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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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Garcia-Mollá, V. M., Liberos, A., Vidal, A. M., et al. (2013) · Computers in Biology and Medicine · doi:10.1016/j.compbiomed.2013.10.023 · Cited by 46
“More detailed descriptions can be found in [9,20].”
BackgroundMethodInfluentialCardoso-Ribeiro, F. L., Matignon, D., & Pommier‐Budinger, V. (2017) · Journal of Fluids and Structures · doi:10.1016/j.jfluidstructs.2016.12.007 · Cited by 46
“Kunkel, P., Mehrmann, V., 2006.”
BackgroundInfluentialMethods for suspensions of passive and active filaments
Schoeller, S. F., Townsend, A. K., Westwood, T. A., & Keaveny, E. E. (2020) · Journal of Computational Physics · doi:10.1016/j.jcp.2020.109846 · Cited by 50
“We show in Appendix A.4 that re-centring this yields the simpler second-order BDF scheme uj+1n = 1 3u j n + 2 3∆t dexp −1 uj+1n ( Ωj+1n ) , (60) for the Lie algebra element of segment n. Combining the Lie algebra element update with that for the positions, as well as the constraints linking the segment orientations an…”
MethodInfluentialA New Class of Highly Accurate Solvers for Ordinary Differential Equations
Gläser, A., & Rokhlin, V. (2008) · Journal of Scientific Computing · doi:10.1007/s10915-008-9245-1 · Cited by 46
“As a consequence, a broad variety of such algorithms exists and their construction and analysis are a well-established fields (see, for example, [10, 11, 14, 2, 15]).”
BackgroundRecursive Kinematics and Inverse Dynamics for a Planar 3R Parallel Manipulator
Khan, W. A., Krovi, V., Saha, S. K., et al. (2004) · Journal of Dynamic Systems Measurement and Control · doi:10.1115/1.2098890 · Cited by 46
Improved DAE formulation for inverse dynamics simulation of cranes
Blajer, W., & Kołodziejczyk, K. (2010) · Multibody System Dynamics · doi:10.1007/s11044-010-9227-6 · Cited by 45
“An important characteristic of a DAE system is its index, which is a measure of singularity of DAEs and indicates ‘how far’ the DAE system is from an equivalent set of ODEs (ordinary differential-equations) [ 20 ].”
BackgroundA unified approach to the power law and the critical state modeling of superconductors in 2D
Morandi, A., & Fabbri, M. (2014) · Superconductor Science and Technology · doi:10.1088/0953-2048/28/2/024004 · Cited by 45
Flexible Nonhomogeneous Markov Models for Panel Observed Data
Titman, A. (2011) · Biometrics · doi:10.1111/j.1541-0420.2010.01550.x · Cited by 45
Modal Locomotion: Animating Virtual Characters with Natural Vibrations
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A Three-Spring Pseudorigid-Body Model for Soft Joints With Significant Elongation Effects
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On the numerical solution of differential-algebraic equations by Padé series
Çeli̇k, E., & Bayram, M. (2002) · Applied Mathematics and Computation · doi:10.1016/s0096-3003(02)00093-0 · Cited by 45
Piskorz, W., Zasada, F., Stelmachowski, P., et al. (2011) · The Journal of Physical Chemistry C · doi:10.1021/jp2070826 · Cited by 45
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On the Determination of Joint Reactions in Multibody Mechanisms
Blajer, W. (2004) · Journal of Mechanical Design · doi:10.1115/1.1667944 · Cited by 44
Symbolic computation of differential equivalences
Cardelli, L., Tribastone, M., Tschaikowski, M., & Vandin, A. (2016) · ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL) · doi:10.1145/2837614.2837649 · Cited by 43
“[3].”
Cappellini, N., Tamarozzi, T., Blockmans, B., et al. (2017) · Meccanica · doi:10.1007/s11012-017-0710-5 · Cited by 44
Qureshi, S., & Ramos, H. (2018) · International Journal of Nonlinear Sciences and Numerical Simulation · doi:10.1515/ijnsns-2017-0267 · Cited by 44
“On the other hand, in virtually, all modern codes for ODEs, the step-size is selected automatically to achieve reliability and efficiency [2].”
BackgroundLugon, L., Sartelet, K., Kim, Y., et al. (2020) · Atmospheric chemistry and physics · doi:10.5194/acp-20-7717-2020 · Cited by 44
“Note that as in the regional-scale model, chemistry processes are solved with the Rosenbrock algorithm using time steps that may be smaller than the splitting time step defined by the ETR algorithm.”
