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Adaptive LOOCV-based kernel methods for solving time-dependent BVPs

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  • Cavoretto, Roberto

Abstract

In this article we propose an adaptive algorithm for the solution of time-dependent boundary value problems (BVPs). To solve numerically these problems, we consider the kernel-based method of lines that allows us to split the spatial and time derivatives, dealing with each separately. This adaptive scheme is based on a leave-one-out cross validation (LOOCV) procedure, which is employed as an error indicator. By this technique, we can first detect the domain areas where the error is estimated to be too large – generally due to steep variations or quick changes in the solution – and then accordingly enhance the numerical solution by applying a two-point refinement strategy. Numerical experiments show the efficacy and performance of our adaptive refinement method.

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  • Cavoretto, Roberto, 2022. "Adaptive LOOCV-based kernel methods for solving time-dependent BVPs," Applied Mathematics and Computation, Elsevier, vol. 429(C).
    • Handle: RePEc:eee:apmaco:v:429:y:2022:i:c:s0096300322003022
    DOI: 10.1016/j.amc.2022.127228
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    References listed on IDEAS

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    1. Cavoretto, Roberto & De Rossi, Alessandra, 2020. "Error indicators and refinement strategies for solving Poisson problems through a RBF partition of unity collocation scheme," Applied Mathematics and Computation, Elsevier, vol. 369(C).
    2. Yao, Guangming & Duo, Jia & Chen, C.S. & Shen, L.H., 2015. "Implicit local radial basis function interpolations based on function values," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 91-102.
    3. Yang, Xiaojia & Ge, Yongbin & Zhang, Lin, 2019. "A class of high-order compact difference schemes for solving the Burgers’ equations," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 394-417.
    4. Karageorghis, Andreas, 2022. "A time–efficient variable shape parameter Kansa–radial basis function method for the solution of nonlinear boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 413(C).
    5. Guo, Yan & Shi, Yu-feng & Li, Yi-min, 2016. "A fifth-order finite volume weighted compact scheme for solving one-dimensional Burgers’ equation," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 172-185.
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    Cited by:

    1. Li, Yang & Liu, Dejun & Yin, Zhexu & Chen, Yun & Meng, Jin, 2023. "Adaptive selection strategy of shape parameters for LRBF for solving partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 440(C).

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