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 achieve limiting in time. In addition, we propose a novel conservative flux-based 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硬性，由于严格的CFL稳定性条件限制，时间步长必须非常小。在此情况下，可以利用隐式时间积分优越的稳定性特性，仅根据精度要求选择时间步长，从而避免使用过小的时间步长。我们提出一种无需针对特定问题定制的通用高效框架，用于构建刚硬双曲系统的高阶隐式格式。高阶格式的非线性（源于控制非物理振荡的空间和时间限制过程）使得隐式时间积分变得困难，例如即使对于线性问题，离散系统也会呈现非线性。针对标量守恒律，我们采用Puppo等人（Comm.~Appl.~Math.~\& Comput., 2023）提出的方法规避格式的非线性：通过计算一阶隐式预测子来冻结本质无振荡空间重构中的非线性系数，同时实现时间限制。此外，我们提出一种新颖的基于守恒通量的后验时间限制方法，利用数值熵指标检测问题单元。数值测试涉及使用气体动力学欧拉系统的经典问题和人工构造的刚硬问题。