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Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Abstract
From the Publisher: Designed for those people who want to gain a practical knowledge of modern techniques, this book contains all the material necessary for a course on the numerical solution of differential equations. Written by two of the field's leading authorities, it provides a unified presentation of initial value and boundary value problems in ODEs as well as differential-algebraic equations. The approach is aimed at a thorough understanding of the issues and methods for practical computation while avoiding an extensive theorem-proof type of exposition. It also addresses reasons why existing software succeeds or fails. This book is a practical and mathematically well informed introduction that emphasizes basic methods and theory, issues in the use and development of mathematical software, and examples from scientific engineering applications. Topics requiring an extensive amount of mathematical development, such as symplectic methods for Hamiltonian systems, are introduced, motivated, and included in the exercises, but a complete and rigorous mathematical presentation is referenced rather than included. Audience This book is appropriate for senior undergraduate or beginning graduate students with a computational focus and practicing engineers and scientists who want to learn about computational differential equations. A beginning course in numerical analysis is needed, and a beginning course in ordinary differential equations would be helpful. About the Authors Uri M. Ascher is a Professor in the Department of Computer Science at the University of British Columbia, Vancouver. He is also Director of the Institute of Applied Mathematics there. Linda R. Petzold is a Professor in the Departments of Mechanical and Environmental Engineering and Computer Science at the University of California at Santa Barbara. She is also Director of the Computational Science and Engineering Program there.
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“Through extensive experimental investigations, Karras et al. (2022) [111] show that Heun's 2nd order method [5] provides an excellent trade off between sample quality and sampling speed.”
From Diffusion Models: A Comprehensive Survey of Methods and Applications (2024) · cites this paper as [5]
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Lacoursière, C. (2007) · KTH Publication Database DiVA (KTH Royal Institute of Technology) · Cited by 70
Reversible Architectures for Arbitrarily Deep Residual Neural NetworksPreprint
Chang, B., Meng, L., Haber, E., et al. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1709.03698 · Cited by 70
“(7) is the Verlet method (Ascher and Petzold 1998) that reads We choose Eq.”
MethodInfluentialComputational Model Predicts the Effects of Targeting Cellular Metabolism in Pancreatic Cancer
Roy, M., & Finley, S. D. (2017) · Frontiers in Physiology · doi:10.3389/fphys.2017.00217 · Cited by 70
“We used structural parameter identifiability analysis (Maly and Petzold, 1996; Ascher and Petzold, 1998; Shampine et al., 1999; Finley et al., 2011; Berthoumieux et al., 2013) to reduce the number of model parameters being fit to the training data.”
MethodErickson, B. A., Birnir, B., & Lavallée, D. (2011) · Geophysical Journal International · doi:10.1111/j.1365-246x.2011.05123.x · Cited by 68
“(12) We discretize the PDE using the method of lines (Ascher & Petzold 1998) and the spatial derivative ∂ 2 ū ∂ x̄2 is approximated using finite differences.”
MethodShadid, J. N., Pawlowski, R. P., Cyr, E. C., et al. (2016) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2016.01.019 · Cited by 68
A trust region filter method for glass box/black box optimization
Eason, J. P., & Biegler, L. T. (2016) · AIChE Journal · doi:10.1002/aic.15325 · Cited by 68
Nadeem, S., Masood, S., Mehmood, R., et al. (2015) · PLoS ONE · doi:10.1371/journal.pone.0124016 · Cited by 68
Optimization of Fractional Order Dynamic Chemical Processing Systems
Flores‐Tlacuahuac, A., & Biegler, L. T. (2013) · Industrial & Engineering Chemistry Research · doi:10.1021/ie401317r · Cited by 67
Effect of inlet modelling on surface drainage in coupled urban flood simulation
Jang, J., Chang, T.-H., & Chen, W. (2018) · Journal of Hydrology · doi:10.1016/j.jhydrol.2018.05.010 · Cited by 67
PETSc/TAO Users Manual (Rev. 3.19)
Argonne National Laboratory (ANL), A., Balay, S., USDOE Office of Science (SC), A. S. C. R. (., Abhyankar, S., Adams, M., Benson, S., Brown, J., Brune, P., Buschelman, K., Constantinescu, E., et al. (2023) · doi:10.2172/1968587 · Cited by 67
A nonlinear parametric model reduction method for efficient gear contact simulations
Blockmans, B., Tamarozzi, T., Naets, F., et al. (2014) · International Journal for Numerical Methods in Engineering · doi:10.1002/nme.4831 · Cited by 66
“In the present work, these equations are solved using an explicit fourth-order Runge–Kutta solver [32], requiring Equation (30) to be converted into first-order form:”
MethodConsolidation of unorganized point clouds for surface reconstruction
Huang, H. F., Li, D., Zhang, H., Ascher, U. M., & Cohen‐Or, D. (2009) · ACM SIGGRAPH Conference and Exhibition on Computer Graphics and Interactive Techniques in Asia (SIGGRAPH Asia) · doi:10.1145/1661412.1618522 · Cited by 65
“We then estimate particle normals via an iterative predictor-corrector scheme (see [Ascher and Petzold 1998] for origins and analogies for this term).”
