On the uniform accuracy of implicit-explicit backward differentiation formulas (IMEX-BDF) for stiff hyperbolic relaxation systems and kinetic equations
On the uniform accuracy of implicit-explicit backward differentiation formulas (IMEX-BDF) for stiff hyperbolic relaxation systems and kinetic equations
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DOI：
10.1090/mcom/3602
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发表时间：
2019-12
期刊：
影响因子：
--
通讯作者：
Jingwei Hu;Ruiwen Shu
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文献类型：
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作者：
Jingwei Hu;Ruiwen Shu
许多双曲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）。为了解决此类问题，隐式-显式 (IMEX) 多步方法已被广泛使用，并且它们在非刚性状态 ($\varepsilon=O(1)$) 和限制状态 ($\varepsilon\rightarrow 0$) 中的性能得到了很好的理解。然而，在中间状态（例如，$\varepsilon=O(\Delta t)$）中，在没有完整的理论依据的情况下以数值方式报告了统一的精度（除了一些渐近或稳定性分析）。在这项工作中，我们证明了对于具有刚性松弛的线性双曲系统，一类 IMEX 多步方法、IMEX 后向微分公式 (IMEX-BDF) 的统一精度——最优{先验}误差界。该证明基于采用新乘法器技术的能量估计。对于非线性双曲方程和动力学方程，我们使用一系列示例在数值上验证了相同的性质。
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.