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

Previous Article
Discrete gradient methods have an energy conservation law
DCDS Home
This Issue
Next Article
Analysis of the 3DVAR filter for the partially observed Lorenz'63 model

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

Yoonsang Lee ^1, and Bjorn Engquist ^2,

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

Abstract
Related Papers

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.

Keywords: seamless method, highly oscillatory., multiscale, Numerical ODE, HMM.

Mathematics Subject Classification: 34E13, 65LXX, 37MX.

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.

[1]	Thierry Colin, Boniface Nkonga. Multiscale numerical method for nonlinear Maxwell equations. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 631-658. doi: 10.3934/dcdsb.2005.5.631
[2]	Philippe Chartier, Norbert J. Mauser, Florian Méhats, Yong Zhang. Solving highly-oscillatory NLS with SAM: Numerical efficiency and long-time behavior. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1327-1349. doi: 10.3934/dcdss.2016053
[3]	Philippe Chartier, Nicolas Crouseilles, Mohammed Lemou, Florian Méhats. Averaging of highly-oscillatory transport equations. Kinetic and Related Models, 2020, 13 (6) : 1107-1133. doi: 10.3934/krm.2020039
[4]	Claude Le Bris, Frédéric Legoll. Integrators for highly oscillatory Hamiltonian systems: An homogenization approach. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 347-373. doi: 10.3934/dcdsb.2010.13.347
[5]	Emmanuel Frénod, Sever A. Hirstoaga, Eric Sonnendrücker. An exponential integrator for a highly oscillatory vlasov equation. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 169-183. doi: 10.3934/dcdss.2015.8.169
[6]	Hermann Brunner. On Volterra integral operators with highly oscillatory kernels. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 915-929. doi: 10.3934/dcds.2014.34.915
[7]	Yahong Peng, Yaguang Wang. Reflection of highly oscillatory waves with continuous oscillatory spectra for semilinear hyperbolic systems. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1293-1306. doi: 10.3934/dcds.2009.24.1293
[8]	Wenlei Li, Shaoyun Shi. Singular perturbed renormalization group theory and its application to highly oscillatory problems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1819-1833. doi: 10.3934/dcdsb.2018089
[9]	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
[10]	Kamil Aida-Zade, Jamila Asadova. Numerical solution to optimal control problems of oscillatory processes. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021166
[11]	Mikhaël Balabane, Mustapha Jazar, Philippe Souplet. Oscillatory blow-up in nonlinear second order ODE's: The critical case. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 577-584. doi: 10.3934/dcds.2003.9.577
[12]	Frederike Kissling, Christian Rohde. The computation of nonclassical shock waves with a heterogeneous multiscale method. Networks and Heterogeneous Media, 2010, 5 (3) : 661-674. doi: 10.3934/nhm.2010.5.661
[13]	Paweł Pilarczyk. Topological-numerical approach to the existence of periodic trajectories in ODE's. Conference Publications, 2003, 2003 (Special) : 701-708. doi: 10.3934/proc.2003.2003.701
[14]	Olivier Pironneau, Alexei Lozinski, Alain Perronnet, Frédéric Hecht. Numerical zoom for multiscale problems with an application to flows through porous media. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 265-280. doi: 10.3934/dcds.2009.23.265
[15]	Annalisa Iuorio, Christian Kuehn, Peter Szmolyan. Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1269-1290. doi: 10.3934/dcdss.2020073
[16]	Alberto Zingaro, Ivan Fumagalli, Luca Dede, Marco Fedele, Pasquale C. Africa, Antonio F. Corno, Alfio Quarteroni. A geometric multiscale model for the numerical simulation of blood flow in the human left heart. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022052
[17]	Thomas Abballe, Grégoire Allaire, Éli Laucoin, Philippe Montarnal. Application of a coupled FV/FE multiscale method to cement media. Networks and Heterogeneous Media, 2010, 5 (3) : 603-615. doi: 10.3934/nhm.2010.5.603
[18]	Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052
[19]	Eric Chung, Yalchin Efendiev, Ke Shi, Shuai Ye. A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media. Networks and Heterogeneous Media, 2017, 12 (4) : 619-642. doi: 10.3934/nhm.2017025
[20]	Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Yoonsang Lee Bjorn Engquist
on Google Scholar
Yoonsang Lee Bjorn Engquist

[Back to Top]