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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

 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.
Citation: 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
##### References:
 [1] W. E, A. Abdulle, B. Engquist and E. Vanden-Eijnden, The heterogeneous multiscale method,, Acta Numer., 21 (2012), 1. 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. 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., (2009). 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. 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. 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. 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, (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. 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. 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. doi: 10.1017/S0956792507007188. Google Scholar [11] A. H. Robbins and W. Miller, Circuit Analysis: Theory and Practice,, $5^{nd}$ edition, (2012). Google Scholar [12] J. M. Sanz-Serna, Modulated Fourier expansions and heterogeneous multiscale methods,, IMA J. Numer. Anal., 29 (2009), 595. 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. 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. 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, (2004). Google Scholar

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##### References:
 [1] W. E, A. Abdulle, B. Engquist and E. Vanden-Eijnden, The heterogeneous multiscale method,, Acta Numer., 21 (2012), 1. 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. 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., (2009). 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. 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. 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. 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, (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. 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. 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. doi: 10.1017/S0956792507007188. Google Scholar [11] A. H. Robbins and W. Miller, Circuit Analysis: Theory and Practice,, $5^{nd}$ edition, (2012). Google Scholar [12] J. M. Sanz-Serna, Modulated Fourier expansions and heterogeneous multiscale methods,, IMA J. Numer. Anal., 29 (2009), 595. 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. 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. 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, (2004). Google Scholar
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