American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate
  • DCDS Home
  • This Issue
  • Next Article
    On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions
September  2016, 36(9): 4813-4837. doi: 10.3934/dcds.2016008

On asymptotic expansion solvers for highly oscillatory semi-explicit DAEs

Marissa Condon 1, , Jing Gao 2, and  Arieh Iserles 3,

1. 

School of Electronic Engineering, Dublin City University, Dublin 9
2. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, Shaanxi, China
3. 

Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WA

Received  February 2014 Revised  March 2016 Published  May 2016

The paper is concerned with the discretization and solution of DAEs of index $1$ and subject to a highly oscillatory forcing term. Separate asymptotic expansions in inverse powers of the oscillatory parameter are constructed to approximate the differential and algebraic variables of the DAEs. The series are truncated to enable practical implementation. Numerical experiments are provided to illustrate the effectiveness of the method.
Keywords: modulated Fourier expansions, Highly oscillatory problems, differential algebraic equations, numerical analysis., oscillatory frequency.
Mathematics Subject Classification: Primary: 65L80, 34E05; Secondary: 34A0.
Citation: Marissa Condon, Jing Gao, Arieh Iserles. On asymptotic expansion solvers for highly oscillatory semi-explicit DAEs. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4813-4837. doi: 10.3934/dcds.2016008
References:
[1]

W. E, A. Abdulle, B. Engquist and E. Vanden-Eijnden, The heterogeneous multiscale method, Acta Numer., 21 (2012), 1-87. doi: 10.1017/S0962492912000025.  Google Scholar

[2]

M. Condon, A. Deaño, J. Gao and A. Iserles, Asymptotic numerical algorithm for second order differential equations with multiple frequencies, Calcolo, 21 (2013), 1-31. Google Scholar

[3]

M. Condon, A. Deaño and A. Iserles, On Asymptotic-Numerical Solvers for Differential Equations with Highly Oscillatory Forcing Terms, DAMTP Tech. Rep. 2009/NA05. Google Scholar

[4]

M. Condon, A. Deaño and A. Iserles, On systems of differential equations with extrinsic oscillation, Discr. Cont. Dynamical Sys., 28 (2010), 1345-1367. doi: 10.3934/dcds.2010.28.1345.  Google Scholar

[5]

M. Condon, A. Deaño, A. Iserles and K. Kropielnicka, Efficient computation of delay differential equations with highly oscillatory terms, ESAIM Math. Model. Numer. Anal., 46 (2012), 1407-1420. doi: 10.1051/m2an/2012004.  Google Scholar

[6]

A. Iserles and S. P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. Royal Soc. A., 461 (2005), 1383-1399. doi: 10.1098/rspa.2004.1401.  Google Scholar

[7]

D. E. Johnson, J. R. Johnson and J. L. Hilburn, Electric Circuit Analysis, $2^{nd}$ edition, Prentice-Hall, New Jersey, 1992. Google Scholar

[8]

L. Malesani and P. Tenti, Three-phase AC/DC PWM converter with sinusoidal AC currents and minimum filter requirements, IEEE Trans. Ind. Appl., IA-23 (1987), 71-77. doi: 10.1109/TIA.1987.4504868.  Google Scholar

[9]

R. Pulch, Finite difference methods for multi time scale differential algebraic equations, ZAMM-Z Angew Math., 83 (2003), 571-583. doi: 10.1002/zamm.200310042.  Google Scholar

[10]

R. Pulch, M.Günther and S. Knorr, Multirate partial differential algebraic equations for simulating radio frequency signals, Eur. J. Appl. Math., 18 (2007), 709-743. doi: 10.1017/S0956792507007188.  Google Scholar

[11]

A. H. Robbins and W. Miller, Circuit Analysis: Theory and Practice, $5^{nd}$ edition, Cengage Learning, Boston, 2012. Google Scholar

[12]

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

[13]

R. E. Scheid, The accurate numerical solution of highly oscillatory ordinary differential equations, Math. Comp., 41 (1983), 487-509. doi: 10.1090/S0025-5718-1983-0717698-9.  Google Scholar

[14]

M. Selva Soto M. and C. Tischendorf, Numerical analysis of DAEs from coupled circuit and semiconductor simulation, Appl. Numer. Math., 53 (2005), 471-488. doi: 10.1016/j.apnum.2004.08.009.  Google Scholar

[15]

C. Tischendorf, Coupled Systems of Differential Algebraic and Partial Differential Equations in Circuit and Device Simulation, Modeling and numerical analysis, Habilitationsschrift, Inst. für Math., Humboldt-Univ. zu Berlin, 2004. Google Scholar

show all references

References:
[1]

W. E, A. Abdulle, B. Engquist and E. Vanden-Eijnden, The heterogeneous multiscale method, Acta Numer., 21 (2012), 1-87. doi: 10.1017/S0962492912000025.  Google Scholar

[2]

M. Condon, A. Deaño, J. Gao and A. Iserles, Asymptotic numerical algorithm for second order differential equations with multiple frequencies, Calcolo, 21 (2013), 1-31. Google Scholar

[3]

M. Condon, A. Deaño and A. Iserles, On Asymptotic-Numerical Solvers for Differential Equations with Highly Oscillatory Forcing Terms, DAMTP Tech. Rep. 2009/NA05. Google Scholar

[4]

M. Condon, A. Deaño and A. Iserles, On systems of differential equations with extrinsic oscillation, Discr. Cont. Dynamical Sys., 28 (2010), 1345-1367. doi: 10.3934/dcds.2010.28.1345.  Google Scholar