MethodSelf-citationMPEC strategies for optimization of a class of hybrid dynamic systems
Baumrucker, B. T., & Biegler, L. T. (2009) · Journal of Process Control · doi:10.1016/j.jprocont.2009.02.006 · Cited by 65
The holonomic gradient method for the distribution function of the largest root of a Wishart matrix
Hashiguchi, H., Numata, Y., Takayama, N., et al. (2013) · Journal of Multivariate Analysis · doi:10.1016/j.jmva.2013.03.011 · Cited by 65
A Compact Fourth-Order Finite Difference Scheme for Unsteady Viscous Incompressible Flows
Li, M., & Tang, T. (2001) · Journal of Scientific Computing · doi:10.1023/a:1011146429794 · Cited by 65
“Combined with (2.3), we have a system of differential-algebraic equations (DAE) which seemingly can be solved by typical numerical techniques for DAEs (see [ 2 ]).”
BackgroundMethodInfluentialIntroduction to Scientific Computing
Holmes, M. H. (2016) · Texts in computational science and engineering · doi:10.1007/978-3-319-30256-0_1 · Cited by 65
Ropp, D. L., & Shadid, J. N. (2009) · Journal of Computational Physics · doi:10.1016/j.jcp.2009.02.001 · Cited by 64
Port-based modeling and control for efficient bipedal walking robots
Duindam, V. (2006) · University of Twente Research Information · Cited by 64
Large-scale CFD simulations of airflow and particle deposition in lung airway
Soni, B., & Aliabadi, S. K. (2013) · Computers & Fluids · doi:10.1016/j.compfluid.2013.06.015 · Cited by 63
A Method of Lines Based on Immersed Finite Elements for Parabolic Moving Interface Problems
Lin, T., Lin, Y., & Zhang, X. (2013) · Advances in Applied Mathematics and Mechanics · doi:10.4208/aamm.13-13s11 · Cited by 63
“The multi-step method is the following fourth order BDF scheme [1]: u= 1 25 ( 48un−36un−1+16un−2−3un−3+12τFn+1 ) .”
MethodInfluentialComputational ecology as an emerging science
Petrovskii, S., & Petrovskaya, N. B. (2012) · Interface Focus · doi:10.1098/rsfs.2011.0083 · Cited by 62
Abramov, R. V., Kovačič, G., & Majda, A. J. (2002) · Communications on Pure and Applied Mathematics · doi:10.1002/cpa.3032 · Cited by 62
“We use the 6th-order Adams-Bashforth linear multistep method [5] for the time stepping in the method.”
MethodAsynchronous contact mechanics
Harmon, D., Vouga, E., Smith, B., et al. (2009) · ACM SIGGRAPH Conference Papers (SIGGRAPH) · doi:10.1145/1576246.1531393 · Cited by 62
“Using only internal damping, high-frequency vibrations introduced by the prescribed particle are quickly damped out, while the curtains’ graceful swinging continues; using a non-geometric integrator such as backward Euler or BDF2 [Ascher and Petzold 1998], the swinging motion would also be damped.”
