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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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Sefkow, A. B., & Davidson, R. C. (2006) · Physical Review Special Topics - Accelerators and Beams · doi:10.1103/physrevstab.9.090101 · Cited by 14
“There are two primary steps to the method of lines: first, the spatial dimension is discretized and, second, the semidiscrete temporal problem is integrated as a system of ordinary differential equations (ODEs) [34,35].”
A Parallel Direct Solver for the Simulation of Large-Scale Power/Ground Networks
Cauley, S., Balakrishnan, V., & Koh, C. (2010) · IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems · doi:10.1109/tcad.2010.2042901 · Cited by 14
“A classical algorithm for the numerical integration of ordinary differential equations such as (1) is the trapezoidal method [8].”
MethodIterative solver techniques in the dynamic simulation of power systems
Chaniotis, D., & Pai, M. A. (2002) · Power Engineering Society Summer Meeting · doi:10.1109/pess.2000.867656 · Cited by 14
One implementation variant of the finite difference method for solving ODEs/DAEs
Wu, B., & White, R. E. (2003) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2003.06.002 · Cited by 14
“the Euler method, the Runge-Kutta method, and the backward differentiation formulae (BDF) method, have been widely used and extensively studied (Rice & Do, 1995; Holland & Liapis, 1983; Ascher & Petzold, 1998; Brenan, Campbell & Petzold, 1989; Hairer & Wanner, 1991).”
MethodInfluentialGeometric completion of differential systems using numeric-symbolic continuation
Reid, G. J., Smith, C. J., & Verschelde, J. (2002) · ACM SIGSAM Bulletin · doi:10.1145/581316.581317 · Cited by 14
“For example, differential-algebraic equations (DAE) arise from constrained Lagrangian mechanics [2, 28, 30].”
BackgroundMethodInfluentialPath Planning Methods for Autonomous Underwater Vehicles
Yiğit, K. (2011) · Cited by 14
PythNon : A PSE for the Numerical Solution of Nonlinear Algebraic Equations
Spiteri, R. J., & Ter, T.-P. (2008) · Cited by 14
Modelling of physical systems for the design and control of mechatronic systems
van Amerongen, J., & Breedveld, P. C. (2003) · University of Twente Research Information · Cited by 14
Time Integrating Articulated Body Dynamics Using Position-Based Collocation MethodPreprint
Pan, Z., & Manocha, D. (2019) · EasyChair preprint · doi:10.29007/v4ch · Cited by 14
A Survey of Multimodal Controllable Diffusion Models
Jiang, R., Zheng, G., Li, T., Yang, T., Wang, J., & Li, X. (2024) · Journal of Computer Science and Technology · doi:10.1007/s11390-024-3814-0 · Cited by 22
Ren, X., Men, Z., Wang, P., et al. (2025) · Applied Catalysis B: Environmental · doi:10.1016/j.apcatb.2025.125403 · Cited by 13
Non-linear behavior of a face-sheet debonded sandwich panel – Thermal effects
Frostig, Y. (2014) · International Journal of Non-Linear Mechanics · doi:10.1016/j.ijnonlinmec.2014.03.001 · Cited by 13
Fuzzy permutation entropy derived from a novel distance between segments of time series
Zhang, Z., Xiang, Z., Chen, Y., et al. (2020) · AIMS Mathematics · doi:10.3934/math.2020402 · Cited by 13
Ramos, H., & Rufai, M. A. (2022) · Journal of Mathematical Chemistry · doi:10.1007/s10910-022-01427-7 · Cited by 13
Bellotti, T. (2024) · Journal of Computational Physics · doi:10.1016/j.jcp.2024.112871 · Cited by 13
Hoang, M. T. (2024) · Numerical Algorithms · doi:10.1007/s11075-024-01792-1 · Cited by 13
“Remark 2 By using the approaches in [4, 8] , it is easy to verify that the NSFD model (6) is also convergence of order p under the conditions of Theorem 2 .”
