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Split generalized-α method: A linear-cost solver for multi-dimensional second-order hyperbolic systems
Computer Methods in Applied Mechanics and Engineering ( IF 7.2 ) Pub Date : 2021-01-25 , DOI: 10.1016/j.cma.2020.113656
Pouria Behnoudfar , Quanling Deng , Victor M. Calo

We propose a variational splitting technique for the generalized-α method to solve hyperbolic partial differential equations. We use tensor-product meshes to develop the splitting method, which has a computational cost that grows linearly with the total number of degrees of freedom for multi-dimensional problems. We use the generalized-α method for the temporal discretization while standard C0 finite elements as well as isogeometric elements for spatial discretization. . We perform spectral analysis on the amplification matrix to establish the unconditional stability of the method and to show how splitting affects the overall behavior of the time marching scheme for finite time step sizes, including the standard stability analysis limits 0 and as particular cases. We use various examples to demonstrate the performance of the method and the optimal approximation accuracy. In these examples, we compute the L2 and H1 norms of the error to show the optimal convergence of the discrete method in space and second-order accuracy in time. Lastly, we also use these tests to demonstrate the linear cost of the solver as the number of degrees of freedom grows.



中文翻译:

广义分割α 方法:用于多维二阶双曲系统的线性成本求解器

我们提出了一种变分分裂技术,用于α解双曲型偏微分方程的方法。我们使用张量积网格来开发分割方法,该方法的计算成本随着多维问题的自由度总数线性增长。我们使用广义的α 标准时态离散的方法 C0用于空间离散化的有限元和等几何元。。我们对放大矩阵执行频谱分析,以建立该方法的无条件稳定性,并说明分割如何影响有限时间步长的时间行进方案的整体行为,包括标准稳定性分析极限0和作为特殊情况。我们使用各种示例来证明该方法的性能和最佳逼近精度。在这些示例中,我们计算大号2H1个误差的范数表明了离散方法在空间上的最优收敛性和时间的二阶精度。最后,我们还使用这些测试来证明随着自由度数量的增加,求解器的线性成本。

更新日期:2021-01-25
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