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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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An alternative for predicting real-time water levels of urban drainage systems
Huang, P., & Lee, K. T. (2023) · Journal of Environmental Management · doi:10.1016/j.jenvman.2023.119099 · Cited by 26
Rigid body simulation with contact and constraints
Cline, M. B. (2009) · doi:10.14288/1.0051676 · Cited by 26
Elastica of sandwich panels with a transversely flexible core—A high-order theory approach
Frostig, Y. (2008) · International Journal of Solids and Structures · doi:10.1016/j.ijsolstr.2008.05.007 · Cited by 26
“The elastica response is described ahead through the numerical solution of the non-linear set of differential equations that can be solved using numerical schemes such as the multiple-shooting points method, see Stoer and Bulirsch (1980), or the finite-difference (FD) approach using trapezoid or mid-point methods with…”
MethodModeling and Simulations of Ferroresonance by Using BDF/NDF Numerical Methods
Tokić, A., & Smajić, J. (2014) · IEEE Transactions on Power Delivery · doi:10.1109/tpwrd.2014.2346766 · Cited by 26
“The methods offer a higher level of accuracy and they are and stable, with stability angles of 90 , 90 , 86 , 73 , and 51 , respectively [31], [32].”
MethodInfluentialConstant-Parameter Voltage-Behind-Reactance Induction Machine Model Including Main Flux Saturation
Therrien, F., Chapariha, M., & Jatskevich, J. (2014) · IEEE Transactions on Energy Conversion · doi:10.1109/tec.2014.2342258 · Cited by 26
“(56) The error introduced by the BDF approximations is thus proportional to the step size Δt for BDF1 and Δt2 for BDF2, which also proves the consistency [31] of these approaches.”
BackgroundMethodResultInfluentialDynamic modeling in the simulation, optimization, and control of bipedal and quadrupedal robots
Hardt, M., & von Stryk, O. (2003) · ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik · doi:10.1002/zamm.200310068 · Cited by 26
“This approach, also known as coordinate partitioning [1], projects the dynamics (1) onto a reduced set of independent states thus converting the DAE contact system (1) into an ODE system of minimal size.”
MethodMultidisciplinary Design Optimization of Dynamic Engineering Systems
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Numerical solution of differential–algebraic equations with index-2
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Landry, C., Caboussat, A., & Hairer, E. (2009) · SIAM Journal on Scientific Computing · doi:10.1137/080740611 · Cited by 26
“Efficient techniques to solve DAE systems relying on implicit Runge-Kutta methods have been developed in [4, 15, 16].”
MethodMoeckel, R. B. (2014) · Scholarpedia · doi:10.4249/scholarpedia.10667 · Cited by 26
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“fr/) with a BDF time-stepping method (Ascher and Petzold 1988; Curtiss and Hirschfelder 1952) and a tolerance termination technique.”
MethodMulti-Phase-Field Model of Intergranular Corrosion Kinetics in Sensitized Metallic Materials
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Adaptive step-size approach for Simpson’s-type block methods with time efficiency and order stars
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de Perre, G. V., Damme, M. V., Lefeber, D., & Vanderborght, B. (2015) · Advanced Robotics · doi:10.1080/01691864.2015.1031697 · Cited by 24
“The desired joint angles q are then calculated by integrating q̇ with the Runge–Kutta algorithm.[33] Since the complete configuration has the same rotational possibili-”
MethodElectro-thermo-chemical computational models for 3D heterogeneous semiconductor device simulation
Mauri, A. G., Sacco, R., & Verri, M. (2014) · Applied Mathematical Modelling · doi:10.1016/j.apm.2014.12.008 · Cited by 25
“, [4, 23, 1]) will be considered in a future extension of the computational scheme proposed in the present article.”
Li, P., Li, Y., & Seem, J. E. (2011) · Journal of Building Performance Simulation · doi:10.1080/19401493.2010.547949 · Cited by 25
“For numerical simulation of thermo-fluid systems which often include sets of differential algebraic equations (DAEs), consistent initialization is often difficult to obtain (Ascher and Petzold 1998).”
BackgroundGros, S., Srinivasan, B., Chachuat, B., et al. (2009) · International Journal of Control · doi:10.1080/00207170802460024 · Cited by 25
Characterization and boriding kinetics of AISI T1 steel
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“This resulting DAE system can be solved numerically [40] if the diffusion coefficient of boron in Fe2B is known.”
BackgroundStartup Operation of Middle-Vessel Batch Distillation Column: Modeling and Simulation
Gruetzmann, S., & Fieg, G. (2008) · Industrial & Engineering Chemistry Research · doi:10.1021/ie070667v · Cited by 25
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“Eventually, we get a system of so-called semi-explicit differential-algebraic equations (DAE) [35] ∂n ∂t = λn fn(n,R) + ξn(t) 0 = G(n,R; t) (44) In this particular form of DAE one can distinguish between the differential variable n(t) and the algebraic variable R(t).”
BackgroundInfluentialViel, A. (2014) · Linköping electronic conference proceedings · doi:10.3384/ecp14096213 · Cited by 24
“• Reference implementation : transient simulation of the continously coupled system, using the LSODA variable-step solver [2]”
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On A -stable one-leg methods for solving nonlinear Volterra functional differential equations
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Evolution of the L1 halo family in the radial solar sail CRTBP
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“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).”
Particles in coronary circulation: A review on modelling for drug carrier design
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“The same scheme, applied to DAEs (18), results in (qu)n+1 − (qu)n − t (vu)n+1 = 0 (vu)n+1 − (vu)n − t b(qn+1, vn+1, un+1) = 0 η̄(qn+1, vn+1, un+1, tn+1) = 0 γ(qn+1, vn+1, tn+1) = 0 φ(qn+1, tn+1) = 0 (20) The rough computational schemes are of acceptable accuracy for appropriately small values of t , and the accuracy c…”
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Review of explicit Falkner methods and its modifications for solving special second-order I.V.P.s
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Lyapunov Analysis of Sample-and-Hold Hybrid Feedbacks
Sanfelice, R. G., & Teel, A. R. (2006) · IEEE Conference on Decision and Control (CDC) · doi:10.1109/cdc.2006.377831 · Cited by 24
“[20], [ 1 ]) and in the construction of approximate models for discrete-time systems (see e.g.”
BackgroundComputational Fluid Dynamics of Catalytic Reactors
Janardhanan, V. M., & Deutschmann, O. (2011) · doi:10.1002/9783527639878.ch8 · Cited by 24
“For the underlying theory of the numerical solution of DAE systems and software implementation it is referred to the textbook by Ascher and Petzold ([60]).”
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“To obtain discrete-time dynamics of the form (1a), the equations of motion (2) are discretized using the Runge-Kutta fourth-order method RK4 [32].”
MethodAWEbox: An Optimal Control Framework for Single- and Multi-Aircraft Airborne Wind Energy Systems
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“This representation does not allow for the deployment of classical integration methods within the optimal control problem [38].”
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“is found by solving the equations involving the states of the system at both the current and next time step [13], unrealistically damps the system.”
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Krämer, S., Rothe, S., & Hartmann, S. (2013) · Engineering With Computers · doi:10.1007/s00366-013-0337-7 · Cited by 23
“In order to solve DAE-systems there are a number of schemes proposed in [4, 15, 16, 54].”
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Symbolic integration of dynamical systems by collocation methods
Meliopoulos, A. P. S., Cokkinides, G. J., & Stefopoulos, G. K. (2005) · IEEE Power Engineering Society General Meeting · doi:10.1109/pes.2005.1489565 · Cited by 23
“Note that Equations (3) correspond one-to-one with the external system states while Equations ( 4 ) correspond one-”
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An adaptive-step time integration method applied to transient magnetic field problems
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Computational Hydraulics: Numerical Methods and Modelling
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Index reduction by minimal extension for the inverse dynamics simulation of cranes
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Thermionic graphene/silicon Schottky infrared photodetectors
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Investigating mass transfer around spatially-decoupled electrolytic bubbles
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“Moreover, the Backward Differentiation Formula (BDF) solver was used for calculating the time step [69].”
MethodComparison of 4 numerical solvers for stiff and hybrid systems simulation
Liu, L., Felgner, F., & Frey, G. (2010) · IEEE Conference on Emerging Technologies & Factory Automation (ETFA) · doi:10.1109/etfa.2010.5641330 · Cited by 22
“Moreover, this system is a so-called high-index DAE problem; it must undergo a reduction algorithm reducing the index to 1 or 0 before, the problem is accessible to common DAE or ODE solvers [2].”
BackgroundGorder, R. A. V., Prasad, K. V., & Vajravelu, K. (2012) · Heat and Mass Transfer · doi:10.1007/s00231-012-0973-2 · Cited by 22
“, [36] and [37] for details on the numerical routine), or another relevant boundary value problem solver.”
BackgroundPishbin, S. (2014) · Journal of Computational and Applied Mathematics · doi:10.1016/j.cam.2014.11.012 · Cited by 22
Fast All-Pairs SimRank Assessment on Large Graphs and Bipartite Domains
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“To iteratively compute ^ S , the conventional way is to use the Euler method [2] for approximating ^ S ð t Þ at time t ¼ C .”
MethodDouble SIB points in differential-algebraic systems
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“Differential-algebraic equations (DAEs) have deserved much recent attention, coming from problems in fields such as controls, nonlinear circuit theory, mechanics, power systems, or chemical processes, among others [ 1 ], [5], [6], [9]‐[11], [14], [15].”
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On the Uncertainty of a Mathematical Model for Drying of a Wood Particle
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“(2016) that the index 3 equations of motion they use to evaluate similar models of friction appear to be sti (Ascher and Petzold, 1998).”
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“numerically using the method such as Runge–Kutta algorithm [23], [24] or using the time-domain simulation software such as PSCAD or electromagnetic transients program.”
MethodOPTIMAL PERFORMANCE OF THE HEAT-SHOCK GENE REGULATORY NETWORK
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“Therefore, the DAE system is of index one (Ascher and Petzold, 1998) and is solvable by Backward Differentiation Formulas as implemented in specialized software packages such as DASSL (Petzold, 1983).”
MethodSelf-citationModeling of underactuated mechanical systems in partly specified motion
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“Such inversions are computationally very costly and so efficient solvers rely on reuse of intermediate calculation results [25].”
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ODE Transformer: An Ordinary Differential Equation-Inspired Model for Sequence Generation
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Numerical solutions of chemical differential-algebraic equations
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“This technique has been studied and successfully applied to practical mechanical models by several researchers [12,22,1,13, 20,5,7,9,10].”
MethodNakshatrala, K. B., Nagarajan, H., & Shabouei, M. (2016) · Communications in Computational Physics · doi:10.4208/cicp.180615.280815a · Cited by 21
“, see references [6, 27, 28]).”
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Iterative solver techniques in fast dynamic calculations of power systems
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A fixed time-step approach for multibody dynamics with contact and friction
Anitescu, M. (2004) · IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) · doi:10.1109/iros.2003.1249734 · Cited by 21
“If the simulation has only joint constraints, then the problem is a differential algebraic equation (DAE) [1], [2], which is a widely studied and used computational paradigm.”
Background
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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