Chowell, G., Bleichrodt, A., & Luo, R. (2024) · Statistics in Medicine · doi:10.1002/sim.10036 · Cited by 13
“For example, some studies have evaluated explicit vs implicit methods, 37 adaptive time-stepping methods that adjust the time step size based on the solution behavior, 38,39 and the impact on computational complexity.”
BackgroundRobust Symmetry Detection via Riemannian Langevin Dynamics
Je, J., Liu, J., Yang, G., Deng, B., Cai, S., Wetzstein, G., Litany, O., & Guibas, L. J. (2024) · SIGGRAPH Asia Conference Papers · doi:10.1145/3680528.3687682 · Cited by 17
“…solver minimizes discretization errors at the expense of an additional evaluation of the learned score function per timestep and shows that using Heun’s method [Ascher and Petzold 1998] can produce samples with quality on par with or better than Euler’s method within fewer sampling steps.”
An Event-Triggered $\boldsymbol{k}$-WTA Model With Experimental Verification on Multirobot System
Jin, L., Li, Y., Chen, Y., et al. (2025) · IEEE Transactions on Industrial Electronics · doi:10.1109/tie.2025.3531457 · Cited by 13
“Besides, as proved in [31], the convergence guarantee of the model can be deduced from the consistency and zero stability.”
Use of Tikhonov Regularization to Improve the Accuracy of Position Estimates in Inertial Navigation
Nieminen, T., Kangas, J., & Kettunen, L. (2011) · International Journal of Navigation and Observation · doi:10.1155/2011/450269 · Cited by 13
“More specifically, we will consider two-point BVPs where the necessary constraints are given at the beginning and at the end of the interesting time interval [3].”
MethodAn algorithm for simulation of electrochemical systems with surface–bulk coupling strategies
Buoni, M., & Petzold, L. R. (2009) · Journal of Computational Physics · doi:10.1016/j.jcp.2009.09.032 · Cited by 13
Self-citationA new partitioned staggered scheme for flexible multibody interactions with strong inertial effects
Yenduri, A., Ghoshal, R., & Jaiman, R. K. (2016) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2016.10.044 · Cited by 13
Amiri, N., Ebrahimi, S., Jatskevich, J., et al. (2019) · IEEE Transactions on Energy Conversion · doi:10.1109/tec.2019.2907268 · Cited by 13
“This leads to numerical stiffness [28], which requires small time steps or implicit solvers for the model to remain stable [29].”
BackgroundDecoding the Time-Dependent Response of Bioluminescent Metal-Detecting Whole-Cell Bacterial Sensors
Duval, J. F. L., & Pagnout, C. (2019) · ACS Sensors · doi:10.1021/acssensors.9b00349 · Cited by 13
“1 Bn , M K , u J and ads ), for given n , K , Hi K and fixed values of the Deborah numbers e a k , r a k and q a / , the dimensionless forms of eqs 4-6 and 8 were solved in Mathcad 15 calculus environment using an hybrid solver based on either Adams or backward differentiation methods depending on the sti…”
MethodEffective dispersion in the focusing nonlinear Schrödinger equation
Leisman, K. P., Zhou, D., Banks, J. W., et al. (2019) · Physical review. E · doi:10.1103/physreve.100.022215 · Cited by 13
Boriding Kinetics of FeB and Fe2B Layers on AISI M2 Steel by the Integral Diffusion Model
Zouzou, C., & Keddam, M. (2019) · Annales de Chimie Science des Matériaux · doi:10.18280/acsm.430304 · Cited by 13
“After determining the boron diffusivity in each boride layer, a computer program was created by using the Petzold’s DAE solver DASPK [16] with the purpose of evaluating the output variables u(t) and v(t) for a given treatment time and temperature.”
MethodInfluentialThe Discontinuous Galerkin Material Point Method for variational hyperelastic–plastic solids
Renaud, A., Heuzé, T., & Stainier, L. (2020) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2020.112987 · Cited by 13
“This requires the employment of fractional-step methods such as Godunov’s splitting [4], that come with their own stability restrictions [49].”
