论文标题
浅水方程的自适应稳定时空FE方法
An Adaptive Stable Space-Time FE Method for the Shallow Water Equations
论文作者
论文摘要
我们通过考虑使用无条件稳定的Fe方法来确定空间和时间的离散化，来考虑浅水方程（SWE）的有限元（Fe）近似。特别是，我们考虑了一种自动变异稳定的Fe（AVS-FE）方法，一种不连续的Petrov-Galerkin（DPG）方法。 DPG方法的理念使我们能够打破测试空间并实现无条件稳定的Fe近似值，并在解决方程组的鞍点系统时准确地进行后验误差估计器。由此产生的错误指标使我们能够采用网格自适应策略并执行时空网格的精炼，即当地时间步进。 我们得出了AVS-FE方法和线性化SWE的先验误差估计，并执行数值验证以确认相应的渐近收敛行为。为了保持计算成本较低，我们考虑了一种替代时空方法，其中将时空域分配到有限尺寸的时空切片中。因此，我们可以对每个单独的切片执行适应性，以根据需要的特定应用程序预设错误公差。比较两种替代方案的数值验证表明，长期使用时空切片对于模拟而言是优越的，而解决方案在短时间内是无法区分的。多次数值验证显示了AVS-FE方法的自适应网格改进能力，以及该方法在SWE的常用基准中的应用。
We consider the finite element (FE) approximation of the shallow water equations (SWE) by considering discretizations in which both space and time are established using an unconditionally stable FE method. Particularly, we consider the automatic variationally stable FE (AVS-FE) method, a type of discontinuous Petrov-Galerkin (DPG) method. The philosophy of the DPG method allows us to break the test space and achieve unconditionally stable FE approximations as well as accurate a posteriori error estimators upon solution of a saddle point system of equations. The resulting error indicators allow us to employ mesh adaptive strategies and perform space-time mesh refinements, i.e., local time stepping. We derive a priori error estimates for the AVS-FE method and linearized SWE and perform numerical verifications to confirm corresponding asymptotic convergence behavior. In an effort to keep the computational cost low, we consider an alternative space-time approach in which the space-time domain is partitioned into finite sized space-time slices. Hence, we can perform adaptivity on each individual slice to preset error tolerances as needed for a particular application. Numerical verifications comparing the two alternatives indicate the space-time slices are superior for simulations over long times, whereas the solutions are indistinguishable for short times. Multiple numerical verifications show the adaptive mesh refinement capabilities of the AVS-FE method, as well the application of the method to commonly applied benchmarks for the SWE.
