IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v466y2024ics0096300323005957.html
   My bibliography  Save this article

On improving the efficiency of ADER methods

Author

Listed:
  • Han Veiga, Maria
  • Micalizzi, Lorenzo
  • Torlo, Davide

Abstract

The (modern) arbitrary derivative (ADER) approach is a popular technique for the numerical solution of differential problems based on iteratively solving an implicit discretization of their weak formulation. In this work, focusing on an ODE context, we investigate several strategies to improve this approach. Our initial emphasis is on the order of accuracy of the method in connection with the polynomial discretization of the weak formulation. We demonstrate that precise choices lead to higher-order convergences in comparison to the existing literature. Then, we put ADER methods into a Deferred Correction (DeC) formalism. This allows to determine the optimal number of iterations, which is equal to the formal order of accuracy of the method, and to introduce efficient p-adaptive modifications. These are defined by matching the order of accuracy achieved and the degree of the polynomial reconstruction at each iteration. We provide analytical and numerical results, including the stability analysis of the new modified methods, the investigation of the computational efficiency, an application to adaptivity and an application to hyperbolic PDEs with a Spectral Difference (SD) space discretization.

Suggested Citation

  • Han Veiga, Maria & Micalizzi, Lorenzo & Torlo, Davide, 2024. "On improving the efficiency of ADER methods," Applied Mathematics and Computation, Elsevier, vol. 466(C).
    • Handle: RePEc:eee:apmaco:v:466:y:2024:i:c:s0096300323005957
    DOI: 10.1016/j.amc.2023.128426
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300323005957
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2023.128426?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Laura Río-Martín & Saray Busto & Michael Dumbser, 2021. "A Massively Parallel Hybrid Finite Volume/Finite Element Scheme for Computational Fluid Dynamics," Mathematics, MDPI, vol. 9(18), pages 1-41, September.
    2. Gaburro, Elena & Öffner, Philipp & Ricchiuto, Mario & Torlo, Davide, 2023. "High order entropy preserving ADER-DG schemes," Applied Mathematics and Computation, Elsevier, vol. 440(C).
    3. Franco, Sebastião Romero & Gaspar, Francisco José & Villela Pinto, Marcio Augusto & Rodrigo, Carmen, 2018. "Multigrid method based on a space-time approach with standard coarsening for parabolic problems," Applied Mathematics and Computation, Elsevier, vol. 317(C), pages 25-34.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ciallella, Mirco & Gaburro, Elena & Lorini, Marco & Ricchiuto, Mario, 2023. "Shifted boundary polynomial corrections for compressible flows: high order on curved domains using linear meshes," Applied Mathematics and Computation, Elsevier, vol. 441(C).
    2. Boscheri, Walter & Tavelli, Maurizio, 2022. "High order semi-implicit schemes for viscous compressible flows in 3D," Applied Mathematics and Computation, Elsevier, vol. 434(C).
    3. Busto, S. & Dumbser, M. & Río-Martín, L., 2023. "An Arbitrary-Lagrangian-Eulerian hybrid finite volume/finite element method on moving unstructured meshes for the Navier-Stokes equations," Applied Mathematics and Computation, Elsevier, vol. 437(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:466:y:2024:i:c:s0096300323005957. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.