Fast Numerical Approximation of Linear, Second-Order Hyperbolic Problems Using Model Order Reduction and the Laplace Transform
<|reference_start|>Fast Numerical Approximation of Linear, Second-Order Hyperbolic Problems Using Model Order Reduction and the Laplace Transform: 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.<|reference_end|>
@article{henriquez2024fast,
    title={Fast Numerical Approximation of Linear, Second-Order Hyperbolic Problems
  Using Model Order Reduction and the Laplace Transform},
    author={Fernando Henriquez and Jan S. Hesthaven},
    journal={arXiv preprint arXiv:2405.19896},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.19896},
    primaryClass={math.NA cs.NA}
}