MethodZanardi, I., Venturi, S., & Panesi, M. (2022) · AIAA SciTech Forum · doi:10.2514/6.2022-1639 · Cited by 13
Parker, R., & Biegler, L. T. (2022) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2022.07.476 · Cited by 13
“More information the index of a DAE is given by Ascher and Petzold (1998).”
Evaluation of Differentially Constrained Motion Models for Graph-Based Trajectory Prediction
Westny, T., Oskarsson, J., Olofsson, B., & Frisk, E. (2023) · IEEE Intelligent Vehicles Symposium · doi:10.1109/iv55152.2023.10186615 · Cited by 18
“The classic fourth-order Runge-Kutta method is one such example [25]:”
MethodInfluentialLiu, X. (2009) · Optics Communications · doi:10.1016/j.optcom.2008.12.051 · Cited by 13
Brandolin, A., Sarmoria, C., Failla, M. D., et al. (2007) · Industrial & Engineering Chemistry Research · doi:10.1021/ie070013m · Cited by 13
Johnson, R. W., Hansen, G. A., & Newman, C. (2011) · Applied Mathematics and Computation · doi:10.1016/j.amc.2011.03.101 · Cited by 13
An efficient time-integration method for nonlinear dynamic analysis of solids and structures
Liu, T.-Y., Li, Q., & Zhao, C. (2013) · Science China Physics Mechanics and Astronomy · doi:10.1007/s11433-013-5021-9 · Cited by 13
Gros, S., Chachuat, B., & Bonvin, D. (2009) · International Journal of Control · doi:10.1080/00207170802460032 · Cited by 13
“In particular, efficient procedures exist for fast and reliable solution of linear TPBVPs (Pesch 1989b ; Ascher and Petzold 1998 ).”
MethodGąsiorowski, D. (2012) · Acta Geophysica · doi:10.2478/s11600-012-0087-8 · Cited by 13
“Such a problem corresponds to the integration of ordinary differential equation over time using one of numerous well known methods suitable for initial value problem (Dahlquist and Björck 1974, Ascher and Petzold 1998).”
MethodChain Drive Simulation Using Spatial Multibody Dynamics
Omar, M. A. (2014) · Advances in Mechanical Engineering · doi:10.1155/2014/378030 · Cited by 13
Arslantürk, C. (2010) · DergiPark (Istanbul University) · Cited by 13
Langevin incremental mixture importance sampling
Fasiolo, M., de Melo, F. E., & Maskell, S. R. (2017) · Statistics and Computing · doi:10.1007/s11222-017-9747-5 · Cited by 13
“This is a relatively mild problem, which can be addressed by using a numerical integrator, such as a Runge-Kutta method (Ascher and Petzold, 1998).”
MethodViscoelastic modeling of fiber preform compaction in vacuum infusion process
Yenilmez, B., Çağlar, B., & Sozer, E. M. (2017) · Journal of Composite Materials · doi:10.1177/0021998317699983 · Cited by 13
“DAEs are commonly solved by transforming the algebraic equation in the system to a differential equation through differentiation if the nonlinearity parameter in the algebraic equation is an integer (in this case it is m, non-linearity parameter of the spring).(25,26) It should be noted that, in addition to Maxwell, t…”
BackgroundEnhancement of operating flux in a membrane bio-reactor coupled with a mechanical sieve unit
Park, S., Yeon, K., Moon, S., et al. (2017) · Chemosphere · doi:10.1016/j.chemosphere.2017.10.079 · Cited by 13
Mathematical model of natural circulation biomass boilers during start-up
Bilde, K. G., Sørensen, K., & Condra, T. (2019) · International Journal of Heat and Mass Transfer · doi:10.1016/j.ijheatmasstransfer.2019.118477 · Cited by 13
Submerged Arches Optimal Design With a Multi-Method Ensemble Meta-Heuristic Approach
Pérez‐Aracil, J., Camacho-Gómez, C., Hernández-Díaz, A. M., et al. (2020) · IEEE Access · doi:10.1109/access.2020.3040479 · Cited by 13
On unified framework for nonlinear grey system models: An integro-differential equation perspective
Yang, L., Xie, N., Wei, B., et al. (2022) · Communications in Nonlinear Science and Numerical Simulation · doi:10.1016/j.cnsns.2022.106250 · Cited by 13
Rivera‐Toledo, M., García-Crispín, L. E., Flores‐Tlacuahuac, A., et al. (2006) · Industrial & Engineering Chemistry Research · doi:10.1021/ie060206u · Cited by 13
Dynamics of a pneumatic artificial muscle actuation system driving a trailing edge flap
Woods, B. K. S., Kothera, C. S., Wang, G., et al. (2014) · Smart Materials and Structures · doi:10.1088/0964-1726/23/9/095014 · Cited by 13
Hart, G. D., & Anitescu, M. (2003) · doi:10.1145/948542.948550 · Cited by 13
“The advantage of this approach is that it is easy to set up and results in a DAE, for which both analytical and software tools are in a fairly mature state of development.”
