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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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Approximate network loading and dual-time-scale dynamic user equilibrium
Friesz, T. L., Kim, T., Kwon, C., et al. (2010) · Transportation Research Part B Methodological · doi:10.1016/j.trb.2010.05.003 · Cited by 114
“(1996), Ascher and Petzold (1998) and Pang and Stewart (2008) for more details regarding DAEs.”
BackgroundArqub, O. A. (2016) · Mathematical Methods in the Applied Sciences · doi:10.1002/mma.3884 · Cited by 110
Method of conditional moments (MCM) for the Chemical Master Equation
Hasenauer, J., Wolf, V., Kazeroonian, A., & Theis, F. J. (2013) · Journal of Mathematical Biology · doi:10.1007/s00285-013-0711-5 · Cited by 100
“As the conditional moment equation is a DAE, the numerical treatment is however non-trivial (Ascher and Petzold 1998).”
BackgroundModeling the effect of insoluble corrosion products on pitting corrosion kinetics of metals
Ansari, T. Q., Luo, J., & Shi, S. (2019) · npj Materials Degradation · doi:10.1038/s41529-019-0090-5 · Cited by 107
James, D. L., & Pai, D. K. (2003) · ACM Transactions on Graphics · doi:10.1145/588272.588278 · Cited by 107
“Implicit integration methods [Ascher and Petzold 1998] (appropriate for numerically stiff equations) have been revived in graphics [Terzopoulos and Fleischer 1988; Hauth and Etzmuß 2001] and successfully applied to off-line cloth animation [Baraff and Witkin 1998].”
MethodAdjoint sensitivity analysis for differential-algebraic equations: algorithms and software
Cao, Y., Li, S., & Petzold, L. R. (2002) · Journal of Computational and Applied Mathematics · doi:10.1016/s0377-0427(02)00528-9 · Cited by 106
Self-citationPredicting crystals of Janus colloids
Vissers, T., Preisler, Z., Smallenburg, F., et al. (2013) · The Journal of Chemical Physics · doi:10.1063/1.4801438 · Cited by 105
Numerical methods for nonlinear equations
Kelley, C. T. (2018) · Acta Numerica · doi:10.1017/s0962492917000113 · Cited by 105
“See Petzold (1983) or Ascher and Petzold (1998) for examples of how this can be used in an initial value problem integration, and Tocci, Kelley and Miller (1997) for an example of the limitations of this idea.”
BackgroundSemi-implicit projection methods for incompressible flow based on spectral deferred corrections
Minion, M. L. (2003) · Applied Numerical Mathematics · doi:10.1016/j.apnum.2003.11.005 · Cited by 105
“[ 3 ]). When applying any ODE method to the solution of PDEs where F (t; ) typically contains spatial derivatives, one must remember that the derivative is an unbounded linear operator and hence the Lipschitz constant for F (t; ) typically goes to innit y as the grid spacing goes to zero.”
MethodA VLSI Analog Computer/Digital Computer Accelerator
Cowan, G. E. R., Melville, R. C., & Tsividis, Y. (2005) · IEEE Journal of Solid-State Circuits · doi:10.1109/jssc.2005.858618 · Cited by 105
“, method of lines [15]) for approximating partial differential equations (PDEs) by a system of ODEs of the above form.”
Kinetic Modeling of Biological Systems
Resat, H., Petzold, L. R., & Pettigrew, M. F. (2009) · Methods in molecular biology · doi:10.1007/978-1-59745-243-4_14 · Cited by 119
Self-citationSanderse, B., & Koren, B. (2011) · Journal of Computational Physics · doi:10.1016/j.jcp.2011.11.028 · Cited by 104
“[8,9,11]), where u plays the role of the differential variable, and p the role of the algebraic variable.”
BackgroundArbitrary order Krylov deferred correction methods for differential algebraic equations
Huang, J., Jia, J., & Minion, M. L. (2006) · Journal of Computational Physics · doi:10.1016/j.jcp.2006.06.040 · Cited by 104
“Finally for this example, note that the KDC method using 9 Radau IIa nodes allows a step-size of 1 (one time step from 0 to 1) for 12 digits of accuracy in y1 and y2, with a total number of 162 function evaluations (compared to more than 1000 for the BDF methods in [3]).”