BackgroundThe Secrets to the Success of the Rush–Larsen Method and its Generalizations
Marsh, M. E., Ziaratgahi, S. T., & Spiteri, R. J. (2012) · IEEE Transactions on Biomedical Engineering · doi:10.1109/tbme.2012.2205575 · Cited by 61
Gimbun, J., Nagy, Z. K., & Rielly, C. D. (2009) · Industrial & Engineering Chemistry Research · doi:10.1021/ie900548s · Cited by 65
Shi, Y., Green, W. H., Wong, H., et al. (2011) · Combustion and Flame · doi:10.1016/j.combustflame.2011.01.024 · Cited by 61
Modeling and control of HVDC grids: A key challenge for the future power system
Beerten, J., Gomis‐Bellmunt, O., Guillaud, X., et al. (2014) · Power Systems Computation Conference · doi:10.1109/pscc.2014.7038505 · Cited by 61
“The application of a variable time step-size is an option [41], though this is a non-trivial feature of the most commonly used commercial software, and often at the cost of an enfeebled numerical stability [42].”
BackgroundA fully coupled air foil bearing model considering friction – Theory & experiment
von Osmanski, S., Larsen, J. S., & Santos, I. F. (2017) · Journal of Sound and Vibration · doi:10.1016/j.jsv.2017.04.008 · Cited by 61
“If this foil model was coupled directly to the shaft and fluid film ODEs, the coupled equation system would obtain a singular Jacobian matrix, meaning that instead of being a system of ODEs, it would belong to the wider class of differential/algebraic equation (DAE) systems [49].”
MethodGoyal, P., & Benner, P. (2022) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.2021.0883 · Cited by 60
YANG, W., & KEE, R. (2002) · Combustion and Flame · doi:10.1016/s0010-2180(02)00382-6 · Cited by 60
A Hybrid Systems Model for Simple Manipulation and Self-Manipulation Systems
Johnson, A. M., Burden, S., & Koditschek, D. E. (2016) · ScholarlyCommons (University of Pennsylvania) · Cited by 60
Arqub, O. A. (2016) · Fundamenta Informaticae · doi:10.3233/fi-2016-1384 · Cited by 59
Akyildiz, F. T., Bellout, H., Vajravelu, K., et al. (2011) · Nonlinear Analysis Real World Applications · doi:10.1016/j.nonrwa.2011.02.017 · Cited by 59
A convergent optimization method using pattern search algorithms with adaptive precision simulation
Wetter, M., & Polak, E. (2004) · Building Services Engineering Research and Technology · doi:10.1191/0143624404bt097oa · Cited by 59
Blajer, W. (2011) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2011.01.007 · Cited by 59
Benchmarking of numerical integration methods for ODE models of biological systems
Städter, P., Schälte, Y., Schmiester, L., Hasenauer, J., & Stapor, P. L. (2021) · Scientific Reports · doi:10.1038/s41598-021-82196-2 · Cited by 58
Dynamical properties of hybrid systems simulators
Sanfelice, R. G., & Teel, A. R. (2009) · Automatica · doi:10.1016/j.automatica.2009.09.026 · Cited by 58
“The theory of numerical simulation for differential equations is well developed and several textbooks on the subject are available, including [32] and [3].”
BackgroundA general-purpose process modelling framework for marine energy systems
Dimopoulos, G. G., Georgopoulou, C. A., Stefanatos, I. C., et al. (2014) · Energy Conversion and Management · doi:10.1016/j.enconman.2014.04.046 · Cited by 58
Solving Differential Equations in R
Soetaert, K., Petzoldt, T., & Woodrow Setzer, R. (2010) · The R Journal · doi:10.32614/rj-2010-013 · Cited by 58
Optimum design of space radiators with temperature-dependent thermal conductivity
Arslantürk, C. (2005) · Applied Thermal Engineering · doi:10.1016/j.applthermaleng.2005.10.038 · Cited by 58
Modeling of Airborne Wind Energy Systems in Natural Coordinates
Gros, S., & Diehl, M. (2013) · Green energy and technology · doi:10.1007/978-3-642-39965-7_10 · Cited by 57
“2) [2].”
Jacobi spectral solution for integral algebraic equations of index-2
Hadizadeh, M., Ghoreishi, F., & Pishbin, S. (2010) · Applied Numerical Mathematics · doi:10.1016/j.apnum.2010.08.009 · Cited by 57
“[1,12,13]).”
