We extend our previous work [F. Henr\'iquez and J. S. Hesthaven, arXiv:2403.02847 (2024)] to the linear, second-order wave equation in bounded domains. This technique, referred to as the Laplace Transform Reduced Basis (LT-RB) method, uses two widely known mathematical tools to construct a fast and efficient method for the solution of linear, time-dependent problems: The Laplace transform and the Reduced Basis method, hence the name. The application of the Laplace transform yields a time-independent problem parametrically depending on the Laplace variable. Following the two-phase paradigm of the RB method, firstly in an offline stage we sample the Laplace parameter, compute the full-order or high-fidelity solution, and then resort to a Proper Orthogonal Decomposition (POD) to extract a basis of reduced dimension. Then, in an online phase, we project the time-dependent problem onto this basis and proceed to solve the evolution problem using any suitable time-stepping method. We prove exponential convergence of the reduced solution computed by the LT-RB method toward the high-fidelity one as the dimension of the reduced space increases. Finally, we present a set of numerical experiments portraying the performance of the method in terms of accuracy and, in particular, speed-up when compared to the full-order model.

翻译：本文将我们先前的工作 [F. Henríquez 与 J. S. Hesthaven, arXiv:2403.02847 (2024)] 拓展至有界域中的线性二阶波动方程。该技术称为拉普拉斯变换降基（LT-RB）方法，它结合两种广为人知的数学工具——拉普拉斯变换与降基方法，构建了一种用于求解线性时变问题的快速高效算法。应用拉普拉斯变换可得到一个在参数上依赖于拉普拉斯变量的时不变问题。遵循降基方法的两阶段范式：首先在离线阶段，对拉普拉斯参数进行采样，计算全阶或高保真解，随后采用本征正交分解（POD）提取降维基。接着在在线阶段，将时变问题投影至此基上，并采用任意合适的时步进方法求解演化问题。我们证明了随着降维空间维度的增加，LT-RB 方法计算得到的降阶解将以指数速度收敛于高保真解。最后，我们通过一组数值实验展示了该方法在精度方面的表现，特别是相较于全阶模型所实现的加速效果。