BackgroundInfluentialParticle Swarm-assisted state feedback control: From pole selection to state estimation
Wang, J., Brackett, B. T., & Harley, R. G. (2009) · Proceedings of the ... American Control Conference/Proceedings of the American Control Conference · doi:10.1109/acc.2009.5160514 · Cited by 13
“The Runge-Kutta method [14] is typically used for control systems simulations, but in our approach, we chose the Euler method [15], a basic method of numerical integration for ordinary differential equations, to obtain the control response, where for x ( t 0 ) = x 0 .”
MethodKhan, W. A., Tang, C. P., & Krovi, V. (2006) · Mechanism and Machine Theory · doi:10.1016/j.mechmachtheory.2006.05.002 · Cited by 13
“The dynamics of mechanical systems with closed loops can be formulated as a system of ODEs whose solutions are required to satisfy additional holonomic (algebraic) equations resulting from cutting the loops [19].”
MethodInfluentialNonlinear Observer for Temperatures and Emissivities in a Strip Annealing Furnace
Strommer, S., Niederer, M., Steinboeck, A., et al. (2017) · IEEE Transactions on Industry Applications · doi:10.1109/tia.2017.2669327 · Cited by 13
CHEMSIMUL: A chemical kinetics software package
Kirkegaard, P., Bjergbakke, E., & Olsen, J. V. (2008) · Cited by 13
Fast simulation of VLSI interconnects
Jain, J., Koh, C., & Balakrishnan, V. (2005) · doi:10.1109/iccad.2004.1382549 · Cited by 13
“A standard algorithm for the numerical integration of differential equations such as (1) is the trapezoidal method [1].”
MethodHardt, M., von Stryk, O., Wollherr, D., et al. (2004) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/robot.2003.1241780 · Cited by 13
“This approach, also known as coordinate partitioning [2], 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.”
MethodMultiscale Modeling and Design of Electrochemical Systems
Braatz, R. D., Seebauer, E. G., & Alkire, R. C. (2008) · Advances in electrochemical science and engineering · doi:10.1002/9783527625307.ch4 · Cited by 13
Benderskaya, G. (2008) · Technischen Universität Darmstadt · doi:10.26083/tuprints-00000807 · Cited by 13
Mathematics in Chemical Engineering
Finlayson, B. A., Biegler, L. T., & Grossmann, I. E. (2006) · Ullmann's Encyclopedia of Industrial Chemistry · doi:10.1002/14356007.b01_01.pub2 · Cited by 13
Generalized Compensation Principle
Tsyvinski, A., & Werquin, N. (2017) · National Bureau of Economic Research · doi:10.3386/w23509 · Cited by 21
“1The DAE theory is much more recent than the theory of ordinary differential equations (Ascher and Petzold [1998], p.”