Zapata‐González, I., Saldívar‐Guerra, E., & Ortiz‐Cisneros, J. (2011) · Macromolecular Theory and Simulations · doi:10.1002/mats.201100023 · Cited by 56
Single-Event Microkinetic Model for Fischer−Tropsch Synthesis on Iron-Based Catalysts
Lozano‐Blanco, G., Thybaut, J. W., Surla, K., et al. (2008) · Industrial & Engineering Chemistry Research · doi:10.1021/ie071587u · Cited by 56
Option Hedging with Smooth Market Impact
Almgren, R. F., & Li, T. M. (2016) · Market Microstructure and Liquidity · doi:10.1142/s2382626616500027 · Cited by 56
“The action was similar to a forward Euler discretization of an ordinary differential equation [Ascher and Petzold, 1998] which can introduce instability into a stable problem. As an additional example, on Oct. 15, 2014, the US Treasury market underwent the largest intraday move since 2009 [Almgren, 2015]. Devasabai [2…”
MethodInfluentialPETSc/TAO Users Manual (Rev. 3.20)
Balay, S., Abhyankar, S. G., Adams, M. F., et al. (2023) · doi:10.2172/2205494 · Cited by 56
Milano, F. (2021) · IEEE Transactions on Power Systems · doi:10.1109/tpwrs.2021.3107501 · Cited by 55
“The condition (14) implies that the set of DAEs in (12) is assumed to be index 1 [15], which is the form of DAEs that describes most physical systems, including power systems [12].”
BackgroundMonogamy and haplodiploidy act in synergy to promote the evolution of eusociality
Fromhage, L., & Kokko, H. (2011) · Nature Communications · doi:10.1038/ncomms1410 · Cited by 55
How to Get There From Here: Ecological and Economic Dynamics of Ecosystem Service Provision
Sanchirico, J. N., & Springborn, M. (2010) · Environmental and Resource Economics · doi:10.1007/s10640-010-9410-5 · Cited by 59
Direct and Adjoint Sensitivity Analysis of Ordinary Differential Equation Multibody Formulations
Dopico, D., Zhu, Y., Sandu, A., & Sandu, C. (2014) · Journal of Computational and Nonlinear Dynamics · doi:10.1115/1.4026492 · Cited by 54
Kechagiopoulos, P. N., Thybaut, J. W., & Marin, G. B. (2013) · Industrial & Engineering Chemistry Research · doi:10.1021/ie403160s · Cited by 55
Initialization strategies for optimization of dynamic systems
Safdarnejad, S. M., Hedengren, J. D., Lewis, N. R., & Haseltine, E. L. (2015) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2015.04.016 · Cited by 55
Colclasure, A. M., Sanandaji, B. M., Vincent, T. L., et al. (2010) · Journal of Power Sources · doi:10.1016/j.jpowsour.2010.06.074 · Cited by 54
Finite element simulation of rate-dependent magneto-active polymer response
Haldar, K., Kiefer, B., & Menzel, A. (2016) · Smart Materials and Structures · doi:10.1088/0964-1726/25/10/104003 · Cited by 54
A fixed-point iteration approach for multibody dynamics with contact and small friction
Anitescu, M., & Hart, G. D. (2004) · Mathematical Programming · doi:10.1007/s10107-004-0535-6 · Cited by 54
“Projection methods have been successfully used for constraint stabilization for differential algebraic equations, and can be shown to not affect the order of convergence [9].”
MethodNedialkov, N. S., & Jackson, K. R. (2001) · Springer eBooks · doi:10.1007/978-3-7091-6282-8_13 · Cited by 54
A conservative discretization of the Poisson–Nernst–Planck equations on adaptive Cartesian grids
Mirzadeh, M., & Gibou, F. (2014) · Journal of Computational Physics · doi:10.1016/j.jcp.2014.06.039 · Cited by 53
“Implicit–Explicit (IMEX) methods [3,2] are quite popular and provide an effective treatment of time discretizations to problems that have both linear and nonlinear parts.”
MethodA state event detection algorithm for numerically simulating hybrid systems with model singularities
Esposito, J. M., & Kumar, V. R. (2007) · ACM Transactions on Modeling and Computer Simulation · doi:10.1145/1189756.1189757 · Cited by 53
“The error is estimated using Milne s method [Ascher and Petzold 1998] p Err Cxk+1 - xk+1, (16) A State Event Detection Algorithm 11 where C is a known constant which is unique to the particular predictorcorrector method.”
