{
  "id": "2405.19896",
  "version": "v1",
  "published": "2024-05-30T09:53:10.000Z",
  "updated": "2024-05-30T09:53:10.000Z",
  "title": "Fast Numerical Approximation of Linear, Second-Order Hyperbolic Problems Using Model Order Reduction and the Laplace Transform",
  "authors": [
    "Fernando Henriquez",
    "Jan S. Hesthaven"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2403.02847",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "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, 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-30T09:53:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "laplace transform",
      "model order reduction",
      "second-order hyperbolic problems",
      "fast numerical approximation",
      "time-dependent problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}