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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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A numerical analysis of variational valuation techniques for derivative securities
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Sanfelice, R. G. (2007) · Cited by 21
Optimization of Compliant Path-Generating Mechanisms Based on Non-linear Analytical Modeling
Henning, S. W., Linß, S., Gräser, P., et al. (2020) · Mechanisms and machine science · doi:10.1007/978-3-030-61652-6_3 · Cited by 20
Stiffness in numerical initial-value problems: A and L-stability of numerical methods
Fazio, R. (2001) · International Journal of Mathematical Education in Science and Technology · doi:10.1080/002073901753124619 · Cited by 20
Dynamically consistent nonstandard finite difference schemes for a virus-patch dynamic model
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Electromagnetic Simulation of No-Insulation Coils Using H – $\phi$ Thin Shell Approximation
Schnaubelt, E., Woźniak, M., Schöps, S., et al. (2023) · IEEE Transactions on Applied Superconductivity · doi:10.1109/tasc.2023.3258905 · Cited by 20
Investigation of turbo-molecular pumps in strong magnetic fields
Wolf, J., Bornschein, B., Drexlin, G., et al. (2011) · Vacuum · doi:10.1016/j.vacuum.2011.07.063 · Cited by 20
Stochastic growth reduces population fluctuations inDaphnia–algal systems
Ananthasubramaniam, B., Nisbet, R. M., Nelson, W. A., et al. (2010) · Ecology · doi:10.1890/09-2346.1 · Cited by 20
3D Euler spirals for 3D curve completion
Harary, G., & Tal, A. (2010) · Annual Symposium on Computational Geometry (SoCG) · doi:10.1145/1810959.1811025 · Cited by 20
Antonini, G., & Romano, D. (2015) · IET Microwaves Antennas & Propagation · doi:10.1049/iet-map.2014.0149 · Cited by 20
Bohl exponent for time-varying linear differential-algebraic equations
Berger, T. (2012) · International Journal of Control · doi:10.1080/00207179.2012.688872 · Cited by 20
“…to be the perfect tool for modelling a vast variety of problems, e.g. in mechanical engineering (Führer and Leimkuhler 1991; Petzold 1992; Ascher and Petzold 1998), multibody dynamics (Simeon, Führer, and Rentrop 1991; Eich-Soellner and Führer 1998), electrical networks (Newcomb 1981;…”
Tkachuk, A., Wohlmuth, B. I., & Bischoff, M. (2013) · International Journal for Numerical Methods in Engineering · doi:10.1002/nme.4457 · Cited by 20
“Standard spatial discretization of dynamic contact problems using Lagrange multipliers leads to systems of differential algebraic equations (DAE) with differential index 3 [1] M R UC f .”
BackgroundMethodGas Turbine Modeling for Diagnosis and Control
Larsson, E., Åslund, J., Frisk, E., & Eriksson, L. (2014) · Journal of Engineering for Gas Turbines and Power · doi:10.1115/1.4026598 · Cited by 20
Boundary-Value Problems for Differential-Algebraic Equations: A Survey
Lamour, R., März, R., & Weinmüller, E. B. (2015) · Differential-algebraic equations forum · doi:10.1007/978-3-319-22428-2_4 · Cited by 20
“, in [12, 9, 51].”
Integral equation methods for vesicle electrohydrodynamics in three dimensions
Veerapaneni, S. (2016) · Journal of Computational Physics · doi:10.1016/j.jcp.2016.08.052 · Cited by 20
“The semi-implicit scheme can be generalized to achieve high-order accuracy in time via backward difference formulas [2] or spectral-deferred correction methods [34].”
MethodAutomatically Generating SystemC Code from HCSP Formal Models
Yan, G., Jiao, L., Wang, S., et al. (2020) · ACM Transactions on Software Engineering and Methodology · doi:10.1145/3360002 · Cited by 20
“, invertible), and then F (ṡ, s) = 0 becomes an implicit ODE system rather than Differential-Algebraic Equations whose Jacobian matrix is singular [14].”
BackgroundMarquis, D., Pavageau, M., Guillaume, É., et al. (2012) · Fire and Materials · doi:10.1002/fam.2136 · Cited by 20
“The choice was made for the backward differentiation formula method [38], a variation of the numerical differentiation formula.”
MethodAn evaluation of the FCT method for high-speed flows on structured overlapping grids
Banks, J. W., Henshaw, W. D., & Shadid, J. N. (2009) · Journal of Computational Physics · doi:10.1016/j.jcp.2009.04.033 · Cited by 20
“Detail concerning the implementation of these time integrators can be found for example in [29,25].”