MethodA Second-Order Algorithm for Solving Dynamic Cell Membrane Equations
Sundnes, J., Artebrant, R., Skavhaug, O., et al. (2009) · IEEE Transactions on Biomedical Engineering · doi:10.1109/tbme.2009.2014739 · Cited by 53
“The leftmost column shows results for the explicit midpoint method, which is a two-step RK method of order 2 (see, e.g., [ 2 ]), the middle column is the original Rush–Larsen scheme, and the rightmost column is the second-order method proposed here.”
MethodDigital twins in process engineering: An overview on computational and numerical methods
Peterson, L., Gosea, I. V., Benner, P., & Sundmacher, K. (2024) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2024.108917 · Cited by 79
Nonlinear Path Following Method
Gates, D. J. (2010) · Journal of Guidance Control and Dynamics · doi:10.2514/1.46679 · Cited by 52
“It is possible to solve these equations by numerical methods [23], but they will be transformed into ordinary differential equations (Sec.”
MethodA method for determination of optimal gaits with application to a snake-like serial-link structure
Hicks, G., & Ito, K. (2005) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2005.854583 · Cited by 52
“For a conversational explanation of just how these discretizations and some of the corresponding notations come into existence, see [ 12 ].”
BackgroundQuitian, A., & Ancheyta, J. (2016) · Energy & Fuels · doi:10.1021/acs.energyfuels.5b01953 · Cited by 52
On the modified conjugate gradient method in cloth simulation
Ascher, U. M., & Boxerman, E. (2003) · The Visual Computer · doi:10.1007/s00371-003-0220-4 · Cited by 52
“in the absence of other reasons to make them small) a discretization method with stiff decay [ 2 ] is advantageous, at least for the stretching energy term.”
MethodSelf-citationDelay-differential-algebraic equations in control theory
Shampine, L. F., & Gahinet, P. M. (2005) · Applied Numerical Mathematics · doi:10.1016/j.apnum.2005.04.025 · Cited by 52
Solving differential equations in R
(2013) · Choice Reviews Online · doi:10.5860/choice.50-3321 · Cited by 52
Non-linear analytical modeling of planar compliant mechanisms
Henning, S. W., Linß, S., Gräser, P., et al. (2020) · Mechanism and Machine Theory · doi:10.1016/j.mechmachtheory.2020.104067 · Cited by 51
Gu, B., & Asada, H. H. (2004) · Journal of Dynamic Systems Measurement and Control · doi:10.1115/1.1648307 · Cited by 51
Closed-Loop Control of Variable Stiffness Actuated Robots via Nonlinear Model Predictive Control
Zhakatayev, A., Rubagotti, M., & Varol, H. A. (2015) · IEEE Access · doi:10.1109/access.2015.2418157 · Cited by 51
“Remark 1: In order to numerically solve the OCP, the evolution of the state is obtained by discretizing its continuous evolution with a given number of sub-intervals, and by employing numerical integration algorithms, such as Runge-Kutta methods [32], which are implemented in ACADO.”
MethodMulti-Rate Time Integration for Large Scale Multibody System Models
Arnold, M. (2007) · Springer eBooks · doi:10.1007/978-1-4020-5981-0_1 · Cited by 51
“The stability analysis for classical ODE time integration methods, see, e.g., [ 3 ], may be extended to show that the multi-rate method is stable if all subsystems are integrated by stable time integration methods and the condition”
MethodMulti-Adaptive Galerkin Methods for ODEs I
Logg, A. (2003) · SIAM Journal on Scientific Computing · doi:10.1137/s1064827501389722 · Cited by 51
“For a presentation of the traditional approach to error estimation in ODE codes, we refer to [2], where the following rather pessimistic view is presented: Here we just note that a precise error bound is often unknown and not really needed.”
