论文标题
单层多晶体方案的自适应时步控制，耦合热和波方程
Adaptive time-step control for a monolithic multirate scheme coupling the heat and wave equation
论文作者
论文摘要
我们考虑抛物线方程和双曲线方程的动力学结合在公共接口上，并开发时间步长方案，这些方案可以为每个子问题使用不同的时步尺寸。该问题是在强耦合（整体上）时空框架中提出的。耦合两个不同的步骤尺寸单一构成产生大型方程式系统，其中必须立即解决多个子问题的多个状态。为了有效地求解这些代数系统，我们从分区政权中继承了思想，并提出了两种脱钩方法，即一种分区的放松方案和一种拍摄方法。此外，我们开发了一个后验误差估计量，该估计量是自适应时间稳定过程的平均值。目标是最佳平衡两个子问题的时间步长。误差估计器基于双重加权残差方法，依赖于耦合问题的时空盖尔金公式。例如，我们采用线性设置，并与热方程式耦合到波方程。我们使用时空框架以整体式的方式提出问题。在数值测试用例中，我们证明了解决方案过程的效率，还验证了A后验误差估计器的准确性及其用于控制时间步长大小的使用。
We consider the dynamics of a parabolic and a hyperbolic equation coupled on a common interface and 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 where multiple states of the subproblems must be solved at once. For efficiently solving these algebraic systems, we inherit ideas from the partitioned regime and 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.
