体精度。最后进行了数值测试，以显示有限体积方案的收敛性。