Imhauser, C. W., Siegler, S., Udupa, J. K., et al. (2008) · Journal of Biomechanics · doi:10.1016/j.jbiomech.2007.12.017 · Cited by 51
Sartori, A., Baroli, D., Cammi, A., et al. (2014) · Annals of Nuclear Energy · doi:10.1016/j.anucene.2014.03.043 · Cited by 50
Invited Article: VEDA: A web-based virtual environment for dynamic atomic force microscopy
Melcher, J. T., Hu, S., & Raman, A. (2008) · Review of Scientific Instruments · doi:10.1063/1.2938864 · Cited by 50
Liwarska-Bizukojć, E., & Gendaszewska, D. (2012) · Journal of Bioscience and Bioengineering · doi:10.1016/j.jbiosc.2012.08.002 · Cited by 50
A PDE Based Approach to Multidomain Partitioning and Quadrilateral Meshing
Kowalski, N., Ledoux, F., & Frey, P. (2012) · Springer eBooks · doi:10.1007/978-3-642-33573-0_9 · Cited by 50
“Then we use the Heun’s method [20], an easy-to-use variation of a Runge-Kutta method, to integrate the ODE Y = f ( X ) over T , starting from X i .”
Methodhp-Adaptive time integration based on the BDF for viscous flows
Hay, A., Étienne, S., Pelletier, D., et al. (2015) · Journal of Computational Physics · doi:10.1016/j.jcp.2015.03.022 · Cited by 50
Acary, V. (2015) · ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik · doi:10.1002/zamm.201400231 · Cited by 50
“Classical approaches to study the stability of numerical scheme for ordinary differential equations through its amplification matrix or the roots of the stability function were also extended with success to the case of differential algebraic equations [4, 12, 28].”
Numerical solution of the bidomain equations
Linge, S. O., Sundnes, J., Hanslien, M., et al. (2009) · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rsta.2008.0306 · Cited by 50
“A time-adaptive scheme with error control was suggested by Rousseau & Kapral (2000), who used the classical technique of step doubling (e.g. Ascher & Petzold 1998) to estimate the local error in a forward Euler scheme and chose the local time step to satisfy a given error tolerance.”
InfluentialDunn, S.-J., Appleton, P. L., Nelson, S. A., Näthke, I. S., Gavaghan, D. J., & Osborne, J. M. (2012) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1002515 · Cited by 48
“The system is evolved in small intervals, and cell positions are updated at each timestep using the Forward Euler method [52].”
MethodOn the benefits of the LDLT factorization for large-scale differential matrix equation solvers
Lang, N., Mena, H., & Saak, J. (2015) · Linear Algebra and its Applications · doi:10.1016/j.laa.2015.04.006 · Cited by 49
Luo, S., Yu, J., Ansari, T. Q., et al. (2020) · Applied Materials Today · doi:10.1016/j.apmt.2020.100882 · Cited by 49
Rathinam, M., & Samad, H. E. (2007) · Journal of Computational Physics · doi:10.1016/j.jcp.2006.10.034 · Cited by 49
“This ensures that the method is consistent in the sense of numerical analysis terminology [26,14], keeping in mind that in general ~cj may depend on the current state b X ðtÞ 1⁄4 x.”
MethodNguyen, T., Tayakout‐Fayolle, M., Ropars, M., et al. (2013) · Chemical Engineering Science · doi:10.1016/j.ces.2013.02.036 · Cited by 48
Adams–Bashforth and Adams–Moulton methods for solving differential Riccati equations
Peinado, J., Ibáñez, J., Arias, E., et al. (2010) · Computers & Mathematics with Applications · doi:10.1016/j.camwa.2010.10.002 · Cited by 48
Banks, J. W., Berger, R. L., Brunner, S., et al. (2011) · Physics of Plasmas · doi:10.1063/1.3577784 · Cited by 48
Time-Domain Simulation of Nonlinear Circuits Through Implicit Runge–Kutta Methods
Maffezzoni, P., Codecasa, L., & D’Amore, D. (2007) · IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications · doi:10.1109/tcsi.2006.887476 · Cited by 48
A reference model for airborne wind energy systems for optimization and control
Malz, E. C., Koenemann, J., Sieberling, S., et al. (2019) · Renewable Energy · doi:10.1016/j.renene.2019.03.111 · Cited by 48
“Finally, λ is the algebraic variable related to that constraint in Lagrange modeling91 [15], and is proportional to the tether tension.92 Since (3) is holonomic, i.e. purely position-dependent, one has to differentiate it twice in order93 to obtain an index-1 DAE [14], i.e. we use94 Ċ = v>p− ll̇ = 0 (4a) C̈ = v̇>p +…”
BackgroundInfluentialInteractions of Microbial-Enhanced Oil Recovery Processes
Li, J., Liu, J., Trefry, M. G., et al. (2010) · Transport in Porous Media · doi:10.1007/s11242-010-9669-6 · Cited by 48
A Roadmap for Optimal Control: The Right Way to Commute
Ross, I. M. (2005) · Annals of the New York Academy of Sciences · doi:10.1196/annals.1370.015 · Cited by 48
3D Euler spirals for 3D curve completion
Harary, G., & Tal, A. (2011) · Computational Geometry · doi:10.1016/j.comgeo.2011.10.001 · Cited by 48
What kinds of dynamics are there? Lie pseudogroups, dynamical systems and geometric integration
McLachlan, R. I., & Quispel, G. R. W. (2001) · Nonlinearity · doi:10.1088/0951-7715/14/6/315 · Cited by 48
A comparison of high-order time integrators for thermal convection in rotating spherical shells
Garcia, F., Net, M., Garcı́a-Archilla, B., et al. (2010) · Journal of Computational Physics · doi:10.1016/j.jcp.2010.07.004 · Cited by 47
Juárez-Abad, J. A., Flores, J. L., Guzmán‐Ramírez, E., et al. (2013) · IEEE Transactions on Industrial Informatics · doi:10.1109/tii.2013.2242085 · Cited by 47
“For this purpose, we have used the Euler method [29], [30].”
