-
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
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 |
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
[Back to Top]