基于 telescopic projective integration 的具有多重弛豫时间的动能方程求解
Telescopic projective integration for kinetic equations with multiple relaxation times
AI总结：
本文提出一种高阶显式数值方法，用于模拟具有多重弛豫时间的动能方程，通过构造投影层次结构提高效率，适用于具有多个特征值簇的刚性问题。
摘要：
我们研究了一种通用的、高阶的、完全显式数值方法，用于模拟具有BGK型碰撞模型且具有多重弛豫时间的动能方程。在这种情况下，问题具有刚性，其谱由多个分离的特征值簇组成。投影积分方法是一种显式积分方案，首先进行几次小的（内层）步骤，使用简单的显式方法，然后将解向前时间 extrapolate 过一个大的（外层）时间步长。这些方案在仅存在两个特征值簇时非常高效。Telescopic projective integration methods 通过构造投影层次结构扩展了投影积分方法的思想。本文展示了如何利用 telescopic projective integration methods 有效积分具有多重弛豫时间的动能方程。我们证明所需投影层次的数量取决于簇的数量，而簇的数量又取决于BGK源项的刚性。外层时间步长的大小仅取决于模型中存在的最慢时间尺度，与问题的刚性无关。我们讨论了稳定性，并通过一维和二维空间中的模拟示例说明了该方法。
英文摘要：
We study a general, high-order, fully explicit numerical method for simulating kinetic equations with a BGK-type collision model with multiple relaxation times. In that case, the problem is stiff and its spectrum consists of multiple separated eigenvalue clusters. Projective integration methods are explicit integration schemes that first take a few small (inner) steps with a simple, explicit method, after which the solution is extrapolated forward in time over a large (outer) time step. These are very efficient schemes, provided there are only two clusters of eigenvalues. Telescopic projective integration methods generalize the idea of projective integration methods by constructing a hierarchy of projective levels. Here, we show how telescopic projective integration methods can be used to efficiently integrate kinetic equations with multiple relaxation times. We show that the required number of projective levels depends on the number of clusters, which in turn depends on the stiffness of the BGK source term. The size of the outer level time step only depends on the slowest time scale present in the model and is independent of the stiffness of the problem. We discuss stability and illustrate the approach with simulations in one and two spatial dimensions.
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