论文标题
多层潮汐模型的加权标准预处理
Weighted-norm preconditioners for a multi-layer tide model
论文作者
论文摘要
我们得出线性化的旋转浅水系统建模潮汐，可以通过混合有限元素离散。与以前的模型不同，该模型允许按密度分层的多层。像单层案例〜\ cite {kirby2021-trecentioning}加权标记预处理给出了一种（几乎）参数式的方法，用于在每个时间步骤求解所得的线性系统，但是模型中的所有图层之间的全部耦合对效率构成了重大挑战。忽略层间耦合会给预处理，该预处理随着层数增加而迅速降解。通过仔细分析伴侣层的矩阵，我们得出了一种可靠的方法，该方法需要求解仅涉及相邻层之间耦合的重新制定系统。使用Firedrake获得的数值结果证实了这一理论。
We derive a linearized rotating shallow water system modeling tides, which can be discretized by mixed finite elements. Unlike previous models, this model allows for multiple layers stratified by density. Like the single-layer case~\cite{kirby2021preconditioning} a weighted-norm preconditioner gives a (nearly) parameter-robust method for solving the resulting linear system at each time step, but the all-to-all coupling between the layers in the model poses a significant challenge to efficiency. Neglecting the inter-layer coupling gives a preconditioner that degrades rapidly as the number of layers increases. By a careful analysis of the matrix that couples the layers, we derive a robust method that requires solving a reformulated system that only involves coupling between adjacent layers. Numerical results obtained using Firedrake confirm the theory.
