Fast Numerical Approximation of Linear, Second-Order Hyperbolic Problems Using Model Order Reduction and the Laplace Transform
This work provides an efficient computational method for time-dependent wave problems, but it is incremental as it builds directly on prior research.
The authors tackled the problem of solving linear, second-order wave equations by extending a previous method that combines the Laplace transform and model-order reduction, achieving exponential convergence and significant speed-up in numerical experiments.
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.