BackgroundModeling microbial-mediated reduction in batch reactors
Mohamed, M. M., & Hatfield, K. (2005) · Chemosphere · doi:10.1016/j.chemosphere.2004.12.013 · Cited by 20
Stochastic DAEs in Circuit Simulation
Römisch, W., & Winkler, R. (2003) · Birkhäuser Basel eBooks · doi:10.1007/978-3-0348-8065-7_19 · Cited by 20
“If a DAE is of higher index, the constraints are not locally solvable for the algebraic compo nents, and there exist solution components that are determined only by a hidden differentiation step, which may cause serious difficulties in the numerical solution of such problems (see, e.g., [1, 11]).”
Convergence of a Class of Semi-Implicit Time-Stepping Schemes for Nonsmooth Rigid Multibody Dynamics
Gavrea, B., Anitescu, M., & Potra, F. A. (2008) · SIAM Journal on Optimization · doi:10.1137/060675745 · Cited by 20
“We note that the same reasons have prompted the search for methods that are specific to differential algebraic, as opposed to ordinary differential, equations [7].”
BackgroundConvolution complementarity problems with application to impact problems
Stewart, D. E. (2005) · IMA Journal of Applied Mathematics · doi:10.1093/imamat/hxh087 · Cited by 20
“Note that the DAE’s related to LCS’s can have arbitrarily high index [3, Chap.”
BackgroundInfluentialAn algorithm for solving DAEs with mechanization
Wang, W. (2004) · Applied Mathematics and Computation · doi:10.1016/j.amc.2004.08.010 · Cited by 20
Reduction and Approximation in Large and Infinite Potential-driven Flow Networks
Mayes, J. (2022) · Figshare · doi:10.7274/ng451g07r7z · Cited by 20
Development and application of a transient analysis code for heat pipe cooled reactor systems
Guo, Z., Liu, L., Ziyin, L., et al. (2024) · Nuclear Engineering and Design · doi:10.1016/j.nucengdes.2024.112979 · Cited by 19
Bounded-error optimal experimental design via global solution of constrained min–max program
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A fast time domain solver for the equilibrium Dyson equation
Kaye, J., & Strand, H. U. R. (2023) · Advances in Computational Mathematics · doi:10.1007/s10444-023-10067-7 · Cited by 24
An approach for fast cascading failure simulation in dynamic models of power systems
Gharebaghi, S., Chaudhuri, N. R., He, T., et al. (2022) · Applied Energy · doi:10.1016/j.apenergy.2022.120534 · Cited by 19
Trajectory optimization for autonomous mobile robots in ITER
Vale, A., Fonte, D., Valente, F., et al. (2014) · Robotics and Autonomous Systems · doi:10.1016/j.robot.2014.01.007 · Cited by 19
“The simpler Euler integration method tends to be numerically unstable and less accurate for small rates [12], meaning that the path may oscillate widely and not reaching a stable configuration.”
MethodAn implicit discretization scheme for linear-quadratic control problems with bang–bang solutions
Alt, W., & Seydenschwanz, M. (2013) · Optimization methods & software · doi:10.1080/10556788.2013.821612 · Cited by 19
“This is an implicit method which is known to be A-stable [6].”
MethodTorabi, M., & Yaghoobi, H. (2013) · Arabian Journal for Science and Engineering · doi:10.1007/s13369-013-0636-8 · Cited by 19
Gashi, B., & Pantelous, A. A. (2015) · Probabilistic Engineering Mechanics · doi:10.1016/j.probengmech.2015.02.003 · Cited by 19
On simplified handling of state events in time-domain simulation
Fabozzi, D., Chieh, A. S., Panciatici, P., et al. (2011) · ORBi (University of Liège) · Cited by 19
An adaptive step size GPU ODE solver for simulating the electric cardiac activity
García, V. M., Liberos, A., Climent, A. M., et al. (2011) · Computing in cardiology · Cited by 19
Licitra, G., Sieberling, S., Engelen, S., et al. (2016) · European Control Conference (ECC) · doi:10.1109/ecc.2016.7810515 · Cited by 19
Generic method for generating blended gestures and affective functional behaviors for social robots
de Perre, G. V., Cao, H.-L., Beir, A. D., Esteban, P. G., Lefeber, D., & Vanderborght, B. (2017) · Autonomous Robots · doi:10.1007/s10514-017-9650-0 · Cited by 19
“Using the Runge–Kutta algorithm (Ascher and Petzold 1998), for every block, the angle values q are obtained from their derivatives ˙ q , calculated by the following algorithm (Sciavicco 2009): Here, x d is the desired end effector pose.”