MethodSensitivity analysis of differential-algebraic equations and partial differential equations
Petzold, L., Li, S., Cao, Y., et al. (2006) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2006.05.015 · Cited by 104
“In [11] we derived the adjoint sensitivity system for DAEs of index [ 3 ] up to two (Hessenberg) and investigated some of its fundamental properties.”
MethodSelf-citationComposable Models for Simulation-Based Design
Paredis, C. J. J., Díaz-Calderón, A., Sinha, R., et al. (2001) · Engineering With Computers · doi:10.1007/pl00007197 · Cited by 104
“However, in our current research, we have applied this framework only to system-level modeling of mechatronic systems [7,8]; that is, modeling of computer-controlled electro-mechanical systems using Differential Algebraic Equations (DAEs) [9] and/or discrete event systems specifications (DEVS) [10].”
MethodContinuous-time point-queue models in dynamic network loading
Ban, X. J., Pang, J., Liu, H. X., et al. (2012) · Transportation Research Part B Methodological · doi:10.1016/j.trb.2011.11.004 · Cited by 103
Numerical methods in vehicle system dynamics: state of the art and current developments
Arnold, M., Burgermeister, B., Führer, C., et al. (2011) · Vehicle System Dynamics · doi:10.1080/00423114.2011.582953 · Cited by 103
“Therefore, all common stepsize control algorithms proceed stepwise and adjust the stepsize h n to keep the local error of the current time step below the user defined error bounds [46,48].”
BackgroundMethodInfluentialAnimating developable surfaces using nonconforming elements
English, E., & Bridson, R. (2008) · ACM Transactions on Graphics · doi:10.1145/1360612.1360665 · Cited by 102
Towards a scalable fully-implicit fully-coupled resistive MHD formulation with stabilized FE methods
Shadid, J. N., Pawlowski, R. P., Banks, J. W., et al. (2010) · Journal of Computational Physics · doi:10.1016/j.jcp.2010.06.018 · Cited by 100
“These methods can be designed with various types of stability properties that allow robust integration of multiple timescale systems without the requirement to resolve the stiff modes of the system (which are not of interest since they do not control the accuracy of time integration [2,3]).”
BackgroundMethodSimulation of the Navier–Stokes equations in three dimensions with a spectral collocation method
Subich, C. J., Lamb, K. G., & Stastna, M. (2013) · International Journal for Numerical Methods in Fluids · doi:10.1002/fld.3788 · Cited by 100
“To adapt this algorithm, it is useful to consider the ̨ and ˇ coefficients as the product of a polynomial interpolant in time [34].”
MethodAutomated generation of 0D and 1D reduced‐order models of patient‐specific blood flow
Pfaller, M. R., Pham, J., Verma, A., et al. (2022) · International Journal for Numerical Methods in Biomedical Engineering · doi:10.1002/cnm.3639 · Cited by 100
“The local building blocks, along with these local equations, are assembled to construct the full 0D model, which yields a global system of differential-algebraic equations governing the entire 0D model [57, 58]:”
BackgroundEstimation of Nonlinear Dynamic Systems : Theory and Applications
Schön, T. B. (2006) · KTH Publication Database DiVA (KTH Royal Institute of Technology) · Cited by 100
Alvim, H. G. O., Lima, T. B., Oliveira, A., et al. (2014) · The Journal of Organic Chemistry · doi:10.1021/jo5001498 · Cited by 99
To agent-based simulation from System Dynamics
Macal, C. M. (2010) · Winter Simulation Conference · doi:10.1109/wsc.2010.5679148 · Cited by 99
“Solving a System Dynamics model is equivalent to solving a set of ordinary differential equations using the forward Euler method in which the step size, .t, is set to one (Ascher and Petzold 1998).”
MethodSliding Motion in Filippov Differential Systems: Theoretical Results and a Computational Approach
Dieci, L., & Lopez, L. (2009) · SIAM Journal on Numerical Analysis · doi:10.1137/080724599 · Cited by 99
Post-stabilization for rigid body simulation with contact and constraints
Cline, M. B., & Pai, D. K. (2004) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/robot.2003.1242171 · Cited by 99
Luo, S., Ansari, T. Q., Yu, J., et al. (2021) · Chemical Engineering Journal · doi:10.1016/j.cej.2021.128476 · Cited by 98
Consistency and Stability of Tau-Leaping Schemes for Chemical Reaction Systems
Rathinam, M., Petzold, L. R., Cao, Y., et al. (2005) · Multiscale Modeling and Simulation · doi:10.1137/040603206 · Cited by 98
“It is well known that in such situations, explicit numerical methods perform very poorly compared to implicit methods [2].”
