We propose a central scheme framework for the approximation of hyperbolic systems of conservation laws in any space dimension. The new central schemes are defined so that any convex invariant set containing the initial data can be an invariant domain for the numerical method. The underlying first-order central scheme is the analog of the guaranteed maximum speed method of [J.‑L. Guermond and B. Popov, SIAM J. Anal., 54(4):2466–2489, 2016] adjusted to the finite volume framework. There are three novelties in this work. The first one is that any classical second-order central scheme can be modified to satisfy an invariant domain property of the first-order scheme via a process which we call convex limiting. This is done by using convex flux limiting along the lines of [J. ‑L. Guermond, B. Popov and I. Tomas, Comput. Meth. Appl. Mech. Engrg., 347:143–175, 2019]. The second novelty is the design of a new second-order method based on slope limiting only. The new local slope reconstruction technique is based on convex limiting so that the cell interface values are corrected to fit into a local invariant domain of the hyperbolic system. This new type of slope limiting depends on the hyperbolic system and to the best of our knowledge is the only one to guarantee local invariant domain preservation. Both schemes, flux and slope limiting based, are shown to be secondorder accurate for smooth solutions in the $L^\infty$-norm and robust in all test cases. The third novelty is a new second-order method based on the MAPR limiter from [I. Christov and B. Popov, J. Comput. Phys., 227(11):5736–5757, 2008] and adaptive slope limiting in the spirit of [A. Kurganov, G. Petrova and B. Popov, SIAM J. Sci. Comput., 29(6):2381–2401, 2007] but based on an entropy commutator. This new method can be used as an underlying high-order method and combined with convex flux limiting to guarantee a local invariant domain property. The time stepping of all methods is done by using strong stability preserving Runge–Kutta methods and the invariant domain property is proved under a standard CFL condition.

中文翻译：

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非线性双曲系统的不变域保集中方案

我们提出了一个中心方案框架，用于近似任何空间维度上的双曲守恒律系统。定义了新的中心方案，以便任何包含初始数据的凸不变集都可以成为数值方法的不变域。基本的一阶中央方案类似于[J.‑L.]的保证最大速度方法。Guermond和B. Popov，SIAM J. Anal。，54（4）：2466-2489，2016]调整为有限体积框架。这项工作有三个新颖之处。第一个是任何经典的二阶中心方案都可以通过我们称为凸极限的过程进行修改，以满足一阶方案的不变域性质。。这是通过使用沿[J.]的线的凸通量限制来完成的。‑L。Guermond，B。Popov和I.Tomas，计算机。方法 应用 机甲。gr。，347：143–175，2019]。第二个新颖之处是仅基于斜率限制的新的二阶方法的设计。新的局部斜率重构技术基于凸限制，因此可以对单元界面值进行校正以适合双曲系统的局部不变域。这种新型的边坡限制取决于双曲线系统，据我们所知，它是唯一可以保证局部不变域保留的方法。对于$ L ^ \ infty $范数的平滑解，这两种方案都基于通量和斜率限制，被证明是二阶精度的，并且在所有测试案例中都非常可靠。第三种新颖性是基于[I.]的MAPR限制器的新的二阶方法。Christov和B. Popov，J。Comput。物理，227（11）：5736-5757，2008]和[A. Kurganov，G。Petrova和B. Popov，SIAM J. Sci。计算 ，29（6）：2381-2401，2007]，但基于熵换向器。该新方法可以用作基础的高阶方法，并与凸通量限制相结合以保证局部不变域的性质。所有方法的时间步长都是通过使用保持强稳定性的Runge-Kutta方法完成的，并且在标准CFL条件下证明了不变域的性质。