Fast Numerical Approximation of Linear, Second-Order Hyperbolic Problems Using Model Order Reduction and the Laplace Transform

TL;DR

This paper introduces an efficient LT-MOR method combining Laplace transform and model reduction to solve second-order wave equations, achieving exponential convergence and significant speed-up over full models.

Contribution

It extends previous work to second-order wave equations, integrating Laplace transform with reduced basis methods for fast, accurate solutions.

Findings

Exponential convergence of the reduced solution to the high-fidelity solution.

Numerical experiments show improved speed and accuracy over full-order models.

Abstract

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 uses two widely known mathematical tools to construct a fast and efficient method for the solution of linear, time-dependent problems: the Laplace transform (LT) and the model-order reduction (MOR) techniques, hence the name LT-MOR method. The application of the Laplace transform yields a time-independent problem parametrically depending on the Laplace variable. Following the two-phase paradigm of the reduced basis method, first in an offline stage we sample the Laplace parameter, compute the 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 proposed method toward the high-fidelity one as the dimension of the reduced space increases. Finally, we present numerical experiments illustrating the performance of the method in terms of accuracy and, in particular, speed-up when compared to the full-order model.