BackgroundSelf-citation3-D inversion of transient EM data with topography using unstructured tetrahedral grids
Liu, Y., Yin, C., Qiu, C., et al. (2019) · Geophysical Journal International · doi:10.1093/gji/ggz014 · Cited by 97
Analytic Eigensystems for Isotropic Distortion Energies
Smith, B., de Goes, F., & Kim, T. (2019) · ACM Transactions on Graphics · doi:10.1145/3241041 · Cited by 99
“Our production volume simulator is a semi-implicit solver (Baraff and Witkin 1998) that uses a BDF2 time discretization (Ascher and Petzold 1998), combined with continuous collision detection (e.g., see Tang et al. (2010)), and with impulsebased and penalty-based collision resolution (Bridson et al.”
MethodNoda, T., Takenaka, K., & Inoue, T. (2008) · IEEE Transactions on Power Delivery · doi:10.1109/tpwrd.2008.923397 · Cited by 97
“The 2S-DIRK method is A-stable [6], and all of the backward Euler, trapezoidal, and Gear‐Shichman methods are also A-stable [ 2 ].”
BackgroundMethodInfluentialEASY-FIT: a software system for data fitting in dynamical systems
Schittkowski, K. (2002) · Structural and Multidisciplinary Optimization · doi:10.1007/s00158-002-0174-6 · Cited by 97
“Ascher and Petzold [2].”
Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics
Layton, A. T., & Minion, M. L. (2003) · Journal of Computational Physics · doi:10.1016/j.jcp.2003.09.010 · Cited by 97
“In contrast, semi-implicit or implicit-explicit methods can be applied which treat the advection (and possibly diffusion) term explicitly and the reaction (and possibly diffusion) terms implicitly [2,16].”
BackgroundAir charge estimation in turbocharged spark ignition engines
Andersson, P. O. (2005) · Cited by 96
Learning Deep Transformer Models for Machine TranslationPreprint
Wang, Q., Li, B., Xiao, T., et al. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1906.01787 · Cited by 96
“• Inspired by the linear multi-step method in numerical analysis (Ascher and Petzold, 1998), we propose an approach based on dynamic linear combination of layers (D LCL ) to memorizing the features extracted from all preceding layers.”
MethodInfluentialControl of underactuated mechanical systems with servo-constraints
Blajer, W., & Kołodziejczyk, K. (2007) · Nonlinear Dynamics · doi:10.1007/s11071-007-9231-4 · Cited by 95
Modeling Electrochemical Impedance Spectra in SOFC Button Cells with Internal Methane Reforming
Zhu, H., & Kee, R. J. (2006) · Journal of The Electrochemical Society · doi:10.1149/1.2220065 · Cited by 95
Modelling of physical systems for the design and control of mechatronic systems
van Amerongen, J., & Breedveld, P. C. (2003) · Annual Reviews in Control · doi:10.1016/s1367-5788(03)00010-5 · Cited by 95
Zurita-Bustamante, E. W., Flores, J. L., Guzmán‐Ramírez, E., et al. (2011) · IEEE Transactions on Industrial Electronics · doi:10.1109/tie.2011.2123857 · Cited by 95
“In the solution of this problem, we have used the Adams–Bashforth method of the second order [31], [32].”
MethodMukherjee, S. P., & Byrne, H. J. (2012) · Nanomedicine Nanotechnology Biology and Medicine · doi:10.1016/j.nano.2012.05.002 · Cited by 97
A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side
Dieci, L., & Lopez, L. (2012) · Journal of Computational and Applied Mathematics · doi:10.1016/j.cam.2012.02.011 · Cited by 94
“As alternative to projecting onto Σ , in order to remain close to Σ one may modify the system as in x = fF (x) − μh(x)hx(x), x(0) = x̄, (μ is a suitable positive constant) which is a technique used in numerical methods for differential–algebraic equations (see [50]).”
