March  2014, 34(3): 1079-1097. doi: 10.3934/dcds.2014.34.1079

Variable step size multiscale methods for stiff and highly oscillatory dynamical systems

1. 

Department of Mathematics, the University of Texas at Austin, Austin, TX 78712, United States

2. 

Department of Mathematics and ICES, the University of Texas at Austin, Austin, TX 78712, United States

Received  November 2012 Revised  February 2013 Published  August 2013

We present a new numerical multiscale integrator for stiff and highly oscillatory dynamical systems. The new algorithm can be seen as an improved version of the seamless Heterogeneous Multiscale Method by E, Ren, and Vanden-Eijnden and the method FLAVORS by Tao, Owhadi, and Marsden. It approximates slowly changing quantities in the solution with higher accuracy than these other methods while maintaining the same computational complexity. To achieve higher accuracy, it uses variable mesoscopic time steps which are determined by a special function satisfying moment and regularity conditions. Detailed analytical and numerical comparison between the different methods are given.
Citation: Yoonsang Lee, Bjorn Engquist. Variable step size multiscale methods for stiff and highly oscillatory dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1079-1097. doi: 10.3934/dcds.2014.34.1079
References:
[1]

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

[2]

G. Ariel, B. Engquist, S. Kim, Y. Lee and R. Tsai, A multiscale method for highly oscillatory dynamical systems using a Poincaré map type technique, Journal of Scientific Computing, 54 (2013), 247-268. doi: 10.1007/s10915-012-9656-x.

[3]

G. Ariel, B. Engquist and R. Tsai, A multiscale method for highly oscillatory ordinary differential equations with resonance, Mathematics of Computation, 78 (2009), 929-956. doi: 10.1090/S0025-5718-08-02139-X.

[4]

G. Ariel, B. Engquist and R. Tsai, Oscillatory systems with three separated time scales analysis and computation, Numerical Analysis of Multiscale Computations, 82 (2012), 23-45. doi: 10.1007/978-3-642-21943-6_2.

[5]

M. Condon, A. Deaño and A. Iserles, On second-order differential equations with highly oscillatory forcing terms, Proc. R. Soc. Lond. Ser. A Math. Eng. Sci., 466 (2010), 1809-1828. doi: 10.1098/rspa.2009.0481.

[6]

W. E, W. Ren and E. Vanden-Eijnden, A general strategy for designing seamless multiscale methods, Journal of Computational Physics, 228 (2009), 5437-5453. doi: 10.1016/j.jcp.2009.04.030.

[7]

B. Engquist and Y. H. Tsai, Heterogeneous multiscale methods for stiff ordinary differential equations, Mathematics of Computation, 74 (2005), 1707-1742. doi: 10.1090/S0025-5718-05-01745-X.

[8]

I. Fatkullin and E. Vanden-Eijnden, A computational strategy for multiscale systems with applications to Lorenz 96 model, Journal of Computational Physics, 200 (2004), 606-638. doi: 10.1016/j.jcp.2004.04.013.

[9]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II," Springer Series in Computational Mathematics, 14, Springer Berlin Heidelberg, 1996.

[10]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," Springer Series in Computational Mathematics, 31, Springer Berlin Heidelberg, 2006.

[11]

M. Hochbruck, C. Lubich and H. Selhofer, Exponential integrators for large systems of differential equations, SIAM Journal on Scientific Computing, 19 (1998), 1552-1574. doi: 10.1137/S1064827595295337.

[12]

M. Hochbruck and C. Lubich, A Gautschi-type method for oscillatory second-order differential equations, Numerische Mathematik, 83 (1999), 403-426. doi: 10.1007/s002110050456.

[13]

J. Kevorkian and J. D. Cole, "Perturbation Methods in Applied Mathematics," Applied Mathematical Sciences, 34, Springer New York, 1981.

[14]

J. Kevorkian and J. D. Cole, "Multiple Scale and Singular Perturbation Methods," Applied Mathematical Sciences, 114, Springer New York, 1996. doi: 10.1007/978-1-4612-3968-0.

[15]

Y. Lee and B. Engquist, Seamless multiscale methods for advection enhanced diffusion in incompressible turbulent velocity fields, preprint.

[16]

Y. Lee and B. Engquist, Fast integrators for several well-separated scales, preprint.

[17]

A. J. Majda and M. J. Grote, Mathematical test models for superparametrization in anisotropic turbulence, Proceedings of the National Academy of Sciences of the United States of America, 106 (2009), 5470-5474. doi: 10.1073/pnas.0901383106.

[18]

G. A. Pavliotis and A. Stuart, "Multiscale Methods: Averaging and Homogenization," Texts in Applied Mathematics, 53, Springer New York, 2008.

[19]

J. A. Sanders, F. Verhulst and J. Murdock, "Averaging Methods in Nonlinear Dynamical Systems," Applied Mathematical Sciences, 59, Springer New York, 2007.