MethodA Voltage-Behind-Reactance Induction Machine Model for the EMTP-Type Solution
Wang, L., Jatskevich, J., Wang, C., et al. (2008) · IEEE Transactions on Power Systems · doi:10.1109/tpwrs.2008.926423 · Cited by 47
“At the same time, for better numerical accuracy, it is desirable to have eigenvalues with smaller magnitude [32]‐[ 33 ].”
BackgroundComputational Dynamics of Flapping Wings in Hover Flight: A Co-Simulation Strategy
Roccia, B. A., Preidikman, S., & Balachandran, B. (2017) · AIAA Journal · doi:10.2514/1.j055137 · Cited by 47
Advanced optimization strategies for integrated dynamic process operations
Biegler, L. T. (2017) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2017.10.016 · Cited by 47
“Moreover, sequential approaches are known to fail on unstable dynamic systems [3, 22].”
Solving Degenerate Reaction-Diffusion Equations via Variable Step Peaceman--Rachford Splitting
Cheng, H. M., Lin, P., Sheng, Q., et al. (2003) · SIAM Journal on Scientific Computing · doi:10.1137/s1064827501380691 · Cited by 47
Model reduction for constrained mechanical systems via spectral submanifolds
Li, M., Jain, S., & Haller, G. (2023) · Nonlinear Dynamics · doi:10.1007/s11071-023-08300-5 · Cited by 46
Towards efficient SimRank computation on large networks
Yu, W., Lin, X., & Zhang, W. (2013) · Proceedings - International Conference on Data Engineering · doi:10.1109/icde.2013.6544859 · Cited by 46
“To iteratively compute Ŝ, the conventional way is to use the Euler method [2] for approximating Ŝ(t) at time t = C.”
Method
Metadata
- Authors
- Ascher, U. M., & Petzold, L. R.
- Affiliations
- University of British Columbia (CA)
- University of California, Santa Barbara (US)
- Venue
- Society for Industrial and Applied Mathematics eBooks
- Published
- 1998-01
- OpenAlex
- W1597944220
- Cited by
- 2,403
- References
- 0
- Citations by year
- 2026: 51 · 2025: 109 · 2024: 111 · 2023: 90 · 2022: 109 · 2021: 114 · 2020: 102 · 2019: 93 · 2018: 101 · 2017: 118 · 2016: 105 · 2015: 103 · 2014: 139 · 2013: 122 · 2012: 106
- FWCI
- 5.63
- Type
- book
- Language
- en
- Open access
- closed
- Locations
- Society for Industrial and Applied Mathematics eBooks (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Numerical Analysis › Numerical Solvers for Ordinary Differential Equations
- Physical Sciences › Mathematics › Modeling and Simulation › Mathematical and Computational Methods
- Physical Sciences › Mathematics › Modeling and Simulation › Modeling and Simulation Systems
- Keywords
- ordinary differential equations, differential-algebraic equations, numerical solution of differential equations, initial value problems, boundary value problems, ODE solvers, mathematical software, symplectic methods, Hamiltonian systems
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