MethodRest-to-Rest Trajectory Planning for Underactuated Cable-Driven Parallel Robots
Idà, E., Brückmann, T., & Carricato, M. (2019) · IEEE Transactions on Robotics · doi:10.1109/tro.2019.2931483 · Cited by 93
“This approach is similar to a classic iterative shooting method [42], which is a state-of-the-art method for the solution of standard BVPs without free parameters.”
BackgroundMethodAn interaction switch predicts the nested architecture of mutualistic networks
Zhang, F., Hui, C., & Terblanche, J. S. (2011) · Ecology Letters · doi:10.1111/j.1461-0248.2011.01647.x · Cited by 95
Volume contact constraints at arbitrary resolution
Allard, J., Fauré, F., Courtecuisse, H., et al. (2010) · ACM Transactions on Graphics · doi:10.1145/1778765.1778819 · Cited by 93
Leal, A. M. M., Kulik, D. A., Smith, W. R., & Saar, M. O. (2017) · Pure and Applied Chemistry · doi:10.1515/pac-2016-1107 · Cited by 92
“The mathematical literature contains a vast number of algorithms for solving systems of both non-stiff and stiff ordinary differential equations [115–118] such as those arising from chemical kinetics (Eq.”
BackgroundDynamic Modeling and Decentralized Control of Supply Chains
Perea-López, E., Grossmann, I. E., Ydstie, B. E., et al. (2001) · Industrial & Engineering Chemistry Research · doi:10.1021/ie000573k · Cited by 92
Design Exploration of Hybrid CMOS and Memristor Circuit by New Modified Nodal Analysis
Fei, W., Yu, H., Zhang, W., & Yeo, K. S. (2011) · IEEE Transactions on Very Large Scale Integration (VLSI) Systems · doi:10.1109/tvlsi.2011.2136443 · Cited by 91
Thermomechanical Effects Induced by Energy Piles Operation in a Small Piled Raft
Salciarini, D., Ronchi, F., Cattoni, E., et al. (2013) · International Journal of Geomechanics · doi:10.1061/(asce)gm.1943-5622.0000375 · Cited by 91
Numerical Methods for Chemical Engineering: Applications in MATLAB
Beers, K. J. (2006) · Cited by 91
Pérez, L. V., & Pilotta, E. A. (2007) · Mathematics and Computers in Simulation · doi:10.1016/j.matcom.2007.03.006 · Cited by 90
Energy-conserving Runge–Kutta methods for the incompressible Navier–Stokes equations
Sanderse, B. (2012) · Journal of Computational Physics · doi:10.1016/j.jcp.2012.07.039 · Cited by 90
“Firstly, in case of very stiff ODEs order reduction occurs if a method is not stiffly accurate [46,51,53].”
Simulation and stability analysis of impacting systems with complete chattering
Nordmark, A., & Piiroinen, P. T. (2009) · Nonlinear Dynamics · doi:10.1007/s11071-008-9463-y · Cited by 90
“There are many different numerical solvers for smooth systems [1, 33], but the question is how to deal with interactions between solutions and discontinuity surfaces.”
BackgroundChen, G., & Raja, L. L. (2004) · Journal of Applied Physics · doi:10.1063/1.1818354 · Cited by 89
Perspectives on the design and control of multiscale systems
Braatz, R. D., Alkire, R. C., Seebauer, E. G., et al. (2005) · Journal of Process Control · doi:10.1016/j.jprocont.2005.06.001 · Cited by 89
Stochastic differential algebraic equations of index 1 and applications in circuit simulation
Winkler, R. (2003) · Journal of Computational and Applied Mathematics · doi:10.1016/s0377-0427(03)00436-9 · Cited by 89
Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator
Kowalczyk, P., & Piiroinen, P. T. (2008) · Physica D Nonlinear Phenomena · doi:10.1016/j.physd.2007.12.007 · Cited by 88
Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems
Polyanin, A. D., & Zaitsev, V. F. (2017) · Cited by 87
Guillot, L., Lazarus, A., Thomas, O., et al. (2020) · Journal of Computational Physics · doi:10.1016/j.jcp.2020.109477 · Cited by 86
Abad, A., Barrio, R., Blesa, F., et al. (2012) · ACM Transactions on Mathematical Software · doi:10.1145/2382585.2382590 · Cited by 86
“Low-order TSMs (order n ≤ 4) have the same stability domain as any explicit Runge-Kutta method of n stages and also order n [Ascher and Petzold 1998], but this is only valid for very low order, and we are interested in high-order TSMs. From Barrio [2005] and Barrio et al.”
