Many hyperbolic and kinetic equations contain a non-stiff convection/transport part and a stiff relaxation/collision part (characterized by the relaxation or mean free time $\varepsilon$). To solve this type of problems, implicit-explicit (IMEX) multistep methods have been widely used and their performance is understood well in the non-stiff regime ($\varepsilon=O(1)$) and limiting regime ($\varepsilon\rightarrow 0$). However, in the intermediate regime (say, $\varepsilon=O(\Delta t)$), uniform accuracy has been reported numerically without a complete theoretical justification (except some asymptotic or stability analysis). In this work, we prove the uniform accuracy -- an optimal {\it a priori} error bound -- of a class of IMEX multistep methods, IMEX backward differentiation formulas (IMEX-BDF), for linear hyperbolic systems with stiff relaxation. The proof is based on the energy estimate with a new multiplier technique. For nonlinear hyperbolic and kinetic equations, we numerically verify the same property using a series of examples.

中文翻译：

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关于刚性双曲松弛系统和动力学方程的隐式-显式后向微分公式 (IMEX-BDF) 的统一精度

许多双曲线和动力学方程包含非刚性对流/传输部分和刚性松弛/碰撞部分（以松弛或平均自由时间 $\varepsilon$ 为特征）。为了解决这类问题，隐式-显式 (IMEX) 多步方法已被广泛使用，并且它们在非刚性机制 ($\varepsilon=O(1)$) 和限制机制 ($\varepsilon\右箭头 0$）。然而，在中间状态下（例如，$\varepsilon=O(\Delta t)$），在没有完整理论依据的情况下（除了一些渐近或稳定性分析），在数值上报告了一致的精度。在这项工作中，我们证明了一类 IMEX 多步方法、IMEX 后向微分公式 (IMEX-BDF) 的统一精度——最佳{\it a先验}误差界限，对于具有刚性松弛的线性双曲系统。证明基于使用新乘法器技术的能量估计。对于非线性双曲线和动力学方程，我们使用一系列示例在数值上验证了相同的属性。