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，受限于苛刻的CFL稳定性条件而必须采用极小时间步长。在此背景下，可借助隐式时间积分卓越的稳定性特性，仅依据精度要求选择时间步长，从而避免使用微小步长。本文提出一个高效框架，无需针对特定问题进行定制，即可为刚性双曲系统设计高阶隐式格式。高阶格式因控制非物理振荡的空间/时间限制过程而产生的非线性特性（例如，离散系统即便对于线性问题仍呈现非线性），使得隐式时间积分面临挑战。参照Puppo等学者(Comm. Appl. Math. & Comput., 2023)针对标量守恒律提出的策略，本文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，并辅助时间方向的限制过程，从而规避格式的非线性问题。此外，我们提出一种新颖的守恒型通量中心化后验时间限制方法，利用数值熵指示器检测不稳定单元。数值测试涵盖经典及人工设计的刚性问题，采用气体动力学欧拉系统进行验证。