We study the dynamics of a parabolic and a hyperbolic equation coupled on a common interface. We develop time-stepping schemes that can use different time-step sizes for each of the subproblems. The problem is formulated in a strongly coupled (monolithic) space-time framework. Coupling two different step sizes monolithically gives rise to large algebraic systems of equations. There, multiple states of the subproblems must be solved at once. For efficiently solving these algebraic systems, we inherit ideas from the partitioned regime. Therefore we present two decoupling methods, namely a partitioned relaxation scheme and a shooting method. Furthermore, we develop an a posteriori error estimator serving as a mean for an adaptive time-stepping procedure. The goal is to optimally balance the time-step sizes of the two subproblems. The error estimator is based on the dual weighted residual method and relies on the space-time Galerkin formulation of the coupled problem. As an example, we take a linear set-up with the heat equation coupled to the wave equation. We formulate the problem in a monolithic manner using the space-time framework. In numerical test cases, we demonstrate the efficiency of the solution process and we also validate the accuracy of the a posteriori error estimator and its use for controlling the time-step sizes.

我们研究在公共接口上耦合的抛物线方程和双曲线方程的动力学。我们开发了可以针对每个子问题使用不同时间步长的时间步长方案。该问题是在强耦合（单片）时空框架中提出的。整体耦合两个不同的步长会产生大型的代数方程组。在那里，子问题的多个状态必须立即解决。为了有效地解决这些代数系统，我们从分区体制中继承了思想。因此，我们提出了两种解耦方法，即分区松弛方案和射击方法。此外，我们开发了后验误差估计器，作为自适应时间步长过程的均值。目标是最佳地平衡两个子问题的时间步长。误差估计器基于对偶加权残差法，并依赖耦合问题的时空Galerkin公式。例如，我们采用热方程与波动方程耦合的线性设置。我们使用时空框架以整体方式提出问题。在数值测试案例中，我们证明了求解过程的效率，我们还验证了后验误差估计器的准确性及其在控制时间步长方面的用途。