MethodA Second-Order Advection-Reflection Solver
Narain, R., Zehnder, J., & Thomaszewski, B. (2019) · Proceedings of the ACM on Computer Graphics and Interactive Techniques · doi:10.1145/3340257 · Cited by 19
“Projected implicit midpoint is a projected Gauss-Legendre method and hence second-order accurate in time [Ascher and Petzold 1998, Ch. 10.1.5].”
MethodInfluentialMultibody Dynamics on Differentiable Manifolds
Haug, E. J. (2021) · Journal of Computational and Nonlinear Dynamics · doi:10.1115/1.4049995 · Cited by 19
Guichard, R., Tanière, A., Belut, E., & Rimbert, N. (2014) · International Journal of Multiphase Flow · doi:10.1016/j.ijmultiphaseflow.2014.05.003 · Cited by 19
Zhang, L., & Zhang, D. (2016) · Nonlinear Dynamics · doi:10.1007/s11071-016-2682-8 · Cited by 19
On wrinkling of a sandwich panel with a compliant core and self-equilibrated loads
Frostig, Y. (2011) · Journal of Sandwich Structures & Materials · doi:10.1177/1099636211419131 · Cited by 19
“It uses trapezoid or mid-point methods with Richardson extrapolation or deferred corrections [15], as implemented in Maple [16], along with parametric or arc-length continuation methods [17].”
MethodHuang, X., Jézéquel, L., Besset, S., et al. (2018) · Journal of Sound and Vibration · doi:10.1016/j.jsv.2018.07.027 · Cited by 19
Matrix Equation Techniques for Certain Evolutionary Partial Differential Equations
Palitta, D. (2021) · Archivio istituzionale della ricerca (Alma Mater Studiorum Università di Bologna) · doi:10.1007/s10915-021-01515-x · Cited by 19
PETSc Users Manual (Revision 3.15)
Balay, S., Abhyankar, S. G., Adams, M. F., et al. (2021) · doi:10.2172/1814627 · Cited by 19
Immersed finite element method of lines for moving interface problems with nonhomogeneous flux jump
Lin, T., Lin, Y., & Zhang, X. (2013) · Contemporary mathematics - American Mathematical Society · doi:10.1090/conm/586/11633 · Cited by 19
A review of bushing modelling approaches for MultiBody simulations
Adduci, R., Vermaut, M., Perrelli, M., et al. (2023) · Mechanism and Machine Theory · doi:10.1016/j.mechmachtheory.2023.105496 · Cited by 18
Hoang, M. T., & Ehrhardt, M. (2025) · SeMA Journal · doi:10.1007/s40324-025-00404-9 · Cited by 21
“This property can be easily verified using well-known theorems on the positivity of ODEs [228, Lemma 1], [444].”
InfluentialPeinado, J., Ibáñez, J., Hernández, V., et al. (2010) · Procedia Computer Science · doi:10.1016/j.procs.2010.04.290 · Cited by 18
“coefficient strategies [2] by using an interpolating polynomial Pk which interpolates the points (tk,Xk−r), (tk+1,Xk−r+1), .”
MethodAndrés‐Martínez, O., & Ricardez‐Sandoval, L. (2021) · Journal of Process Control · doi:10.1016/j.jprocont.2021.08.017 · Cited by 18
Hazra, S., Mukherjee, S. P., Ravi, S., Sardar, S., & Adhikari, S. (2022) · ChemPhysChem · doi:10.1002/cphc.202200482 · Cited by 19
Dong, S., Otsuka, K., & Makihara, K. (2022) · Journal of Sound and Vibration · doi:10.1016/j.jsv.2022.117535 · Cited by 18
Diffusion-based generation of Histopathological Whole Slide Images at a Gigapixel scale
Harb, R., Pock, T., & Müller, H. (2024) · IEEE Winter Conference on Applications of Computer Vision · doi:10.1109/wacv57701.2024.00505 · Cited by 23
Distilling Diffusion Models Into Conditional GANs
Kang, M., Zhang, R., Barnes, C., Paris, S., Kwak, S., Park, J., Shechtman, E., Zhu, J.-Y., & Park, T. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-73390-1_25 · Cited by 24
Bıyık, E., Brooks, J. D., Sehgal, H., et al. (2015) · Proceedings of the ... American Control Conference/Proceedings of the American Control Conference · doi:10.1109/acc.2015.7170975 · Cited by 18
A numerical method for coupled surface and grain boundary motion
Pan, Z., & Wetton, B. T. R. (2008) · European Journal of Applied Mathematics · doi:10.1017/s0956792508007390 · Cited by 18
Robust residual selection for fault detection
Khorasgani, H., Jung, D., Biswas, G., et al. (2014) · Proceedings of the IEEE Conference on Decision & Control, including the Symposium on Adaptive Processes · doi:10.1109/cdc.2014.7040291 · Cited by 18
“With respect to previous work, a contribution here is a measure for quantifying the performance of residuals in different operating regions by using sensitivity analysis [12].”