MethodInfluentialBlundell, R. W., Horowitz, J. L., & Parey, M. (2012) · Quantitative Economics · doi:10.3982/qe91 · Cited by 83
Controllability of Linear Differential-Algebraic Systems—A Survey
Berger, T., & Reis, T. (2013) · Springer eBooks · doi:10.1007/978-3-642-34928-7_1 · Cited by 85
“DAEs have then been discovered to be the perfect tool for modeling a vast variety of problems in economics [104], demography [35], mechanical systems [7, 29, 55, 62, 120], multibody dynamics [51,62,129,131], electrical networks [7,34,50,99,110], fluid mechanics [7,60,99] and chemical engineering [45,47,48,119], which…”
Greater accuracy and broadened applicability of phase reduction using isostable coordinates
Wilson, D. D., & Ermentrout, B. (2017) · Journal of Mathematical Biology · doi:10.1007/s00285-017-1141-6 · Cited by 82
“These can be calculated, for example, using methods that rely on Newton iteration (Ascher and Petzold 1998).”
MethodSSH-Aerosol v1.1: A Modular Box Model to Simulate the Evolution of Primary and Secondary Aerosols
Sartelet, K., Couvidat, F., Wang, Z., et al. (2020) · Atmosphere · doi:10.3390/atmos11050525 · Cited by 85
Kwok, R., Fung, J. C. H., Lau, A. K. H., et al. (2010) · Journal of Geophysical Research Atmospheres · doi:10.1029/2009jd012809 · Cited by 85
“…option [Byun et al., 1999b; CMAS, 2006], eddy for vertical diffusion, Gear’s chemical solver (smvgear) [Gear, 1971a, 1971b; Jacobson and Turco, 1994; Ascher and Petzold, 1998] for the CBIV mechanism, and the Asymmetric Convection Model (ACM) for in‐cloud aqueous chemistry [Pleim and Chang, 1992;…”
BackgroundSalciarini, D., Ronchi, F., & Tamagnini, C. (2017) · Acta Geotechnica · doi:10.1007/s11440-017-0551-3 · Cited by 85
Re-examination of Synchronous Machine Modeling Techniques for Electromagnetic Transient Simulations
Wang, L., Jatskevich, J., & Dommel, H. W. (2007) · IEEE Transactions on Power Systems · doi:10.1109/tpwrs.2007.901308 · Cited by 84
“Interested readers may find a more detailed analysis of numerical accuracy and system eigenvalues in [34] and [35].”
BackgroundNewman, C., Hansen, G. A., & Gaston, D. R. (2009) · Journal of Nuclear Materials · doi:10.1016/j.jnucmat.2009.03.035 · Cited by 83
“(17)–(19) are integrated in time using the implicit Euler method [21].”
MethodAn overview of the CellML API and its implementation
Miller, A. K., Marsh, J., Reeve, A. M., Garny, A., Britten, R. D., Halstead, M. D. B., Cooper, J., Nickerson, D. P., & Nielsen, P. F. (2010) · BMC Bioinformatics · doi:10.1186/1471-2105-11-178 · Cited by 81
“Systems of differential algebraic equations (DAEs) [1] are one particularly common and useful form of mathematical model.”
BackgroundHu, S., & Raman, A. (2008) · Nanotechnology · doi:10.1088/0957-4484/19/37/375704 · Cited by 82
Mehmood, R., Nadeem, S., & Masood, S. (2015) · Journal of Magnetism and Magnetic Materials · doi:10.1016/j.jmmm.2015.10.102 · Cited by 82
Horowitz, J. L., Blundell, R. W., & Parey, M. (2011) · doi:10.1920/wp.cem.2011.24111 · Cited by 82
“We solve this equation along a grid of points by using Euler’s method (Ascher and Petzold (1998)).”