InfluentialOn the adjoint formulation of design sensitivity analysis of multibody dynamics cs
Schaffer, A. (2005) · doi:10.17077/etd.ufj63owx · Cited by 13
“1 Numerical Integration of Index-1 Differential-Algebraic Equations Using Backward Differentiation Formulas Since numerical integration of DAE has similar stability properties to the numerical integration of stiff ODE [6] and because multibody systems can include stiff elements [41], implicit numerical integration met…”
MethodShu, D., Xie, X., Zhang, S., et al. (2016) · IET Generation Transmission & Distribution · doi:10.1049/iet-gtd.2016.0489 · Cited by 12
Luo, X.-L., & Yao, Y. (2022) · Journal of Computational Mathematics · doi:10.4208/jcm.2101-m2020-0173 · Cited by 12
Moreno, E., Dialami, N., & Cervera, M. (2021) · Journal of Non-Newtonian Fluid Mechanics · doi:10.1016/j.jnnfm.2021.104512 · Cited by 12
Sturm, A. B., Visintin, P., & Bennett, B. (2022) · Engineering Structures · doi:10.1016/j.engstruct.2022.114071 · Cited by 12
Zhang, X., Song, C., You, T., Bai, Q., Wei, W., & Lei Zhang, A. (2023) · IEEE Transactions on Geoscience and Remote Sensing · doi:10.1109/tgrs.2023.3346526 · Cited by 12
“Common choices include the Runge–Kutta family [49] and the Euler method [50].”
MethodChen, L., Jin, L., Shang, M., & Wang, F. (2024) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2024.3387408 · Cited by 12
“The Order of Numerical Methods: A numerical method for solving ODEs is said to be of order p if the global truncation error of the method is o (τ p ) as τ → 0 [61].”
BackgroundMethodInfluentialA GMRES-based BDF method for solving differential Riccati equations
Hernández, V., Ibáñez, J. J., Peinado, J., et al. (2007) · Applied Mathematics and Computation · doi:10.1016/j.amc.2007.06.021 · Cited by 12
“One of these methods is the well-known backward differentiation formula (BDF) [19].”
MethodA new power series solution on the electrostatic pull-in instability of nano cantilever actuators
Ghalambaz, M., Noghrehabadi, A., Abadyan, M., et al. (2011) · Procedia Engineering · doi:10.1016/j.proeng.2011.04.608 · Cited by 12
Bas, E., Erdoğmuş, D., Draft, R. W., et al. (2012) · Journal of Visual Communication and Image Representation · doi:10.1016/j.jvcir.2012.09.003 · Cited by 12
Stable time integration suppresses unphysical oscillations in the bidomain model
Ziaratgahi, S. T., Marsh, M. E., Sundnes, J., et al. (2014) · Frontiers in Physics · doi:10.3389/fphy.2014.00040 · Cited by 12
“, Ascher and Petzold [15] for an explanation of the Butcher tableau for specifying Runge–Kutta methods.”
BackgroundMethodInfluentialReaching and pointing gestures calculated by a generic gesture system for social robots
de Perre, G. V., Beir, A. D., Cao, H.-L., et al. (2016) · Robotics and Autonomous Systems · doi:10.1016/j.robot.2016.06.006 · Cited by 12
“The desired joint angles q are calculated by integrating q̇ with the Runge-Kutta algorithm [3].”
MethodHaug, E. J. (2016) · Mechanics Based Design of Structures and Machines · doi:10.1080/15397734.2016.1273783 · Cited by 12
“The rst-order index 0 DAE that govern motion of mechanical systems with holonomic and nonholonomic constraints are shown to be well suited for integration with explicit and implicit numerical integrators.”
BackgroundInfluentialKim, C.-J., Lee, D. H., & Hur, S. W. (2019) · International Journal of Aeronautical and Space Sciences · doi:10.1007/s42405-019-00160-x · Cited by 12
Luo, X.-L., Xiao, H., Lv, J.-H., et al. (2021) · Applied Numerical Mathematics · doi:10.1016/j.apnum.2021.02.019 · Cited by 12
Chen, S., Peterson, C. W., Parker, J. A., et al. (2021) · Nature Communications · doi:10.1038/s41467-021-22794-w · Cited by 12
Hoang, M. T. (2023) · Computational and Applied Mathematics · doi:10.1007/s40314-023-02230-z · Cited by 12
“Therefore, the NSFD (4) is 0-stability (see (Ascher and Petzold 1998)).”
