On asymptotic expansion solvers for highly oscillatory semi-explicit DAEs

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[7]	D. E. Johnson, J. R. Johnson and J. L. Hilburn, Electric Circuit Analysis, $2^{nd}$ edition, Prentice-Hall, New Jersey, 1992.
[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.
[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.
[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.
[11]	A. H. Robbins and W. Miller, Circuit Analysis: Theory and Practice, $5^{nd}$ edition, Cengage Learning, Boston, 2012.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[7]	D. E. Johnson, J. R. Johnson and J. L. Hilburn, Electric Circuit Analysis, $2^{nd}$ edition, Prentice-Hall, New Jersey, 1992.
[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.
[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.
[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.
[11]	A. H. Robbins and W. Miller, Circuit Analysis: Theory and Practice, $5^{nd}$ edition, Cengage Learning, Boston, 2012.
[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.
[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.
[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.
[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.

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