SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page A2402-A2435, January 2020.
We consider the development of high-order space and time numerical methods based on implicit-explicit multistep time integrators for hyperbolic systems with relaxation. More specifically, we consider hyperbolic balance laws in which the convection and the source term may have very different time and space scales. As a consequence, the nature of the asymptotic limit changes completely, passing from a hyperbolic to a parabolic system. From the computational point of view, standard numerical methods designed for the fluid-dynamic scaling of hyperbolic systems with relaxation present several drawbacks and typically lose efficiency in describing the parabolic limit regime. In this work, in the context of implicit-explicit linear multistep methods we construct high-order space-time discretizations which are able to handle all the different scales and to capture the correct asymptotic behavior, independently from its nature, without time step restrictions imposed by the fast scales. Several numerical examples confirm the theoretical analysis.

中文翻译：

---

多尺度松弛双曲系统的隐式-显式多步方法

SIAM科学计算杂志，第42卷，第4期，第A2402-A2435页，2020年1月。
考虑带松弛的双曲系统基于隐式-显式多步时间积分器的高阶时空数值方法的发展。更具体地说，我们考虑双曲平衡定律，其中对流和源项可能具有非常不同的时空尺度。结果，渐近极限的性质完全改变，从双曲线系统变为抛物线系统。从计算的角度来看，为具有松弛的双曲线系统的流体动力学缩放而设计的标准数值方法存在一些缺点，并且在描述抛物线极限状态时通常会失去效率。在这项工作中 在隐式-显式线性多步方法的背景下，我们构造了高阶时空离散化方法，该方法能够处理所有不同的音阶并捕获正确的渐近行为，而不受其性质的影响，而不受快速音阶施加的时间步长限制。几个数值例子证实了理论分析。