BackgroundMethodHigh-order ALE schemes for incompressible capillary flows
Montefuscolo, F., de Sousa, F. S., & Buscaglia, G. C. (2014) · Journal of Computational Physics · doi:10.1016/j.jcp.2014.08.030 · Cited by 18
“, [19, 20]) 39 are not necessarily the most appropriate methods for the considered problem.”
BackgroundBulatov, M. V., Lima, P. M., & Weinmüller, E. B. (2013) · Open Mathematics · doi:10.2478/s11533-013-0334-5 · Cited by 18
“We illustrate the difference between the above situation and the case where detK(t, t) ̸= 0,∀ t ∈ [0, 1], in the example following Theorem 3.”
BackgroundMethodInfluentialA computationally efficient algorithm for fitting ion channel parameters
Teed, Z. R., & Silva, J. R. (2016) · MethodsX · doi:10.1016/j.mex.2016.11.001 · Cited by 18
“The solution to this ODE can be approximated by using standard solvers such as Runge-Kutta methods [12].”
MethodLinearizing Discrete-Time Hybrid Systems
Alimguzhin, V., Mari, F., Melatti, I., Salvo, I., & Tronci, E. (2017) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2017.2694559 · Cited by 17
“[6]).”
Prediction of the deflagration index for organic dusts as a function of the mean particle diameter
Fumagalli, A., Derudi, M., Rota, R., et al. (2017) · Journal of Loss Prevention in the Process Industries · doi:10.1016/j.jlp.2017.08.015 · Cited by 18
Wang, P., Deng, N., & Zhang, X. (2018) · IEEE Power and Energy Technology Systems Journal · doi:10.1109/jpets.2018.2842041 · Cited by 18
A Synthesis Model With Intuitive Control Capabilities for Rolling Sounds
Conan, S., Derrien, O., Aramaki, M., et al. (2014) · IEEE/ACM Transactions on Audio Speech and Language Processing · doi:10.1109/taslp.2014.2327297 · Cited by 18
“The equations are discretized and solved using an explicit fourth-order RungeKutta method (RK4) [41], which is known to be an accurate method for solving differential equations.”
MethodInfluentialState Constrained Optimal Control Applied to Supervisory Control in HEVs
Pérez, L. V., & García, G. O. (2010) · Oil & Gas Science and Technology – Revue d’IFP Energies nouvelles · doi:10.2516/ogst/2009040 · Cited by 18
“Finally, the problem can be solved in its continuous form using the optimality conditions of the Pontryagin’s Maximum Principle and just performing the discretization to solve the resulting equations at the final stage, using software for differential-algebraic equations (Ascher and Petzold, 1998).”
MethodMultigrid second-order accurate solution of parabolic control-constrained problems
González-Andrade, S., & Borzı̀, A. (2010) · Computational Optimization and Applications · doi:10.1007/s10589-010-9358-y · Cited by 18
“For a detailed discussion of the BDF and CN schemes, see [ 2 , 17]; for higher-order space discretization of optimality systems, we refer to [7].”
BackgroundMethodThe Use of Servo-Constraints in the Inverse Dynamics Analysis of Underactuated Multibody Systems
Blajer, W. (2013) · Journal of Computational and Nonlinear Dynamics · doi:10.1115/1.4025855 · Cited by 18
Marías, F., Benzaoui, A., Vaxelaire, J., et al. (2015) · Energy & Fuels · doi:10.1021/acs.energyfuels.5b01109 · Cited by 18
Verified High-Order Integration of DAEs and Higher-Order ODEs
Hoefkens, J., Berz, M., & Makino, K. (2001) · Springer eBooks · doi:10.1007/978-1-4757-6484-0_23 · Cited by 18
“Since the method can also determine the index (Ascher and Petzold, 1998) and a scheme for transforming DAEs into implicit ODEs, it can be used to compute Taylor model enclosures of the solutions of DAEs.”