[20]

J. M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems," Chapman and Hall Londong and New York, 1994.

[21]

M. Tao, H. Owhadi and J. Marsden, Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and hamiltonian systems with hidden slow dynamics via flow averaging, Multiscale Modeling and Simulation, 8 (2010), 1269-1324. doi: 10.1137/090771648.

[22]

E. Vanden-Eijnden, On HMM-like integrators and projective integration methods for systems with multiple time scales, Communications in Mathematical Sciences, 5 (2007), 495-505.

show all references

References:
[1]

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

[2]

G. Ariel, B. Engquist, S. Kim, Y. Lee and R. Tsai, A multiscale method for highly oscillatory dynamical systems using a Poincaré map type technique, Journal of Scientific Computing, 54 (2013), 247-268. doi: 10.1007/s10915-012-9656-x.

[3]

G. Ariel, B. Engquist and R. Tsai, A multiscale method for highly oscillatory ordinary differential equations with resonance, Mathematics of Computation, 78 (2009), 929-956. doi: 10.1090/S0025-5718-08-02139-X.

[4]

G. Ariel, B. Engquist and R. Tsai, Oscillatory systems with three separated time scales analysis and computation, Numerical Analysis of Multiscale Computations, 82 (2012), 23-45. doi: 10.1007/978-3-642-21943-6_2.

[5]

M. Condon, A. Deaño and A. Iserles, On second-order differential equations with highly oscillatory forcing terms, Proc. R. Soc. Lond. Ser. A Math. Eng. Sci., 466 (2010), 1809-1828. doi: 10.1098/rspa.2009.0481.

[6]

W. E, W. Ren and E. Vanden-Eijnden, A general strategy for designing seamless multiscale methods, Journal of Computational Physics, 228 (2009), 5437-5453. doi: 10.1016/j.jcp.2009.04.030.

[7]

B. Engquist and Y. H. Tsai, Heterogeneous multiscale methods for stiff ordinary differential equations, Mathematics of Computation, 74 (2005), 1707-1742. doi: 10.1090/S0025-5718-05-01745-X.

[8]

I. Fatkullin and E. Vanden-Eijnden, A computational strategy for multiscale systems with applications to Lorenz 96 model, Journal of Computational Physics, 200 (2004), 606-638. doi: 10.1016/j.jcp.2004.04.013.

[9]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II," Springer Series in Computational Mathematics, 14, Springer Berlin Heidelberg, 1996.

[10]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," Springer Series in Computational Mathematics, 31, Springer Berlin Heidelberg, 2006.

[11]

M. Hochbruck, C. Lubich and H. Selhofer, Exponential integrators for large systems of differential equations, SIAM Journal on Scientific Computing, 19 (1998), 1552-1574. doi: 10.1137/S1064827595295337.

[12]

M. Hochbruck and C. Lubich, A Gautschi-type method for oscillatory second-order differential equations, Numerische Mathematik, 83 (1999), 403-426. doi: 10.1007/s002110050456.

[13]

J. Kevorkian and J. D. Cole, "Perturbation Methods in Applied Mathematics," Applied Mathematical Sciences, 34, Springer New York, 1981.

[14]

J. Kevorkian and J. D. Cole, "Multiple Scale and Singular Perturbation Methods," Applied Mathematical Sciences, 114, Springer New York, 1996. doi: 10.1007/978-1-4612-3968-0.

[15]

Y. Lee and B. Engquist, Seamless multiscale methods for advection enhanced diffusion in incompressible turbulent velocity fields, preprint.

[16]

Y. Lee and B. Engquist, Fast integrators for several well-separated scales, preprint.

[17]

A. J. Majda and M. J. Grote, Mathematical test models for superparametrization in anisotropic turbulence, Proceedings of the National Academy of Sciences of the United States of America, 106 (2009), 5470-5474. doi: 10.1073/pnas.0901383106.

[18]

G. A. Pavliotis and A. Stuart, "Multiscale Methods: Averaging and Homogenization," Texts in Applied Mathematics, 53, Springer New York, 2008.

[19]

J. A. Sanders, F. Verhulst and J. Murdock, "Averaging Methods in Nonlinear Dynamical Systems," Applied Mathematical Sciences, 59, Springer New York, 2007.

[20]

J. M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems," Chapman and Hall Londong and New York, 1994.

[21]

M. Tao, H. Owhadi and J. Marsden, Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and hamiltonian systems with hidden slow dynamics via flow averaging, Multiscale Modeling and Simulation, 8 (2010), 1269-1324. doi: 10.1137/090771648.

[22]

E. Vanden-Eijnden, On HMM-like integrators and projective integration methods for systems with multiple time scales, Communications in Mathematical Sciences, 5 (2007), 495-505.

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