MethodDeng, Y., Liu, Z., & Wu, Y. (2012) · Structural and Multidisciplinary Optimization · doi:10.1007/s00158-012-0847-8 · Cited by 81
“For the steady flows, the stationary equations are solved using the stationary solver femstatic of Comsol; for the unsteady flows, the transient equations are solved using the backward differentiation formula method by choosing BDF for the time dependent solver femtime of Comsol (Ascher and Petzold 1998).”
MethodOperator-splitting with ISAT to model reacting flow with detailed chemistry
Singer, M. A., Pope, S. B., & Najm, H. N. (2006) · Combustion Theory and Modelling · doi:10.1080/13647830500307501 · Cited by 80
On computational methods for nonlinear estimation
Schön, T. B. (2003) · Cited by 80
Veerapaneni, S., Gueyffier, D., Biros, G., et al. (2009) · Journal of Computational Physics · doi:10.1016/j.jcp.2009.06.020 · Cited by 79
“2 Semi-Implicit Scheme In semi-implicit schemes, the linear part of the stiffest terms is treated implicitly [2].”
BackgroundMethodInfluentialMenzel, A., & Steinmann, P. (2001) · International Journal of Solids and Structures · doi:10.1016/s0020-7683(01)00136-6 · Cited by 79
General method for atomistic spin-lattice dynamics with first-principles accuracy
Hellsvik, J., Thonig, D., Modin, K., Iuşan, D., Bergman, A., Eriksson, O., Bergqvist, L., & Delin, A. (2019) · Physical review. B./Physical review. B · doi:10.1103/physrevb.99.104302 · Cited by 74
Hua, J., Lin, P., Liu, C., & Wang, Q. (2011) · Journal of Computational Physics · doi:10.1016/j.jcp.2011.05.013 · Cited by 80
Port-based modeling of mechatronic systems
Breedveld, P. C. (2004) · Mathematics and Computers in Simulation · doi:10.1016/j.matcom.2003.11.002 · Cited by 78
“equations (ODE) that can be solved by explicit numerical integration schemes [1], see also the course slides athttp://www.”
BackgroundSvärd, C., & Nyberg, M. (2010) · IEEE Transactions on Systems Man and Cybernetics - Part A Systems and Humans · doi:10.1109/tsmca.2010.2049993 · Cited by 78
“DAE system or descriptor system [28]–[30].”
A hybrid systems model for simple manipulation and self-manipulation systems
Johnson, A. M., Burden, S. A., & Koditschek, D. E. (2016) · The International Journal of Robotics Research · doi:10.1177/0278364916639380 · Cited by 77
“The flow in that particular contact mode, FI (equation (75)), is integrated from the state at the initial time as an ordinary differential equation or DAE (Ascher and Petzold, 1998), as appropriate, using the NDSolve command with default parameters. The integration is stopped when the state reaches any guard, GI (equa…”
MethodInfluentialA comparison of implicit time integration methods for nonlinear relaxation and diffusion
Lowrie, R. B. (2003) · Journal of Computational Physics · doi:10.1016/j.jcp.2003.11.016 · Cited by 77
Determination of mass transfer parameters and swelling factor of CO2-oil systems at high pressures
Rezk, M. G., & Foroozesh, J. (2018) · International Journal of Heat and Mass Transfer · doi:10.1016/j.ijheatmasstransfer.2018.05.043 · Cited by 74
Halo orbit mission correction maneuvers using optimal control
Serban, R., Koon, W. S., Lo, M. W., et al. (2002) · Automatica · doi:10.1016/s0005-1098(01)00279-5 · Cited by 76
“where the DAE is index zero, one, or semi-explicit index two (see Ascher and Petzold [1998], or Brenan, Campbell and Petzold [1995]) and the initial conditions have been chosen so that they are consistent (that is, the constraints of the DAE are satisfied).”
BackgroundSelf-citationKee, R. J., Zhu, H., Hildenbrand, B. W., et al. (2013) · Journal of The Electrochemical Society · doi:10.1149/2.016304jes · Cited by 76
Notes on Triangular Sets and Triangulation-Decomposition Algorithms II: Differential Systems
Hubert, É. (2003) · Lecture notes in computer science · doi:10.1007/3-540-45084-x_2 · Cited by 75
“The difficulty of numerical integration has been measured by the differential index and specific numerical schemes can handle low index cases [14, 31, 3].”
MethodModelling and Discretization of Circuit Problems
Günther, M., Feldmann, U., & ter Maten, J. (2005) · Handbook of numerical analysis · doi:10.1016/s1570-8659(04)13006-8 · Cited by 75
“For this, the reader may consult a bunch of excellent textbooks [7, 26, 111] or the survey article [190].”
BackgroundDongyang, C., Abbas, L. K., Guo-ping, W., Xiaoting, R., & Marzocca, P. (2018) · Nonlinear Dynamics · doi:10.1007/s11071-018-4402-z · Cited by 74
“Runge–Kutta fourth-order (RK4) method [41,42] is employed to solve ODE Eqs.”
MethodNumerical Solution of Ordinary Differential Equations
Szymkiewicz, R. (2009) · Water science and technology library · doi:10.1007/978-90-481-3674-2_3 · Cited by 74
A multi-level spectral deferred correction method
Speck, R., Ruprecht, D., Emmett, M., et al. (2014) · BIT Numerical Mathematics · doi:10.1007/s10543-014-0517-x · Cited by 74
Dynamic Optimization of HIPS Open-Loop Unstable Polymerization Reactors
Flores‐Tlacuahuac, A., Biegler, L. T., & Saldívar‐Guerra, E. (2004) · Industrial & Engineering Chemistry Research · doi:10.1021/ie049534p · Cited by 73
Motion planning and control of gantry cranes in cluttered work environment
Blajer, W., & Kołodziejczyk, K. (2007) · IET Control Theory and Applications · doi:10.1049/iet-cta:20060439 · Cited by 73
Efficient Morse Decompositions of Vector Fields
Chen, G., Mischaikow, K., Laramee, R. S., et al. (2008) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2008.33 · Cited by 73
Sivaselvan, M. V., Lavan, O., Dargush, G. F., et al. (2009) · Earthquake Engineering & Structural Dynamics · doi:10.1002/eqe.895 · Cited by 72
“in the algorithm developed here, the analysis increment can be chosen purely based on truncation error control (see, for example, [29]).”
MethodThree-dimensional modeling of IP effects in time-domain electromagnetic data
Marchant, D., Haber, E., & Oldenburg, D. W. (2014) · Geophysics · doi:10.1190/geo2014-0060.1 · Cited by 71
“Similar equations arise in the context of flow that involves chemical reactions (Ascher and Petzold, 1998).”
BackgroundA Coupled A–H Formulation for Magneto-Thermal Transients in High-Temperature Superconducting Magnets
Bortot, L., Auchmann, B., Garcia, I. C., et al. (2020) · IEEE Transactions on Applied Superconductivity · doi:10.1109/tasc.2020.2969476 · Cited by 71
“he normalized current density components are given respect to a) J x, b) J y and c) J z. scheme with the magnetoquasistatic problem, using an adaptive time-stepping method based on the fifth-order BDF [34]. The formulation is validated by comparison with reference results. In detail, the full 3D formulation proposed in…”
MethodWalters, K., Dean, A. M., Zhu, H., et al. (2003) · Journal of Power Sources · doi:10.1016/s0378-7753(03)00536-6 · Cited by 71
“In either case, the DASSL [18,19] software is used to solve the”
MethodPseudospectral method of solution of the Fitzhugh–Nagumo equation
Olmos‐Liceaga, D., & Shizgal, B. D. (2009) · Mathematics and Computers in Simulation · doi:10.1016/j.matcom.2009.01.001 · Cited by 71
Modeling of Induction Machines Using a Voltage-Behind-Reactance Formulation
Wang, L., Jatskevich, J., & Pekarek, S. D. (2008) · IEEE Transactions on Energy Conversion · doi:10.1109/tec.2008.918601 · Cited by 70
“In general, the negative real part of the eigenvalues is desirable for better damping, whereas positive eigenvalues tend to increase the local errors [ 39 ].”
BackgroundModel-Based Formulation of Biofuel Blends by Simultaneous Product and Pathway Design
Dahmen, M., & Marquardt, W. (2017) · Energy & Fuels · doi:10.1021/acs.energyfuels.7b00118 · Cited by 69
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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