MethodFlood risk estimation under the compound influence of rainfall and tide
Jang, J., & Chang, T.-H. (2022) · Journal of Hydrology · doi:10.1016/j.jhydrol.2022.127446 · Cited by 44
Im, H. G., Raja, L. L., Kee, R. J., et al. (2000) · Combustion Science and Technology · doi:10.1080/00102200008947340 · Cited by 44
“Such a system can be further reduced to an index-one system by a simple variable substitution (Ascher and Petzold, 1998).”
BackgroundMethodInfluentialSelf-citationRigorous dynamic modeling and simulation of wine distillations
Osorio, D., Pérez‐Correa, J. R., Belancic, A., et al. (2003) · Food Control · doi:10.1016/j.foodcont.2003.08.003 · Cited by 44
Simulation platform for pattern recognition based on reservoir computing with memristor networks
Tanaka, G., & Nakane, R. (2022) · Scientific Reports · doi:10.1038/s41598-022-13687-z · Cited by 44
Pyomo.DOE : An open‐source package for model‐based design of experiments in Python
Wang, J., & Dowling, A. W. (2022) · AIChE Journal · doi:10.1002/aic.17813 · Cited by 43
Crockatt, M., Christlieb, A. J., Garrett, C., et al. (2017) · Journal of Computational Physics · doi:10.1016/j.jcp.2017.06.017 · Cited by 43
Order‐Optimal Preconditioners for Implicit Runge–Kutta Schemes Applied to Parabolic PDEs
Mardal, K., Nilssen, T. K., & Staff, G. A. (2007) · SIAM Journal on Scientific Computing · doi:10.1137/05064093x · Cited by 43
Numerical Solution of Differential Algebraic Equations
Thomsen, P. G., & Bendtsen, C. (1999) · Cited by 43
On differential-algebraic control systems
Berger, T. (2013) · Common Library Network (Der Gemeinsame Bibliotheksverbund) · Cited by 43
Index Concepts for Differential-Algebraic Equations
Mehrmann, V. (2015) · Springer eBooks · doi:10.1007/978-3-540-70529-1_120 · Cited by 43
Malenica, L., Gotovac, H., Kamber, G., et al. (2018) · Water · doi:10.3390/w10121787 · Cited by 42
“This can be done by using a large number of methods developed for the numerical integration of ODEs and differential algebraic equations (DAEs) [52].”
MethodSmoothness driven frame field generation for hexahedral meshing
Kowalski, N., Ledoux, F., & Frey, P. (2015) · Computer-Aided Design · doi:10.1016/j.cad.2015.06.009 · Cited by 42
Hydromagnetic stagnation point flow of a viscous fluid over a stretching or shrinking sheet
Gorder, R. A. V., Vajravelu, K., & Pop, I. V. (2010) · Meccanica · doi:10.1007/s11012-010-9402-0 · Cited by 42
“2) by using the boundary value problem solver in Maple 11 (see, for instance, [52, 53]).”
MethodBallestra, L. V., Pacelli, G., & Radi, D. (2016) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2016.04.008 · Cited by 42
Shooting-projection method for two-point boundary value problems
Filipov, S. M., Gospodinov, I., & Faragó, I. (2017) · Applied Mathematics Letters · doi:10.1016/j.aml.2017.04.002 · Cited by 42
3D time-domain airborne EM forward modeling with topography
Yin, C., Qi, Y., Liu, Y., et al. (2016) · Journal of Applied Geophysics · doi:10.1016/j.jappgeo.2016.08.002 · Cited by 42
Global Solution of Optimization Problems with Parameter-Embedded Linear Dynamic Systems
Singer, A. B., & Barton, P. I. (2004) · Journal of Optimization Theory and Applications · doi:10.1023/b:jota.0000037606.79050.a7 · Cited by 42
Constructing and maintaining proper process models
Preisig, H. A. (2010) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2010.02.023 · Cited by 42
Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations
Linh, V. H., & Mehrmann, V. (2009) · Journal of Dynamics and Differential Equations · doi:10.1007/s10884-009-9128-7 · Cited by 42
“Advantages of the discrete algorithm are a simpler implementation and that existing DAE solvers for strangeness-free problems like BDF or implicit Runge-Kutta methods, see [3, 9, 41, 50] can be used.”
MethodSolving boundary value problems in the open source software R: package bvpSolve
Mazzia, F., Cash, J. R., & Soetaert, K. (2014) · Opuscula Mathematica · doi:10.7494/opmath.2014.34.2.387 · Cited by 42
“In order to obtain a more precise feeling for the concept of a stable boundary value problem, we refer the reader to [32] and [3].”
BackgroundPLUME-MoM 1.0: A new integral model of volcanic plumes based on the method of moments
Vitturi, M. D. M., Neri, A., & Barsotti, S. (2015) · Geoscientific model development · doi:10.5194/gmd-8-2447-2015 · Cited by 41
“4 Numerical scheme The plume rise equations are solved with a predictor-corrector Heun’s scheme (Petzold and Ascher, 1998) that guarantees a second–order accuracy, keeping the execution time in the order of seconds.”
MethodComparison of numerical methods for modeling laser mode locking with saturable gain
Wang, S., Docherty, A., Marks, B. S., et al. (2013) · Journal of the Optical Society of America B · doi:10.1364/josab.30.003064 · Cited by 41
“A good overview of many of these methods is presented by Ascher and co-workers [23,24].”
Role of Free Radicals in Modeling the Iodide–Peroxide Reaction Mechanism
Pagnacco, M. C., & Stanisavljev, D. R. (2012) · The Journal of Physical Chemistry A · doi:10.1021/jp303732u · Cited by 41
Guo, R., Xia, Y., & Xu, Y. (2017) · Journal of Computational Physics · doi:10.1016/j.jcp.2017.02.059 · Cited by 41
“For non-stiff ODEs, there exist extremely effective high order explicit methods [1,4,21], such as Runge–Kutta, linear multi-step and predictor corrector methods.”
BackgroundMethodHigh-order time integration applied to metal powder plasticity
Hartmann, S., & Bier, W. (2007) · International Journal of Plasticity · doi:10.1016/j.ijplas.2007.01.014 · Cited by 41
Computational Fluid Dynamics Simulation of Catalytic Reactors
Deutschmann, O. (2008) · Handbook of Heterogeneous Catalysis · doi:10.1002/9783527610044.hetcat0097 · Cited by 41
“For the underlying theory of the numerical solution of DAE systems and software implementation it is referred to the textbook by Ascher and Petzold [104] .”
MethodSpeed it up: How temperature drives toxicokinetics of organic contaminants in freshwater amphipods
Raths, J. O., Švara, V., Lauper, B. B., Fu, Q., & Hollender, J. (2022) · Global Change Biology · doi:10.1111/gcb.16542 · Cited by 40
“Arrhenius temperatures were (1) dC tissue, p ( t ) (3) (5) 1 ) was solved numerically using Heun's method (Ascher & Petzold, 1998).”
MethodSimulation Methods in Systems Biology
Gillespie, D. T. (2008) · Springer eBooks · doi:10.1007/978-3-540-68894-5_5 · Cited by 40
On the choice of the demand and hydraulic modeling approach to WDN real‐time simulation
Creaco, E. F., Pezzinga, G., & Savic, D. A. (2017) · Water Resources Research · doi:10.1002/2016wr020104 · Cited by 40
“In order to obtain a second-order accuracy, these values can be assessed through Heun’s predictor/corrector method [Ascher and Petzold, 1998].”
MethodGuaranteed globally injective 3D deformation processing
Fang, Y., Li, M., Jiang, C., et al. (2021) · ACM Transactions on Graphics · doi:10.1145/3450626.3459757 · Cited by 40
Efficient integration over discontinuities for differential-algebraic systems
Mao, G., & Petzold, L. R. (2002) · Computers & Mathematics with Applications · doi:10.1016/s0898-1221(01)00272-3 · Cited by 40
Self-citationReduced dimensional computational models of polymer electrolyte membrane fuel cell stacks
Chang, P., Kim, G.-S., Promislow, K., & Wetton, B. T. R. (2006) · Journal of Computational Physics · doi:10.1016/j.jcp.2006.10.011 · Cited by 41
“The DAE is of index 1, so except for the point raised below, can be considered to add no real additional difficulty to the computation [1].”
BackgroundComparison of Data Assimilation Techniques for a Coupled Model of Surface and Subsurface Flow
Camporese, M., Paniconi, C., Putti, M., et al. (2009) · Vadose Zone Journal · doi:10.2136/vzj2009.0018 · Cited by 39
“On the other hand, increasing the relaxation time G may result in an improved subsurface storage retrieval, but at the cost of a much larger computational eff ort (Paniconi et al., 2003; Hurkmans et al., 2006) unless more appropriate time integrators are used (Ascher and Petzold, 1998).”
BackgroundShamloo, A., Ebrahimi, S., Amani, A. M., et al. (2020) · Scientific Reports · doi:10.1038/s41598-020-62410-3 · Cited by 39
Recent advances in dynamic modeling of HVAC equipment. Part 2: Modelica-based modeling
Li, P., Li, Y., Seem, J. E., et al. (2013) · Science and Technology for the Built Environment · doi:10.1080/10789669.2013.836876 · Cited by 38
“Generally, for numerical simulation of thermo-fluid systems that often include sets of DAEs, achieving consistent initialization can be difficult (Ascher and Petzold 1998).”
BackgroundStudying radon exhalation rates variability from phosphogypsum piles in the SW of Spain
Coto, I. L., Mas, J. L., Vargas, A., et al. (2014) · Journal of Hazardous Materials · doi:10.1016/j.jhazmat.2014.07.025 · Cited by 38
Jiang, Y. (2004) · IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications · doi:10.1109/tcsi.2004.834503 · Cited by 38
“Now, we present an assumption to ensure (1) is index one (see also [1] and [2]).”
BackgroundPotential landscape of high dimensional nonlinear stochastic dynamics with large noise
Tang, Y., Yuan, R., Wang, G., Zhu, X., & Ao, P. (2017) · Scientific Reports · doi:10.1038/s41598-017-15889-2 · Cited by 36
Vibration-minimizing motion retargeting for robotic characters
Hoshyari, S., Xu, H., Knoop, E., et al. (2019) · ACM Transactions on Graphics · doi:10.1145/3306346.3323034 · Cited by 38
“While a Backward Difference Formula (BDF) discretization of either the semi-implicit, index-1 or the pure, index-2 DAE is stable and would fulfill our requirements, the index-2 is preferable as it does not require second derivatives of our constraints for simulation, or third derivatives for gradient computations for…”
MethodInfluentialAn Efficient Numerical Method for Studying Interfacial Motion in Two-Dimensional Creeping Flows
Kropinski, M.-C. (2001) · Journal of Computational Physics · doi:10.1006/jcph.2001.6787 · Cited by 38
LS-SVM approximate solution to linear time varying descriptor systems
Mehrkanoon, S., & Suykens, J. A. K. (2012) · Automatica · doi:10.1016/j.automatica.2012.06.095 · Cited by 38
“Suykens}@esat.kuleuven.be have been developed for solving DAEs using backward differentiation formulas (BDF) [1] [7] [11] [2] [3] or implicit Runge- Kutta (IRK) methods [7] [3] [4] [19].”
BackgroundMethodSimulation of the Internal Conditions During The Hot-Pressing Process
Zombort, B. G., Kamke, F. A., & Watson, L. T. (2003) · Wood and Fiber Science (Society of Wood Science and Technology) · Cited by 38
NAIS-Net: Stable Deep Networks from Non-Autonomous Differential EquationsPreprint
Ciccone, M., Gallieri, M., Masci, J., Osendorfer, C., & Gomez, F. J. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1804.07209 · Cited by 40
Chapariha, M., Therrien, F., Jatskevich, J., et al. (2013) · IEEE Transactions on Energy Conversion · doi:10.1109/tec.2013.2284774 · Cited by 37
Coupled analysis for response and instability of sloping seabed under wave action
Rafiei, A., Rahman, M. S., & Gabr, M. A. (2019) · Applied Ocean Research · doi:10.1016/j.apor.2019.04.017 · Cited by 37
icepack: a new glacier flow modeling package in Python, version 1.0
Shapero, D., Badgeley, J. A., Hoffman, A. O., et al. (2021) · Geoscientific model development · doi:10.5194/gmd-14-4593-2021 · Cited by 37
A perspective on nonlinear model predictive control
Biegler, L. T. (2021) · Korean Journal of Chemical Engineering · doi:10.1007/s11814-021-0791-7 · Cited by 37
On the numerical solution of stiff systems
Güzel, N., & Bayram, M. (2005) · Applied Mathematics and Computation · doi:10.1016/j.amc.2004.11.035 · Cited by 37
A Decoupled Time-Domain Simulation Method via Invariant Subspace Partition for Power System Analysis
Yang, D., & Ajjarapu, V. (2006) · IEEE Transactions on Power Systems · doi:10.1109/tpwrs.2005.860937 · Cited by 37
Uddin, M. F., Saak, J., & Benner, P. (2016) · Numerical Algebra Control and Optimization · doi:10.3934/naco.2016.6.1 · Cited by 37
“The spatial discretization of such equations using finite difference or finite element methods produces large-scale structured index-2 descriptor systems [24, 5].”
MethodNested direct transcription optimization for singular optimal control problems
Chen, W., & Biegler, L. T. (2016) · AIChE Journal · doi:10.1002/aic.15272 · Cited by 37
Dias, R. C. S., & Costa, M. R. P. F. N. (2003) · Macromolecules · doi:10.1021/ma035030b · Cited by 37
Raja, L. L., Kee, R. J., & Petzold, L. R. (1998) · Symposium (International) on Combustion · doi:10.1016/s0082-0784(98)80074-x · Cited by 37
“It is possible and beneficial to reduce the system to index one by replacing K with a new dependent variable f, where K [ ]f/]t [23].”
BackgroundSelf-citationDynamic optimization for the core-flooding problem in reservoir engineering
Kameswaran, S., Biegler, L. T., & Staus, G. H. (2005) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2005.02.038 · Cited by 36
A VLSI analog computer/math co-processor for a digital computer
Cowan, G. E. R., Melville, R. C., & Tsividis, Y. (2005) · doi:10.1109/isscc.2005.1493879 · Cited by 36
“a[12] a[13] col sel[0] col sel[1] col sel[2] col sel[3] 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 l 0 0 0 1 Table A.”
MethodSteady-state laminar flow solutions for NACA 0012 airfoil
Swanson, R. C., & Langer, S. (2015) · Computers & Fluids · doi:10.1016/j.compfluid.2015.11.009 · Cited by 36
Piskorz, W., Zasada, F., Stelmachowski, P., et al. (2013) · The Journal of Physical Chemistry C · doi:10.1021/jp405459g · Cited by 36
Dynamic Reduced Order Models for Simulating Bubbling Fluidized Bed Adsorbers
Yu, M., Miller, D. C., & Biegler, L. T. (2015) · Industrial & Engineering Chemistry Research · doi:10.1021/acs.iecr.5b01270 · Cited by 36
Multi-resolution adaptive modeling of groundwater flow and transport problems
Gotovac, H., Andričević, R., & Gotovac, B. (2006) · Advances in Water Resources · doi:10.1016/j.advwatres.2006.10.007 · Cited by 36
Modular and Recursive Kinematics and Dynamics for Parallel Manipulators
Khan, W. A., Krovi, V., Saha, S. K., et al. (2005) · Multibody System Dynamics · doi:10.1007/s11044-005-1143-9 · Cited by 36
“The solution of such systems of index-3 DAEs by direct discretization is not possible using explicit finite difference discretization methods [7].”
BackgroundMethodCao, H.-L., de Perre, G. V., Kennedy, J., Senft, E., Esteban, P. G., Beir, A. D., Simut, R., Belpaeme, T., Lefeber, D., & Vanderborght, B. (2018) · IEEE Transactions on Cognitive and Developmental Systems · doi:10.1109/tcds.2018.2795343 · Cited by 37
“The joint angles are then calculated using the Runge-Kutte algorithm [71] by the following equation [72]: where x d is the desired end effector pose, J † A ( q ) is the Moorse-Penrose pseudo inverse of the analytical jacobian J A ( q ) , x e the current end effector pose, and K a positive definite…”
Verrier, P., Waters, T., & Sieber, J. (2014) · Celestial Mechanics and Dynamical Astronomy · doi:10.1007/s10569-014-9575-2 · Cited by 35
“Collocation methods are computationally more expensive but more robust when computing periodic orbits with rapidly varying properties (such as stability or shape) along the trajectory (see e.g. Ascher and Petzold 1998).”
Dynamic wave propagation in infinite saturated porous media half spaces
Heider, Y., Markert, B. A., & Ehlers, W. (2011) · Computational Mechanics · doi:10.1007/s00466-011-0647-9 · Cited by 35
“, [3,37,53] among others.”
