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

Solving the wave equation with multifrequency oscillations

Marissa Condon 1, , Arieh Iserles 2, , Karolina Kropielnicka 3, and  Pranav Singh 4,

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.

Keywords: Wave equation, highly oscillatory, asymptotic-numerical solver, exponential splittings, asymptotic expansion.
Mathematics Subject Classification: Primary: 65M70; Secondary: 41A60, 35L05, 65L05.
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
References:
[1]

S. Altinbasak Üsküplu, M. Condon, A. Deaño and A. Iserles, Highly oscillatory diffusion-type equations, J. Comput. Math., 31 (2013), 549-572.  doi: 10.4208/jcm.1307-m3955.  Google Scholar

[2]

P. Bader, S. Blanes, F. Casas, N. 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.  Google Scholar

[3]

S. Blanes and F. Casas, Splitting and composition methods in the numerical integration of differential equations, Bol. Soc. Esp. Mat. Apl. SeMA, 45 (2008), 89-145.   Google Scholar

[4]

D. Cohen and J. Schweitzer, High order numerical methods for highly oscillatory problems, ESAIM Math. Model. Numer. Anal., 49 (2015), 695-711.  doi: 10.1051/m2an/2014056.  Google Scholar

[5]

M. Condon, A. Deaño and A. Iserles, On systems of differential equations with extrinsic oscillation, Discrete Contin. Dyn. Syst., 28 (2010), 1345-1367.  doi: 10.3934/dcds.2010.28.1345.  Google Scholar

[6]

J. Cooper, G. P. Menzala and W. Strauss, On the scattering frequencies of time-dependent potentials, Math. Methods Appl. Sci., 8 (1986), 576-584.  doi: 10.1002/mma.1670080137.  Google Scholar

[7]

S. Cuccagna, On the wave equation with a potential, Commun. Partial Differential Equations, 25 (2000), 1549-1565.  doi: 10.1080/03605300008821559.  Google Scholar

[8]

E. Faou and K. Schratz, Asymptotic preserving schemes for the Klein-Gordon equation in the non-relativistic limit regime, Numer. Math., 126 (2014), 441-469.  doi: 10.1007/s00211-013-0567-z.  Google Scholar

[9]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-30666-8.  Google Scholar

[10]

A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods, in Acta Numerica, Acta Numer., 9, Cambridge Univ. Press, Cambridge, 2000, 215–365. doi: 10.1017/S0962492900002154.  Google Scholar

[11]

G. Majda and M. Wei, Relationships between a potential and its scattering frequencies, SIAM J. Appl. Math., 55 (1995), 1094-1116.  doi: 10.1137/S0036139992231186.  Google Scholar

[12]

G. F. Roach, Wave scattering by time dependent perturbations, Fract. Calc. Appl. Anal., 4 (2001), 209-236.   Google Scholar

[13]

J. M. Sanz-Serna, Modulated Fourier expansions and heterogeneous multiscale methods, IMA J. Numer. Anal., 29 (2009), 595-605.  doi: 10.1093/imanum/drn031.  Google Scholar

[14]

J. M. Sanz-Serna and A. Portillo, Classical numerical integrators for wave-packet dynamics, J. Chem. Phys., 104 (1996), 2349-2355.  doi: 10.1063/1.470930.  Google Scholar

[15]

G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517.  doi: 10.1137/0705041.  Google Scholar

show all references

References:
[1]

S. Altinbasak Üsküplu, M. Condon, A. Deaño and A. Iserles, Highly oscillatory diffusion-type equations, J. Comput. Math., 31 (2013), 549-572.  doi: 10.4208/jcm.1307-m3955.  Google Scholar

[2]

P. Bader, S. Blanes, F. Casas, N. 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.  Google Scholar

[3]

S. Blanes and F. Casas, Splitting and composition methods in the numerical integration of differential equations, Bol. Soc. Esp. Mat. Apl. SeMA, 45 (2008), 89-145.   Google Scholar

[4]

D. Cohen and J. Schweitzer, High order numerical methods for highly oscillatory problems, ESAIM Math. Model. Numer. Anal., 49 (2015), 695-711.  doi: 10.1051/m2an/2014056.  Google Scholar

[5]

M. Condon, A. Deaño and A. Iserles, On systems of differential equations with extrinsic oscillation, Discrete Contin. Dyn. Syst., 28 (2010), 1345-1367.  doi: 10.3934/dcds.2010.28.1345.  Google Scholar

[6]

J. Cooper, G. P. Menzala and W. Strauss, On the scattering frequencies of time-dependent potentials, Math. Methods Appl. Sci., 8 (1986), 576-584.  doi: 10.1002/mma.1670080137.  Google Scholar

[7]

S. Cuccagna, On the wave equation with a potential, Commun. Partial Differential Equations, 25 (2000), 1549-1565.  doi: 10.1080/03605300008821559.  Google Scholar

[8]

E. Faou and K. Schratz, Asymptotic preserving schemes for the Klein-Gordon equation in the non-relativistic limit regime, Numer. Math., 126 (2014), 441-469.  doi: 10.1007/s00211-013-0567-z.  Google Scholar

[9]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-30666-8.  Google Scholar

[10]

A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods, in Acta Numerica, Acta Numer., 9, Cambridge Univ. Press, Cambridge, 2000, 215–365. doi: 10.1017/S0962492900002154.  Google Scholar

[11]

G. Majda and M. Wei, Relationships between a potential and its scattering frequencies, SIAM J. Appl. Math., 55 (1995), 1094-1116.  doi: 10.1137/S0036139992231186.  Google Scholar

[12]

G. F. Roach, Wave scattering by time dependent perturbations, Fract. Calc. Appl. Anal., 4 (2001), 209-236.   Google Scholar

[13]

J. M. Sanz-Serna, Modulated Fourier expansions and heterogeneous multiscale methods, IMA J. Numer. Anal., 29 (2009), 595-605.  doi: 10.1093/imanum/drn031.  Google Scholar

[14]

J. M. Sanz-Serna and A. Portillo, Classical numerical integrators for wave-packet dynamics, J. Chem. Phys., 104 (1996), 2349-2355.  doi: 10.1063/1.470930.  Google Scholar

[15]

G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517.  doi: 10.1137/0705041.  Google Scholar

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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$ 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
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
Table Options

Download as excel

$ 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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