We propose a new family of high-order explicit generalized-$\alpha$ methods for hyperbolic problems with the feature of dissipation control. Our approach delivers $2k,\, \left(k \in \mathbb{N}\right)$ accuracy order in time by solving $k$ matrix systems explicitly and updating the other $2k$ variables at each time-step. The user can control the numerical dissipation in the discrete spectrum's high-frequency regions by adjusting the method's coefficients. We study the method's spectrum behaviour and show that the CFL condition is independent of the accuracy order. The stability region remains invariant while we increase the accuracy order. Next, we exploit efficient preconditioners for the isogeometric matrix to minimize the computational cost. These preconditioners use a diagonal-scaled Kronecker product of univariate parametric mass matrices; they have a robust performance with respect to the spline degree and the mesh size, and their decomposition structure implies that their application is faster than a matrix-vector product involving the fully-assembled mass matrix. Our high-order schemes require simple modifications of the available implementations of the generalized-$\alpha$ method. Finally, we present numerical examples demonstrating the methodology's performance regarding single- and multi-patch IGA discretizations.

中文翻译：

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结构动力学等几何分析的显式高阶广义$α$方法

针对双曲问题，我们提出了具有耗散控制功能的一类新的高阶显式广义$ \ alpha $方法。我们的方法通过显式求解$ k $矩阵系统并在每个时间步更新其他$ 2k $变量，及时提供$ 2k，\，\ left（k \ in \ mathbb {N} \ right）$精度顺序。用户可以通过调整方法的系数来控制离散频谱高频区域中的数值耗散。我们研究了该方法的光谱行为，并证明了CFL条件与准确度顺序无关。当我们增加精度等级时，稳定区域保持不变。接下来，我们为等几何矩阵开发有效的预处理器，以最大程度地减少计算成本。这些预处理器使用单变量参数质量矩阵的对角线比例Kronecker乘积。它们在样条曲线度和网格大小方面具有强大的性能，并且它们的分解结构意味着它们的应用比涉及完全组装的质量矩阵的矩阵矢量乘积要快。我们的高阶方案需要对generalized-$ \ alpha $方法的可用实现进行简单的修改。最后，我们提供了数值示例，说明了有关单补丁和多补丁IGA离散化方法的性能。我们的高阶方案需要对generalized-$ \ alpha $方法的可用实现进行简单的修改。最后，我们提供了数值示例，说明了有关单补丁和多补丁IGA离散化方法的性能。我们的高阶方案需要对generalized-$ \ alpha $方法的可用实现进行简单的修改。最后，我们提供了数值示例，说明了有关单补丁和多补丁IGA离散化方法的性能。