# American Institute of Mathematical Sciences

December  2019, 6(2): 239-249. doi: 10.3934/jcd.2019012

## Solving the wave equation with multifrequency oscillations

 1 School of Electronic Engineering, Dublin City University, DCU Glasnevin Campus, Dublin 9, Ireland 2 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom 3 Institute of Mathematics, University of Gdańsk, ul. Wit Stwosz 57, 80-308, Gdańsk, Poland 4 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom

Received  March 2019 Revised  October 2019 Published  November 2019

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.

Citation: Marissa Condon, Arieh Iserles, Karolina Kropielnicka, Pranav Singh. Solving the wave equation with multifrequency oscillations. Journal of Computational Dynamics, 2019, 6 (2) : 239-249. doi: 10.3934/jcd.2019012
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[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 0.01 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 110000 152.3 44.3 79.4 0.163 0.03 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 68000 9794 960.2 41.8 0.176 0.037 0.009 $\omega = 50$ $\mathcal{A}$ 0.14 0.062 0.081 0.047 0.033 0.032 0.032 0.032 Error $\mathcal{S}$ 4147 160000 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
 $N$ 1 2 4 8 16 64 256 1024 $h$ 10 5 2.5 1.25 0.625 0.156 0.039 0.01 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 110000 152.3 44.3 79.4 0.163 0.03 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 68000 9794 960.2 41.8 0.176 0.037 0.009 $\omega = 50$ $\mathcal{A}$ 0.14 0.062 0.081 0.047 0.033 0.032 0.032 0.032 Error $\mathcal{S}$ 4147 160000 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
[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 0.38 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 0.02 $\omega = 100$ $\mathcal{A}$ 0.117 0.1 0.053 0.008 0.004 0.003 0.003 0.003 0.003
 $N$ 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 0.38 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 0.02 $\omega = 100$ $\mathcal{A}$ 0.117 0.1 0.053 0.008 0.004 0.003 0.003 0.003 0.003
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