MethodControllable velocity projection for constraint stabilization in multibody dynamics
Orden, J. C. G., & Martín, S. C. (2011) · Nonlinear Dynamics · doi:10.1007/s11071-011-0224-y · Cited by 18
Computational performance of aggregated distillation models
Linhart, A., & Skogestad, S. (2008) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2008.09.014 · Cited by 18
Scalable Differential Analysis of Large Process Algebra Models
Tribastone, M. (2010) · International Conference on the Quantitative Evaluation of Systems (QEST) · doi:10.1109/qest.2010.45 · Cited by 18
“The time-course evolution of the system is obtained from the solution of an initial value problem, for which well-known robust and efficient numerical techniques are available [1].”
MethodReachability analysis and deterministic global optimization of differential-algebraic systems
Scott, J. K. (2012) · DSpace@MIT (Massachusetts Institute of Technology) · Cited by 18
Berry, R. A., Peterson, J. W., Zhang, H., et al. (2018) · doi:10.2172/1498781 · Cited by 18
Solutions of the Wheeler-Feynman equations with discontinuous velocities
de Souza, D. C., & Luca, J. D. (2015) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/1.4905201 · Cited by 17
Baghani, M., Heydarzadeh, N., & Roozbahani, M. M. (2016) · International Journal of Applied Mechanics · doi:10.1142/s1758825116500204 · Cited by 17
Guerrero-Nancuante, C., & Manríquez, R. (2020) · Medwave · doi:10.5867/medwave.2020.04.7898 · Cited by 17
Ulitin, N. V., Kharlampidi, K. E., Tereshchenko, K. A., et al. (2021) · Molecular Catalysis · doi:10.1016/j.mcat.2021.111886 · Cited by 17
Performance of implicit A-stable time integration methods for multibody system dynamics
Zhang, H., Zhang, R., Zanoni, A., et al. (2022) · Multibody System Dynamics · doi:10.1007/s11044-021-09806-9 · Cited by 17
Monroy‐Alonso, A., Ordaz‐Quintero, A., Ramírez, J. C., et al. (2021) · Polymers · doi:10.3390/polym14010160 · Cited by 17
“The equations were solved with a code written in FORTRAN using the routine DDASSL (Differential-Algebraic System Solver) [31] which is a Gear type method based on backward differentiation formulas of variable step–variable order for solving systems of differential-algebraic equations.”
MethodHoang, M. T. (2022) · Mathematics and Computers in Simulation · doi:10.1016/j.matcom.2022.04.003 · Cited by 17
Multiobjective Design Optimization of Lightweight Gears
Cosco, F., Adduci, R., Muzzi, L., et al. (2022) · Machines · doi:10.3390/machines10090779 · Cited by 17
“Lastly, the overall MB problem was fully defined, and the assembled EOMs of Equation (3) were solved with respect to the generalized coordinates and accelerations by means of a nonlinear iterative solver [38].”
MethodConstant-Parameter Circuit-Based Models of Synchronous Machines
Chapariha, M., Therrien, F., Jatskevich, J., et al. (2015) · IEEE Transactions on Energy Conversion · doi:10.1109/tec.2014.2385099 · Cited by 17
“Stiff systems generally require implicit solvers and/or small time-steps to ensure numerical stability [20], which undesirably increases the simulation time.”
BackgroundGorder, R. A. V., & Vajravelu, K. (2010) · Applied Mathematics and Computation · doi:10.1016/j.amc.2010.12.063 · Cited by 17
A Parallel Solver for Incompressible Fluid Flows
Wang, Y., Baboulin, M., Dongarra, J. J., et al. (2013) · Procedia Computer Science · doi:10.1016/j.procs.2013.05.207 · Cited by 17
ON THE NUMERICAL CONTINUATION OF ISOLAS OF EQUILIBRIA
Avitabile, D., Desroches, M., & Rodrigues, S. (2012) · International Journal of Bifurcation and Chaos · doi:10.1142/s021812741250277x · Cited by 17
“The computation of an isola can then be seen as an implicit Differential-Algebraic problem [2].”