Stiffness Analysis of Cardiac Electrophysiological Models
Spiteri, R. J., & Dean, R. C. (2010) · Annals of Biomedical Engineering · doi:10.1007/s10439-010-0100-9 · Cited by 35
“This is similar to the definition of stiffness used by Ascher and Petzold [4].”
ResultNew directions for nonlinear process optimization
Biegler, L. T. (2018) · Current Opinion in Chemical Engineering · doi:10.1016/j.coche.2018.02.008 · Cited by 35
Shamloo, A., Amani, A. M., Forouzandehmehr, M., & Ghoytasi, I. (2019) · International Journal of Pharmaceutics · doi:10.1016/j.ijpharm.2018.12.088 · Cited by 35
Gu, X., Zhao, Y.-L., Zhao, X.-L., et al. (2021) · Numerical Mathematics Theory Methods and Applications · doi:10.4208/nmtma.oa-2020-0020 · Cited by 35
“It is natural that the condition number depends linearly on all main properties of the system: number of time steps, time interval length, and the norm of the Jacobian matrix [24, 25, 37].”
BackgroundMethodState-of-the-Art Railway Vehicle Design with Multi-Body Simulation
WEIDEMANN, C. (2010) · Journal of Mechanical Systems for Transportation and Logistics · doi:10.1299/jmtl.3.12 · Cited by 35
Parachute-Payload System Flight Dynamics and Trajectory Simulation
Guglieri, G. (2012) · International Journal of Aerospace Engineering · doi:10.1155/2012/182907 · Cited by 35
“The solver used in the simulation program is DASSL (differential algebraic system solver) originally developed by Petzold [24–26] for the solution of systems of differential algebraic equations (DAEs).”
MethodAn efficient backward Euler time-integration method for nonlinear dynamic analysis of structures
Liu, T., Zhao, C., Li, Q., et al. (2012) · Computers & Structures · doi:10.1016/j.compstruc.2012.03.019 · Cited by 35
Á, Đ. Q., & Hoang, M. T. (2019) · International Journal of Computer Mathematics · doi:10.1080/00207160.2019.1677895 · Cited by 35
“If τ ∗ is very small (e.g. for stiff problems [4, 17]) then τ 2 opt is also very large.”
InfluentialNonsmooth differential-algebraic equations in chemical engineering
Stechlinski, P. G., Patrascu, M. S., & Barton, P. I. (2017) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2017.10.031 · Cited by 35
Robust, stable time-domain methods for solving MPDEs of fast/slow systems
Mei, T., Roychowdhury, J., Coffey, T. S., et al. (2005) · IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems · doi:10.1109/tcad.2004.841073 · Cited by 35
Solutions of the Boussinesq equation subject to a nonlinear Robin boundary condition
Μουτσόπουλος, Κ. Ν. (2012) · Water Resources Research · doi:10.1029/2012wr012221 · Cited by 35
“…Since (31a)–(31c) and (32a)–(32c) involve time derivatives of the variables c0 and cp, but no derivatives of the wetting front length LF, (31a)–(31c) and (32a)–(32c) are systems of differential algebraic equations [Ascher and Petzold, 1998], they were solved using the NDSolve Mathematica routine.”
MethodSoetaert, K., Cash, J. R., & Mazzia, F. (2012) · Springer eBooks · doi:10.1007/978-3-642-28070-2_10 · Cited by 35
Raja, L. L., Kee, R. J., Serban, R., et al. (2000) · Journal of The Electrochemical Society · doi:10.1149/1.1393595 · Cited by 35
Self-citationPseudospectral collocation methods for the direct transcription of optimal control problems
Pietz, J. (2003) · Rice Research Repository (Rice University) · Cited by 35
Multiphysics Simulations: Challenges and Opportunities
USDOE Office of Science (SC), Keyes, D., McInnes, L., Woodward, C., Gropp, W., Myra, E., & Pernice, M. (2012) · OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) · doi:10.2172/1034263 · Cited by 38
Multi-Resolution Modeling of Power Electronics Circuits Using Model-Order Reduction Techniques
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“Hence, a larger sampling time ∆tk requires an appropriate discretization ∆z. Applying the second-order half-explicit Runge-Kutta method (Ascher and Petzold, 1998), the discrete-time system can be expressed by a predictor step X1,k+1 = x1,k +∆tkfk (x1,k,x2,k,mk, vs,k) (12a) 0 = ak…”
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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