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
References:
[1]

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

[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.

[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. 

[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.

[5]

M. CondonA. 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.

[6]

J. CooperG. 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.

[7]

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

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[15]

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

show all references

References:
[1]

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

[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.

[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. 

[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.

[5]

M. CondonA. 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.

[6]

J. CooperG. 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.

[7]

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

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[15]

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

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
$ 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
$ 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
[1]

Marissa Condon, Jing Gao, Arieh Iserles. On asymptotic expansion solvers for highly oscillatory semi-explicit DAEs. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4813-4837. doi: 10.3934/dcds.2016008

[2]

Emmanuel Frénod, Sever A. Hirstoaga, Eric Sonnendrücker. An exponential integrator for a highly oscillatory vlasov equation. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 169-183. doi: 10.3934/dcdss.2015.8.169

[3]

Kazuhiro Kurata, Yuki Osada. Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4239-4251. doi: 10.3934/cpaa.2021157

[4]

Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations and Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023

[5]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080

[6]

Katharina Schratz, Xiaofei Zhao. On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2841-2865. doi: 10.3934/dcdsb.2020043

[7]

Philippe Chartier, Norbert J. Mauser, Florian Méhats, Yong Zhang. Solving highly-oscillatory NLS with SAM: Numerical efficiency and long-time behavior. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1327-1349. doi: 10.3934/dcdss.2016053

[8]

Tian Zhang, Huabin Chen, Chenggui Yuan, Tomás Caraballo. On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5355-5375. doi: 10.3934/dcdsb.2019062

[9]

Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861

[10]

Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure and Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921

[11]

Guanggan Chen, Jian Zhang. Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1441-1453. doi: 10.3934/dcdsb.2012.17.1441

[12]

Walter Allegretto, Liqun Cao, Yanping Lin. Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 543-576. doi: 10.3934/dcds.2008.20.543

[13]

Anna Geyer, Ronald Quirchmayr. Traveling wave solutions of a highly nonlinear shallow water equation. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1567-1604. doi: 10.3934/dcds.2018065

[14]

Chunqing Lu. Asymptotic solutions of a nonlinear equation. Conference Publications, 2003, 2003 (Special) : 590-595. doi: 10.3934/proc.2003.2003.590

[15]

Philippe Chartier, Nicolas Crouseilles, Mohammed Lemou, Florian Méhats. Averaging of highly-oscillatory transport equations. Kinetic and Related Models, 2020, 13 (6) : 1107-1133. doi: 10.3934/krm.2020039

[16]

Claude Le Bris, Frédéric Legoll. Integrators for highly oscillatory Hamiltonian systems: An homogenization approach. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 347-373. doi: 10.3934/dcdsb.2010.13.347

[17]

Hermann Brunner. On Volterra integral operators with highly oscillatory kernels. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 915-929. doi: 10.3934/dcds.2014.34.915

[18]

Yahong Peng, Yaguang Wang. Reflection of highly oscillatory waves with continuous oscillatory spectra for semilinear hyperbolic systems. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1293-1306. doi: 10.3934/dcds.2009.24.1293

[19]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021

[20]

Florian Monteghetti, Ghislain Haine, Denis Matignon. Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions. Mathematical Control and Related Fields, 2019, 9 (4) : 759-791. doi: 10.3934/mcrf.2019049

 Impact Factor: 

Metrics

  • PDF downloads (350)
  • HTML views (325)
  • Cited by (1)

[Back to Top]