MethodOrdinary Differential Equation-Based Deadlock Detection
Ding, Z., Zhou, M., & Wang, S. (2014) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2014.2311757 · Cited by 17
Mass flow estimation with model bias correction for a turbocharged Diesel engine
Polóni, T., Rohal’-Ilkiv, B., & Johansen, T. A. (2013) · Control Engineering Practice · doi:10.1016/j.conengprac.2013.10.011 · Cited by 17
“(A.7), which is the fixed step explicit Heun’s method (Runge-Kutta method) (Ascher and Petzold, 1998) x̃i+1 = xi + hfc(xi, ui) (A.12) xi+1 = xi + h 2 (fc(xi, ui) + fc(x̃i, ui)) (A.13) where i represents the numerical step index and h is the numerical step size.”
MethodSoftware for modeling and analysis of linear systems with delays
Gahinet, P. M., & Shampine, L. F. (2004) · Proceedings of the ... American Control Conference/Proceedings of the American Control Conference · doi:10.23919/acc.2004.1384746 · Cited by 17
“As explained in the introductory texts [2], [12], we still have much to learn about how to solve numerically DAEs and DDEs by themselves, much less in combination.”
BackgroundThree-Dimensional Aircraft Conflict Resolution Based on Smoothing Methods
Chen, W., Chen, J., Shao, Z., et al. (2016) · Journal of Guidance Control and Dynamics · doi:10.2514/1.g001726 · Cited by 17
“In addition, Radau collocation exhibits the advantage wherein the element endpoint is a collocation point, although the algebraic precision of Radau collocation is one order lower than Gauss collocation [25–27].”
BackgroundSoneson, J. E. (2017) · IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control · doi:10.1109/tuffc.2017.2654125 · Cited by 17
“Similarly, more advanced techniques such as the Radau-collocated implicit Runge–Kutta methods [31], which extend the favorable damping properties of implicit Euler to O( ζ 2) or better accuracy, require the solution of two or more linear systems.”
Collins, J. B., Estep, D., & Tavener, S. (2014) · BIT Numerical Mathematics · doi:10.1007/s10543-014-0534-9 · Cited by 17
“For example, the midpoint quadrature rule yields the explicit midpoint method [18] or explicit approximation to the implicit midpoint method,”
New Method for Fast Transient Simulation of Large Linear Circuits Using High-Order Stable Methods
Farhan, M. A., Gad, E., Nakhla, M. S., et al. (2013) · IEEE Transactions on Components Packaging and Manufacturing Technology · doi:10.1109/tcpmt.2012.2233545 · Cited by 17
“Also, maintaining stability at s = ∞ in the complex plane (L-stability [8]) is a crucial requirement for handling stiff circuits.”
BackgroundPreservation of local dynamics when applying central difference methods: application to SIR model
Roeger, L.-I. W., & Barnard, R. (2007) · The Journal of Difference Equations and Applications · doi:10.1080/10236190601079134 · Cited by 17
“Standard textbooks in numerical methods mention little about the local stability of the central-difference method (see for example, [1,3]).”
BackgroundLinh, V. H., Mehrmann, V., & Vleck, E. S. V. (2010) · Advances in Computational Mathematics · doi:10.1007/s10444-010-9156-1 · Cited by 17
Li, Z., & Shi, C. ‐. R. (2006) · IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems · doi:10.1109/tcad.2005.855943 · Cited by 17
“For numerical integration, several strategies have been proposed on reformulating the backward differentiation formulae (BDF) [2], [19] to keep the leading coefficient constant since it is the leading coefficient that contributes to a Jacobian matrix.”
MethodInfluentialRobustness of stability of time-varying index-1 DAEs
Berger, T. (2013) · Mathematics of Control Signals and Systems · doi:10.1007/s00498-013-0123-5 · Cited by 17
“in mechanical engineering [1, 26, 52], multibody dynamics [23, 57], electrical networks [22, 50, 55] and chemical engineering [15, 19, 51], which often cannot be modeled by standard ordinary differential equations (ODEs).”
A non-equilibrium control oriented model for the pressurizer dynamics
Pini, A., Cammi, A., Colombo, L. P. M., et al. (2018) · Progress in Nuclear Energy · doi:10.1016/j.pnucene.2018.02.017 · Cited by 17
Juárez-Abad, J. A., Sandoval-Garcia, A. P., Flores, J. L., et al. (2018) · IEEE Transactions on Industrial Informatics · doi:10.1109/tii.2018.2865445 · Cited by 17
“The Euler method is used for the integration [42].”
MethodModeling of the energetic ion observations in the vicinity of Rhea and Dione
Kotova, A., Roussos, E., Krupp, N., et al. (2015) · Icarus · doi:10.1016/j.icarus.2015.06.031 · Cited by 17
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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