BackgroundMethodInfluentialEfficient Nonlinear Programming Algorithms for Chemical Process Control and Operations
Biegler, L. T. (2009) · IFIP advances in information and communication technology · doi:10.1007/978-3-642-04802-9_2 · Cited by 12
“Moreover, sequential approaches may fail with unstable dynamics [1].”
BackgroundModelling and Control of a Spherical Inverted Pendulum on a Five-Bar Mechanism
Soto, I., & Campa, R. (2015) · International Journal of Advanced Robotic Systems · doi:10.5772/60027 · Cited by 12
“Explicit Lagrange-multiplier computation, using the unknown accelerations computed from the index-1 DAEs formed by appending the acceleration con‐ straints (computed by differentiating the original constraint) to the system equations (see [23] and [24]).”
MethodA Time-Stepping Complementarity Approach for Frictionless Systems of Rigid Bodies
Tzitzouris, J. A., & Pang, J. (2002) · SIAM Journal on Optimization · doi:10.1137/s1052623400370369 · Cited by 12
“In particular, we focus on Runge–Kutta methods (see [3, 26, 4, 6]).”
MethodThe effect of perception anisotropy on particle systems describing pedestrian flows in corridors
Gulikers, L. L., Evers, J. H. M., Muntean, A., et al. (2013) · Journal of Statistical Mechanics Theory and Experiment · doi:10.1088/1742-5468/2013/04/p04025 · Cited by 12
“to [12, 13].”
Buoni, M., & Petzold, L. R. (2007) · Journal of Computational Physics · doi:10.1016/j.jcp.2007.03.025 · Cited by 12
“To see 0-stability, start with equation (32) and simply refer to the proof for the forward Euler method as given in [ 7 ].”
MethodInfluentialSelf-citationNumerical solution of differential algebraic equations using a multiquadric approximation scheme
Vanani, S. K., & Aminataei, A. (2010) · Mathematical and Computer Modelling · doi:10.1016/j.mcm.2010.10.002 · Cited by 12
Cheung, N. J., Xu, Z., Ding, X., & Shen, H. (2015) · European Journal of Operational Research · doi:10.1016/j.ejor.2015.03.047 · Cited by 12
Extending the inverse vehicle propulsion simulation concept - to improve simulation performance
Fröberg, A. (2005) · Cited by 12
Travelling waves in a network of SIR epidemic nodes with an approximation of weak coupling
SAZONOV, I. A., Kelbert, M., & Gravenor, M. B. (2010) · Mathematical Medicine and Biology A Journal of the IMA · doi:10.1093/imammb/dqq016 · Cited by 12
Hybrid Dynamic Optimization Methods for Systems Biology with Efficient Sensitivities
Lewis, N. R., Hedengren, J. D., & Haseltine, E. L. (2015) · Processes · doi:10.3390/pr3030701 · Cited by 12
“It has been established that sequential approaches have properties of single shooting methods and do not perform well with models that exhibit open loop instability [29,30].”
BackgroundEfficient Solution of the Wigner–Poisson Equations for Modeling Resonant Tunneling Diodes
Costolanski, A. S., & Kelley, C. T. (2010) · IEEE Transactions on Nanotechnology · doi:10.1109/tnano.2010.2053214 · Cited by 12
“Poisson’s equation (6) is also approximated with a fourth-order accurate discretization [29] and the spatial derivative in the kinetic energy term (2) with a fourth-order backward difference formula [30].”
MethodVasco, D. W. (2018) · Water Resources Research · doi:10.1002/2017wr021360 · Cited by 12
“The trajectories xðtÞ are found by solving dx dt 5rSeikonal (20) using an improved Euler’s method known as Heun’s or Ralston’s method (Ascher & Petzold, 1998).”