[5]

M. Condon, A. Deaño, A. Iserles and K. Kropielnicka, Efficient computation of delay differential equations with highly oscillatory terms, ESAIM Math. Model. Numer. Anal., 46 (2012), 1407-1420. doi: 10.1051/m2an/2012004.  Google Scholar

[6]

A. Iserles and S. P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. Royal Soc. A., 461 (2005), 1383-1399. doi: 10.1098/rspa.2004.1401.  Google Scholar

[7]

D. E. Johnson, J. R. Johnson and J. L. Hilburn, Electric Circuit Analysis, $2^{nd}$ edition, Prentice-Hall, New Jersey, 1992. Google Scholar

[8]

L. Malesani and P. Tenti, Three-phase AC/DC PWM converter with sinusoidal AC currents and minimum filter requirements, IEEE Trans. Ind. Appl., IA-23 (1987), 71-77. doi: 10.1109/TIA.1987.4504868.  Google Scholar

[9]

R. Pulch, Finite difference methods for multi time scale differential algebraic equations, ZAMM-Z Angew Math., 83 (2003), 571-583. doi: 10.1002/zamm.200310042.  Google Scholar

[10]

R. Pulch, M.Günther and S. Knorr, Multirate partial differential algebraic equations for simulating radio frequency signals, Eur. J. Appl. Math., 18 (2007), 709-743. doi: 10.1017/S0956792507007188.  Google Scholar

[11]

A. H. Robbins and W. Miller, Circuit Analysis: Theory and Practice, $5^{nd}$ edition, Cengage Learning, Boston, 2012. Google Scholar

[12]

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

[13]

R. E. Scheid, The accurate numerical solution of highly oscillatory ordinary differential equations, Math. Comp., 41 (1983), 487-509. doi: 10.1090/S0025-5718-1983-0717698-9.  Google Scholar

[14]

M. Selva Soto M. and C. Tischendorf, Numerical analysis of DAEs from coupled circuit and semiconductor simulation, Appl. Numer. Math., 53 (2005), 471-488. doi: 10.1016/j.apnum.2004.08.009.  Google Scholar

[15]

C. Tischendorf, Coupled Systems of Differential Algebraic and Partial Differential Equations in Circuit and Device Simulation, Modeling and numerical analysis, Habilitationsschrift, Inst. für Math., Humboldt-Univ. zu Berlin, 2004. Google Scholar

[1]

Philippe Chartier, Nicolas Crouseilles, Mohammed Lemou, Florian Méhats. Averaging of highly-oscillatory transport equations. Kinetic & Related Models, 2020, 13 (6) : 1107-1133. doi: 10.3934/krm.2020039
[2]

Wenlei Li, Shaoyun Shi. Singular perturbed renormalization group theory and its application to highly oscillatory problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1819-1833. doi: 10.3934/dcdsb.2018089
[3]

Kamil Aida-Zade, Jamila Asadova. Numerical solution to optimal control problems of oscillatory processes. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021166
[4]

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

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
[7]

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
[8]

Hermann Brunner. On Volterra integral operators with highly oscillatory kernels. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 915-929. doi: 10.3934/dcds.2014.34.915
[9]

R.S. Dahiya, A. Zafer. Oscillatory theorems of n-th order functional differential equations. Conference Publications, 2001, 2001 (Special) : 435-443. doi: 10.3934/proc.2001.2001.435
[10]

John R. Graef, R. Savithri, E. Thandapani. Oscillatory properties of third order neutral delay differential equations. Conference Publications, 2003, 2003 (Special) : 342-350. doi: 10.3934/proc.2003.2003.342
[11]

Yoonsang Lee, Bjorn Engquist. Variable step size multiscale methods for stiff and highly oscillatory dynamical systems. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1079-1097. doi: 10.3934/dcds.2014.34.1079
[12]

Zuji Guo, Zhaoli Liu. Perturbed elliptic equations with oscillatory nonlinearities. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3567-3585. doi: 10.3934/dcds.2012.32.3567
[13]

Vu Hoang Linh, Volker Mehrmann. Spectral analysis for linear differential-algebraic equations. Conference Publications, 2011, 2011 (Special) : 991-1000. doi: 10.3934/proc.2011.2011.991
[14]

Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure & Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1
[15]

Kyril Tintarev. Positive solutions of elliptic equations with a critical oscillatory nonlinearity. Conference Publications, 2007, 2007 (Special) : 974-981. doi: 10.3934/proc.2007.2007.974
[16]

Yu-Ting Lin, John Malik, Hau-Tieng Wu. Wave-shape oscillatory model for nonstationary periodic time series analysis. Foundations of Data Science, 2021, 3 (2) : 99-131. doi: 10.3934/fods.2021009
[17]

Lutz Recke, Anatoly Samoilenko, Alexey Teplinsky, Viktor Tkachenko, Serhiy Yanchuk. Frequency locking of modulated waves. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 847-875. doi: 10.3934/dcds.2011.31.847
[18]

José M. Arrieta, Manuel Villanueva-Pesqueira. Elliptic and parabolic problems in thin domains with doubly weak oscillatory boundary. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1891-1914. doi: 10.3934/cpaa.2020083
[19]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[20]

José M. Arrieta, Ariadne Nogueira, Marcone C. Pereira. Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4217-4246. doi: 10.3934/dcdsb.2019079

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (84)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences