-
Previous Article
Principal symmetric space analysis
- JCD Home
- This Issue
-
Next Article
A new class of integrable Lotka–Volterra systems
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 |
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.
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. |
[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. |
[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. 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. |
[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. |
[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 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. |
[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. |
[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. 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. |
[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. |
[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. |
1 | 2 | 4 | 8 | 16 | 64 | 256 | 1024 | ||
10 | 5 | 2.5 | 1.25 | 0.625 | 0.156 | 0.039 | 0.010 | ||
Calls | 1 | 2 | 4 | 8 | 16 | 64 | 256 | 1024 | |
to |
2 | 4 | 8 | 16 | 32 | 128 | 512 | 2048 | |
Error | 4147 | 1.1e5 | 152.3 | 44.3 | 79.4 | 0.163 | 0.030 | 0.007 | |
0.593 | 0.254 | 0.251 | 0.206 | 0.252 | 0.249 | 0.248 | 0.248 | ||
Error | 4147 | 6.8e4 | 9794 | 960.2 | 41.8 | 0.176 | 0.037 | 0.009 | |
0.140 | 0.062 | 0.081 | 0.047 | 0.033 | 0.032 | 0.032 | 0.032 | ||
Error | 4147 | 1.6e5 | 7914 | 67.95 | 80.2 | 2.081 | 0.039 | 0.009 | |
0.077 | 0.015 | 0.021 | 0.011 | 0.001 | 0.001 | 0.001 | 0.001 |
1 | 2 | 4 | 8 | 16 | 64 | 256 | 1024 | ||
10 | 5 | 2.5 | 1.25 | 0.625 | 0.156 | 0.039 | 0.010 | ||
Calls | 1 | 2 | 4 | 8 | 16 | 64 | 256 | 1024 | |
to |
2 | 4 | 8 | 16 | 32 | 128 | 512 | 2048 | |
Error | 4147 | 1.1e5 | 152.3 | 44.3 | 79.4 | 0.163 | 0.030 | 0.007 | |
0.593 | 0.254 | 0.251 | 0.206 | 0.252 | 0.249 | 0.248 | 0.248 | ||
Error | 4147 | 6.8e4 | 9794 | 960.2 | 41.8 | 0.176 | 0.037 | 0.009 | |
0.140 | 0.062 | 0.081 | 0.047 | 0.033 | 0.032 | 0.032 | 0.032 | ||
Error | 4147 | 1.6e5 | 7914 | 67.95 | 80.2 | 2.081 | 0.039 | 0.009 | |
0.077 | 0.015 | 0.021 | 0.011 | 0.001 | 0.001 | 0.001 | 0.001 |
1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | ||
5 | 2.5 | 1.25 | 0.625 | 0.313 | 0.156 | 0.078 | 0.039 | 0.02 | ||
Calls | 1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | |
to |
2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 | |
Error | 4.237 | 88.3 | 96.3 | 98.1 | 1.18 | 0.447 | 0.189 | 0.092 | 0.047 | |
2.14 | 1.82 | 1.022 | 0.409 | 0.391 | 0.389 | 0.388 | 0.388 | 0.388 | ||
Error | 2.55 | 88.7 | 82.9 | 8.845 | 13.1 | 0.262 | 0.128 | 0.058 | 0.027 | |
0.452 | 0.380 | 0.198 | 0.045 | 0.038 | 0.039 | 0.039 | 0.039 | 0.039 | ||
Error | 2.39 | 32.5 | 20.1 | 18.1 | 2.5 | 2.9 | 0.106 | 0.045 | 0.020 | |
0.117 | 0.100 | 0.053 | 0.008 | 0.004 | 0.003 | 0.003 | 0.003 | 0.003 |
1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | ||
5 | 2.5 | 1.25 | 0.625 | 0.313 | 0.156 | 0.078 | 0.039 | 0.02 | ||
Calls | 1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | |
to |
2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 | |
Error | 4.237 | 88.3 | 96.3 | 98.1 | 1.18 | 0.447 | 0.189 | 0.092 | 0.047 | |
2.14 | 1.82 | 1.022 | 0.409 | 0.391 | 0.389 | 0.388 | 0.388 | 0.388 | ||
Error | 2.55 | 88.7 | 82.9 | 8.845 | 13.1 | 0.262 | 0.128 | 0.058 | 0.027 | |
0.452 | 0.380 | 0.198 | 0.045 | 0.038 | 0.039 | 0.039 | 0.039 | 0.039 | ||
Error | 2.39 | 32.5 | 20.1 | 18.1 | 2.5 | 2.9 | 0.106 | 0.045 | 0.020 | |
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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - S, 2015, 8 (1) : 169-183. doi: 10.3934/dcdss.2015.8.169 |
[3] |
Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations & Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023 |
[4] |
Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080 |
[5] |
Philippe Chartier, Norbert J. Mauser, Florian Méhats, Yong Zhang. Solving highly-oscillatory NLS with SAM: Numerical efficiency and long-time behavior. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1327-1349. doi: 10.3934/dcdss.2016053 |
[6] |
Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861 |
[7] |
Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure & Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921 |
[8] |
Guanggan Chen, Jian Zhang. Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1441-1453. doi: 10.3934/dcdsb.2012.17.1441 |
[9] |
Tian Zhang, Huabin Chen, Chenggui Yuan, Tomás Caraballo. On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5355-5375. doi: 10.3934/dcdsb.2019062 |
[10] |
Walter Allegretto, Liqun Cao, Yanping Lin. Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 543-576. doi: 10.3934/dcds.2008.20.543 |
[11] |
Chunqing Lu. Asymptotic solutions of a nonlinear equation. Conference Publications, 2003, 2003 (Special) : 590-595. doi: 10.3934/proc.2003.2003.590 |
[12] |
Anna Geyer, Ronald Quirchmayr. Traveling wave solutions of a highly nonlinear shallow water equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1567-1604. doi: 10.3934/dcds.2018065 |
[13] |
Claude Le Bris, Frédéric Legoll. Integrators for highly oscillatory Hamiltonian systems: An homogenization approach. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 347-373. doi: 10.3934/dcdsb.2010.13.347 |
[14] |
Hermann Brunner. On Volterra integral operators with highly oscillatory kernels. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 915-929. doi: 10.3934/dcds.2014.34.915 |
[15] |
Yahong Peng, Yaguang Wang. Reflection of highly oscillatory waves with continuous oscillatory spectra for semilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1293-1306. doi: 10.3934/dcds.2009.24.1293 |
[16] |
Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021 |
[17] |
Florian Monteghetti, Ghislain Haine, Denis Matignon. Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions. Mathematical Control & Related Fields, 2019, 9 (4) : 759-791. doi: 10.3934/mcrf.2019049 |
[18] |
Jorge Ferreira, Mauro De Lima Santos. Asymptotic behaviour for wave equations with memory in a noncylindrical domains. Communications on Pure & Applied Analysis, 2003, 2 (4) : 511-520. doi: 10.3934/cpaa.2003.2.511 |
[19] |
Yan Cui, Zhiqiang Wang. Asymptotic stability of wave equations coupled by velocities. Mathematical Control & Related Fields, 2016, 6 (3) : 429-446. doi: 10.3934/mcrf.2016010 |
[20] |
Jeffrey R. Haack, Cory D. Hauck. Oscillatory behavior of Asymptotic-Preserving splitting methods for a linear model of diffusive relaxation. Kinetic & Related Models, 2008, 1 (4) : 573-590. doi: 10.3934/krm.2008.1.573 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]