MethodKamal, A., & Keaveny, E. E. (2018) · Journal of The Royal Society Interface · doi:10.1098/rsif.2018.0592 · Cited by 12
Tracing Locally Pareto-Optimal Points by Numerical Integration
Bolten, M., Doganay, O. T., Gottschalk, H., et al. (2021) · SIAM Journal on Control and Optimization · doi:10.1137/20m1341106 · Cited by 12
“When higher order is demanded, Runge-Kutta methods [3, 27] can be used.”
MethodGraphical simulation environments for modelling and simulation of integrative physiology
Mangourova, V., Ringwood, J. V., & Vliet, B. V. (2010) · Computer Methods and Programs in Biomedicine · doi:10.1016/j.cmpb.2010.05.001 · Cited by 12
Numerical integration of discrete mechanical systems with mixed holonomic and control constraints
Betsch, P., Quasem, M., & Uhlar, S. (2009) · Journal of Mechanical Science and Technology · doi:10.1007/s12206-009-0331-6 · Cited by 12
“Similar to semi-explicit DAEs (see, for example, Ascher & Petzold [ 1 ]), the set of equations in (18) can be cast into the form”
A fixed point-based BDF method for solving differential Riccati equations
Arias, E., Hernández, V., Ibáñez, J. J., et al. (2007) · Applied Mathematics and Computation · doi:10.1016/j.amc.2006.11.001 · Cited by 12
“One of these methods is the well known backward differentiation formula (BDF) [18].”
MethodPeriodic waveform relaxation solutions of nonlinear dynamic equations
Jiang, Y. (2002) · Applied Mathematics and Computation · doi:10.1016/s0096-3003(01)00324-1 · Cited by 12
Axelsson, O., Blaheta, R., & Kohut, R. (2015) · Numerical Linear Algebra with Applications · doi:10.1002/nla.2015 · Cited by 12
“sC1/: It does not occur for stiffly stable variants, such as based on the Radau quadrature; for further discussions of this, see for example [3, 22].”
BackgroundInfluentialExplicit pseudo-transient continuation
Kelley, C. T., & Liao, L. (2013) · Cited by 12
Analysis of the growth kinetics of Fe2B layers by the integral method
Keddam, M., & Kulka, M. (2018) · Journal of Mining and Metallurgy Section B Metallurgy · doi:10.2298/jmmb180405026k · Cited by 12
“This resulting DAE system can be solved either analytically [23-25] or numerically [32].”
BackgroundEfficient Transient Noise Analysis in Circuit Simulation
Denk, G., Römisch, W., Sickenberger, T., et al. (2008) · Springer eBooks · doi:10.1007/978-3-540-77203-3_5 · Cited by 12
Planning Velocities of Free Sliding Objects as a Free Boundary Value Problem
Li, Q., & Payandeh, S. (2004) · The International Journal of Robotics Research · doi:10.1177/0278364904036094 · Cited by 12
“A recent summary of numerical shooting methods can be found in Ascher and Petzold (1998).”
MethodLi, Z., & Shi, C. ‐. R. (2003) · International Conference on Computer Aided Design · doi:10.5555/996070.1009978 · Cited by 12
Comment on “Computational periodicity as observed in a simple system,” by Edward N. Lorenz (2006a)
Yao, L.-S., & Hughes, D. (2008) · Tellus A Dynamic Meteorology and Oceanography · doi:10.1111/j.1600-0870.2008.00301.x · Cited by 12
“…Euler method, various finite-difference approximations with various levels of implicitness, several off-the-shelve packaged ODE solvers, several of the ODE-solver routines given in Press et al. (1986), and a Taylorseries expansion method (Ascher and Petzold, 1998, 73–74), among other methods.”
MethodInfluentialCurcino, I. V., Júnior, P. R. S. C., Gomez, A. O. C., et al. (2022) · Micro and Nano Engineering · doi:10.1016/j.mne.2022.100165 · Cited by 11
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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