In this work we present multi-derivative implicit-explicit (IMEX) Runge--Kutta schemes. We derive their order conditions up to third order, and show that such methods can preserve positivity (and more generally strong stability) with a time-step restriction independent of the stiff term, under mild assumptions on the operators. We present sufficient conditions under which such methods are positivity preserving and asymptotic preserving (AP) when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar-Gross-Krook (BGK) kinetic equation. Previous efforts to devise such methods have used an IMEX Runge--Kutta framework plus a second derivative final correction. In this work, we extend this approach to include derivative information at any stage of the computation. This multi-derivative IMEX approach allowed us to find a second order AP and positivity preserving method that improves upon previous work in terms of the allowable time-step size. Furthermore, this approach produces a third order method that is AP and positivity preserving for a time-step independent of the stiff term, a feature not possessed by any of the existing third-order IMEX schemes. We present numerical results to support the theoretical results, on a variety of problems.

中文翻译：

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高阶正性保和渐近保多导数方法

在这项工作中，我们提出了多导数隐式显式（IMEX）Runge-Kutta方案。我们推导出它们的阶数条件直到三阶，并表明在对运算符的温和假设下，此类方法可以在不受步阶约束的情况下以时间步长约束保持正性（并且通常具有更强的稳定性）。当将这些方法应用于一系列问题时，包括双曲弛豫系统，Broadwell模型和Bhatnagar-Gross-Krook（BGK）动力学方程，我们提供了充分的条件，在这些条件下，这些方法可以保持正性和渐近性（AP）。以前设计此类方法的努力已使用IMEX Runge-Kutta框架以及第二个导数最终校正。在这项工作中，我们将这种方法扩展为在计算的任何阶段都包括派生信息。这种多导数IMEX方法使我们能够找到二阶AP和阳性保留方法，该方法在允许的时间步长方面比以前的工作有所改进。此外，这种方法产生了一种三阶方法，该方法是为AP和阳性保留一个与刚性项无关的时间步长，而这是任何现有的三阶IMEX方案都不具备的特征。我们针对各种问题提出了数值结果以支持理论结果。