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Solving the wave equation with multifrequency oscillations

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  • We explore a new asymptotic-numerical solver for the time-dependent wave equation with an interaction term that is oscillating in time with a very high frequency. The method involves representing the solution as an asymptotic series in inverse powers of the oscillation frequency. Using the new scheme, high accuracy is achieved at a low computational cost. Salient features of the new approach are highlighted by a numerical example.

    Mathematics Subject Classification: Primary: 65M70; Secondary: 41A60, 35L05, 65L05.

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  • Table 1.  [Example 1] Error and cost for the proposed asymptotic method $ \mathcal{A} $ compared to the Lanczos solver $ \mathcal{S} $. Note that each step of $ \mathcal{A} $ uses $ \mathcal{S} $ twice. $ N $ is the number of time steps ($ h = T/N $, $ T = 10 $)

    $ N $ 1 2 4 8 16 64 256 1024
    $ h $ 10 5 2.5 1.25 0.625 0.156 0.039 0.010
    Calls $ \mathcal{S} $ 1 2 4 8 16 64 256 1024
    to $ \mathcal{S} $ $ \mathcal{A} $ 2 4 8 16 32 128 512 2048
    Error $ \mathcal{S} $ 4147 1.1e5 152.3 44.3 79.4 0.163 0.030 0.007
    $ \omega = 25 $ $ \mathcal{A} $ 0.593 0.254 0.251 0.206 0.252 0.249 0.248 0.248
    Error $ \mathcal{S} $ 4147 6.8e4 9794 960.2 41.8 0.176 0.037 0.009
    $ \omega = 50 $ $ \mathcal{A} $ 0.140 0.062 0.081 0.047 0.033 0.032 0.032 0.032
    Error $ \mathcal{S} $ 4147 1.6e5 7914 67.95 80.2 2.081 0.039 0.009
    $ \omega = 100 $ $ \mathcal{A} $ 0.077 0.015 0.021 0.011 0.001 0.001 0.001 0.001
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    Table 2.  [Example 2] Error and cost for the proposed asymptotic method $ \mathcal{A} $ compared to the Lanczos solver $ \mathcal{S} $. Note that each step of $ \mathcal{A} $ uses $ \mathcal{S} $ twice. $ N $ is the number of time steps ($ h = T/N $, $ T = 5 $)

    $ N $ 1 2 4 8 16 32 64 128 256
    $ h $ 5 2.5 1.25 0.625 0.313 0.156 0.078 0.039 0.02
    Calls $ \mathcal{S} $ 1 2 4 8 16 32 64 128 256
    to $ \mathcal{S} $ $ \mathcal{A} $ 2 4 8 16 32 64 128 256 512
    Error $ \mathcal{S} $ 4.237 88.3 96.3 98.1 1.18 0.447 0.189 0.092 0.047
    $ \omega = 25 $ $ \mathcal{A} $ 2.14 1.82 1.022 0.409 0.391 0.389 0.388 0.388 0.388
    Error $ \mathcal{S} $ 2.55 88.7 82.9 8.845 13.1 0.262 0.128 0.058 0.027
    $ \omega = 50 $ $ \mathcal{A} $ 0.452 0.380 0.198 0.045 0.038 0.039 0.039 0.039 0.039
    Error $ \mathcal{S} $ 2.39 32.5 20.1 18.1 2.5 2.9 0.106 0.045 0.020
    $ \omega = 100 $ $ \mathcal{A} $ 0.117 0.100 0.053 0.008 0.004 0.003 0.003 0.003 0.003
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    [2] P. BaderS. BlanesF. CasasN. Kopylov and E. Ponsoda, Symplectic integrators for second-order linear non-autonomous equations, J. Comput. Appl. Math., 330 (2018), 909-919.  doi: 10.1016/j.cam.2017.03.028.
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