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，因受限于严格的CFL稳定性条件，导致时间步长极小。针对此类问题，可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，从而避免使用极小步长。本文提出一种高效框架，用于设计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。高阶格式的非线性特性（源于控制非物理振荡的空间-时间限制策略）使得隐式时间积分面临困难——例如离散系统即便在线性问题上也呈现非线性。我们借鉴Puppo等人（Comm.~Appl.~Math.~\& Comput., 2023）针对标量守恒律提出的方法规避格式非线性：通过计算一阶隐式预测子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。此外，提出一种新型守恒型通量中心后验时间限制方法，利用数值熵指示器检测问题单元。数值测试涵盖经典及人为设计的刚性问题，以欧拉气体动力学系统为验证对象。