We consider the parallel-in-time solution of hyperbolic partial differential equation (PDE) systems in one spatial dimension, both linear and nonlinear. In the nonlinear setting, the discretized equations are solved with a preconditioned residual iteration based on a global linearization. The linear(ized) equation systems are approximately solved parallel-in-time using a block preconditioner applied in the characteristic variables of the underlying linear(ized) hyperbolic PDE. This change of variables is motivated by the observation that inter-variable coupling for characteristic variables is weak relative to intra-variable coupling, at least locally where spatio-temporal variations in the eigenvectors of the associated flux Jacobian are sufficiently small. For an $\ell$-dimensional system of PDEs, applying the preconditioner consists of solving a sequence of $\ell$ scalar linear(ized)-advection-like problems, each being associated with a different characteristic wave-speed in the underlying linear(ized) PDE. We approximately solve these linear advection problems using multigrid reduction-in-time (MGRIT); however, any other suitable parallel-in-time method could be used. Numerical examples are shown for the (linear) acoustics equations in heterogeneous media, and for the (nonlinear) shallow water equations and Euler equations of gas dynamics with shocks and rarefactions.

翻译：本文研究一维空间中线性与非线性双微分方程系统的并行时间求解。在非线性情形下，离散化方程通过基于全局线性化的预处理残差迭代法求解。线性化方程组采用块预处理器在特征变量空间中进行近似并行时间求解，该预处理器基于底层线性化双曲偏微分方程的特征变量构建。变量变换的动机在于：当相关通量雅可比矩阵特征向量的时空变化足够小时，特征变量的变量间耦合强度相对于变量内耦合较弱。对于ℓ维偏微分方程系统，预处理器的应用需要求解一系列ℓ个类标量线性化平流问题，每个问题对应底层线性化偏微分方程中不同的特征波速。我们采用时间多重网格约简法近似求解这些线性平流问题，但亦可使用其他合适的并行时间方法。数值算例展示了异质介质中的线性声学方程，以及包含激波与稀疏波的非线性浅水方程和气体动力学欧拉方程。