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Local discontinuous Galerkin schemes for an ultrasonic propagation equation with fractional attenuation

  • *Corresponding author: Can Li

    *Corresponding author: Can Li 

This work is supported by the Science Basic Research Plan in Shaanxi Province of China under Grant No.2023-JC-YB-045

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  • The goal of this article is to develop local discontinuous Galerkin (LDG) schemes for solving a time fractional equation describing the ultrasonic wave in a homogeneous isotropic porous material. Two novel semi-discrete LDG schemes are designed for the considered model. The semi-discrete LDG schemes are constructed by splitting the original model into a coupled system. The first semi-discrete scheme follows the traditional LDG method by splitting second-order space derivative. The second one splits the original model for both time and space derivatives. The discontinuous Galerkin is used for the spatial discretization. Two kinds of fully discrete LDG schemes are presented by using the Grünwald-Letnikov and L1 approximation formulas for the time fractional derivatives. The $ L^2 $ norm stability and convergence analysis are carried out for both semi-discrete and fully discrete LDG schemes. The stability analysis reveals that the numerical schemes are unconditionally stable in $ L^2 $ norm and convergence with optimal convergence rate. Finally, numerical examples are presented to test the effectiveness of the proposed schemes and the correctness of the theoretical analysis.

    Mathematics Subject Classification: Primary: 34A08, 74S25; Secondary: 26A33.


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  • Table 1.  The $ L^2 $ errors and convergence orders of scheme (46) and scheme (51) for $ P^{1} $ element with $ h = 1/2000, \gamma = 3 $

    $ \tau $ Scheme (46) Scheme (51)
    $ \|\cdot\| $-error order $ \|\cdot\| $-error order
    1/5 5.0164e-02 $ \ast $ 3.4866e-04 $ \ast $
    1/10 2.4845e-02 1.0137 1.2763e-04 1.4499
    1/20 1.1776e-02 1.0771 3.2934e-05 1.9542
    1/40 5.6086e-03 1.0701 8.3057e-06 1.9874
    1/80 2.7134e-03 1.0475 2.1335e-06 1.9609
     | Show Table
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    Table 2.  The $ L^2 $ errors and convergence orders of scheme (46) for different $ P^{k} $ elements with $ \tau = h^{k+1}, \gamma = 3 $

    $ h $ $ P^{0} $ $ P^{1} $ $ P^{2} $
    $ \|\cdot\| $-error order $ \|\cdot\| $-error order $ \|\cdot\| $-error order
    1/5 2.7810e-01 $ \ast $ 6.7699e-02 $ \ast $ 6.8549e-03 $ \ast $
    1/10 1.3414e-01 1.0518 1.7074e-02 1.9873 8.7451e-04 2.9706
    1/20 6.5965e-02 1.0240 4.2764e-03 1.9973 1.0989e-04 2.9925
    1/40 3.2726e-02 1.0113 1.0696e-03 1.9994 1.3751e-05 2.9984
     | Show Table
    DownLoad: CSV

    Table 3.  The $ L^2 $ errors and convergence orders of scheme (51) for different $ P^{k} $ elements with $ \tau = h^{(k+1)/2}, \gamma = 3 $

    $ h $ $ P^{0} $ $ P^{1} $ $ P^{2} $
    $ \|\cdot\| $-error order $ \|\cdot\| $-error order $ \|\cdot\| $-error order
    1/5 1.1735e+00 $ \ast $ 6.4446e-02 $ \ast $ 8.2638e-03 $ \ast $
    1/10 5.1205e-01 1.1965 1.6766e-02 1.9426 8.8156e-04 3.2287
    1/20 1.7857e-01 1.5198 4.2333e-03 1.9857 1.0884e-04 3.0178
    1/40 8.6859e-02 1.0397 1.0609e-03 1.9964 1.3375e-05 3.0249
     | Show Table
    DownLoad: CSV

    Table 4.  The $ L^2 $ errors and convergence orders of schemes (46) and (51) for $ P^{1} $ element with $ h = 1/2000, \gamma = 2 $

    $ \tau $ Scheme (46) Scheme (51)
    $ \|\cdot\| $-error order $ \|\cdot\| $-error order
    1/5 1.0574e-02 $ \ast $ 3.5121e-04 $ \ast $
    1/10 5.8509e-03 0.8537 3.4066e-04 0.0440
    1/20 2.7058e-03 1.1126 1.3991e-04 1.2838
    1/40 1.2358e-03 1.1306 5.1111e-05 1.4528
    1/80 5.7308e-04 1.1087 1.8255e-05 1.4853
     | Show Table
    DownLoad: CSV
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