Many interesting physical problems described by systems of hyperbolic conservation laws are stiff, and thus impose a very small time-step because of the restrictive CFL stability condition. In this case, one can exploit the superior stability properties of implicit time integration which allows to choose the time-step only from accuracy requirements, and thus avoid the use of small time-steps. We discuss an efficient framework to devise high order implicit schemes for stiff hyperbolic systems without tailoring it to a specific problem. The nonlinearity of high order schemes, due to space- and time-limiting procedures which control nonphysical oscillations, makes the implicit time integration difficult, e.g.~because the discrete system is nonlinear also on linear problems. This nonlinearity of the scheme is circumvented as proposed in (Puppo et al., Comm.~Appl.~Math.~\& Comput., 2023) for scalar conservation laws, where a first order implicit predictor is computed to freeze the nonlinear coefficients of the essentially non-oscillatory space reconstruction, and also to assist limiting in time. In addition, we propose a novel conservative flux-centered a-posteriori time-limiting procedure using numerical entropy indicators to detect troubled cells. The numerical tests involve classical and artificially devised stiff problems using the Euler's system of gas-dynamics.

翻译：许多由双曲守恒律系统描述的物理问题具有刚性特征，由于CFL稳定性条件的严格限制，往往要求极小时步长。此时，可利用隐式时间积分方法优越的稳定性特性，仅根据精度要求选择时步长，从而避免使用过小的时步。本文提出一种高效框架，用于构建适用于刚性双曲系统的高阶隐式格式，且无需针对特定问题进行定制化设计。高阶格式的非线性特性（源于控制非物理振荡的空间与时间限制器）使得隐式时间积分变得困难，例如在线性问题中离散系统仍呈现非线性。借鉴（Puppo等人，Comm. Appl. Math. & Comput., 2023）针对标量守恒律提出的方法，我们通过计算一阶隐式预测子来冻结本质无振荡空间重构的非线性系数，并辅助时间限制过程，从而规避格式的非线性问题。此外，我们提出一种新型守恒型通量中心化后验时间限制方法，利用数值熵指标识别问题单元。数值实验通过欧拉气体动力学系统，对经典及人工设计的刚性问题进行了验证。