The DOI matches its Crossref record.
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.
Overview
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“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]
Citations
Cited by
Sorted by citations
Items 901–1,000 of 1,000
Maretvadakethope, S., Hazel, A. L., Vasiev, B. N., et al. (2023) · Journal of Fluid Mechanics · doi:10.1017/jfm.2023.897 · Cited by 11
“The unsteady system is evolved using a second-order backward difference (BDF2) time stepper (Ascher & Petzold 1998) with time step dt = 10−4.”
MethodA New ParaDiag Time-Parallel Time Integration Method
Gander, M. J., & Palitta, D. (2024) · SIAM Journal on Scientific Computing · doi:10.1137/23m1568028 · Cited by 11
Cuoci, A., Cipriano, E., Saufi, A. E., et al. (2024) · Journal of Computational Science · doi:10.1016/j.jocs.2024.102453 · Cited by 11
Solving the discretised shallow water equations using neural networks
Chen, B., Nadimy, A., Heaney, C. E., et al. (2025) · Advances in Water Resources · doi:10.1016/j.advwatres.2025.104903 · Cited by 11
Adaptive Feedback–Feedforward Control of Building Structures
Ma, T., & Yang, H. T. Y. (2004) · Journal of Engineering Mechanics · doi:10.1061/(asce)0733-9399(2004)130:7(786 · Cited by 11
Localization and latency concepts applied to time simulation of large power systems
Fabozzi, D., & Cutsem, T. V. (2010) · doi:10.1109/irep.2010.5563287 · Cited by 11
“For instance, the results presented in this paper have been obtained with Backward Differentiation Formulae (BDF), known to have the stiff decay property [ 13 ], allowing the step size to be increased [14, 3]. The BDF of order o applied to the i-th injector is:”
MethodGeometric phase of light-induced conical intersections: adiabatic time-dependent approach
Halász, G. J., Badankó, P., & Vibók, Á. (2018) · Molecular Physics · doi:10.1080/00268976.2018.1431410 · Cited by 11
Zhu, Q., Wang, K., Shao, Z., et al. (2020) · Journal of Intelligent & Robotic Systems · doi:10.1007/s10846-020-01188-y · Cited by 11
“Previous works have proposed several mesh refinement strategies for boundary value problems [39, 40], among which the most commonly used is the hp-adaptive mesh refinement strategy, which improves the accuracy of the optimization solution by adaptively adjusting the number of finite elements and…”
MethodSaldívar‐Guerra, E., Infante, R., & Islas-Manzur, J. M. (2020) · Processes · doi:10.3390/pr8111508 · Cited by 11
“The numerical routine DDASSL (Differential-Algebraic System Solver) [40] was used in the two stages (programs) for the integration of the equations, which were coded in Fortran.”
MethodDelay differential equations based models in NONMEM
Yan, X., Bauer, R. J., Koch, G., et al. (2021) · Journal of Pharmacokinetics and Pharmacodynamics · doi:10.1007/s10928-021-09770-z · Cited by 11
Applications of the Dulmage–Mendelsohn decomposition for debugging nonlinear optimization problems
Parker, R., Nicholson, B. L., Siirola, J. D., et al. (2023) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2023.108383 · Cited by 11
Numerical Analysis of Dynamical Systems
Guckenheimer, J. M. (2002) · Handbook of dynamical systems · doi:10.1016/s1874-575x(02)80029-7 · Cited by 11
“Implicit multi-step methods called backwards differentiation formulae are used widely as integrators for stiff systems [87, 10].”
BackgroundMethodNoghrehabadi, A., Ghalambaz, M., Beni, Y. T., et al. (2011) · Procedia Engineering · doi:10.1016/j.proeng.2011.04.610 · Cited by 11
Long-time integration methods for mesoscopic models of pattern-forming systems
Abukhdeir, N. M., Vlachos, D. G., Katsoulakis, M. A., et al. (2011) · Journal of Computational Physics · doi:10.1016/j.jcp.2011.03.052 · Cited by 11
Residual Generation Methods for Fault Diagnosis with Automotive Applications
Svärd, C. (2009) · Cited by 11
Quinto, E. J., Arinder, P., Axelsson, L. T., Heir, E., Holck, A. L., Lindqvist, R., Lindblad, M., Andreou, P., Lauzon, H. L., Marteinsson, V., et al. (2014) · Applied and Environmental Microbiology · doi:10.1128/aem.03791-13 · Cited by 11
“A forward Euler method (50) has been implemented in Visual basic code to simulate equation 1 under fluctuating conditions of temperature, pH, aw, and lactic acid.”
MethodModel abstraction and temporal behavior analysis of genetic regulatory networks
Kuwahara, H. (2007) · Cited by 11
Laplace transform homotopy perturbation method for the approximation of variational problems
Filobello-Nino, U. A., Vázquez-Leal, H., Rashidi, M. M., et al. (2016) · SpringerPlus · doi:10.1186/s40064-016-1755-y · Cited by 11
“SpringerPlus (2016) 5:276 Discussion In order to use a pure numerical solution as reference, we utilized the scheme based on trapezoid combined with Richardson extrapolation (Ascher et al. 1995; Ascher and Petzold 1998) from the built-in numerical routines provided by Maple 15.”
MethodAn Investigation on Thermomechanical Flexural Response of Shape-Memory-Polymer Beams
Molaaghaie-Roozbahani, M., Heydarzadeh, N., Baghani, M., et al. (2016) · International Journal of Applied Mechanics · doi:10.1142/s1758825116500630 · Cited by 11
Ortega, R. A., Orden, J. C. G., Cruchaga, M., et al. (2017) · Multibody System Dynamics · doi:10.1007/s11044-017-9565-8 · Cited by 11
“The numerical stability of the resulting equations (DAE and ODE) [27, 28] strongly depend on the time integration scheme used to solve them [9, 10, 12, 13, 29, 30].”
Carfagna, M., & Grillo, A. (2017) · Computing and Visualization in Science · doi:10.1007/s00791-017-0278-6 · Cited by 12
Viability assessment of a rigid wing airborne wind energy pumping system
Licitra, G., Koenemann, J., Horn, G., et al. (2017) · International Conference on Process Control · doi:10.1109/pc.2017.7976256 · Cited by 11
“Successively, a DAE index-1 formulation can be achieved after an index reduction procedure so that classical integration methods work efficiently [10].”
MethodObject-Oriented Modeling and Control of Delta Robots
Bortoff, S. A. (2018) · IEEE Conference on Control Technology and Applications (CCTA) · doi:10.1109/ccta.2018.8511395 · Cited by 11
“Modern tools such as Dymola use proprietary logic to switch among the selected states and equations, together with index 1 DAE solvers such as DASSL [17], [23] to compute an accurate numerical solution.”
MethodAngelova, V., Hached, M., & Jbilou, K. (2019) · Numerical Linear Algebra with Applications · doi:10.1002/nla.2272 · Cited by 11
“…is given as e Instead of solving the low dimensional nonsymmetric differential Riccati equation (3.4) by using the exponential scheme (3.26), we can use an integration scheme for solving ordinary differential equations such as Rosenbrock [35] or Backward Differentiation Formula (BDF) methods [3, 14].”
MethodInfluentialA coupled HDG-FV scheme for the simulation of transient inviscid compressible flows
Sheshachala, S. K., Sevilla, R., & Hassan, O. (2020) · Computers & Fluids · doi:10.1016/j.compfluid.2020.104495 · Cited by 11
Harirchi, F., Kim, D.-H., Khalil, O. A., et al. (2020) · Fuel · doi:10.1016/j.fuel.2020.118204 · Cited by 11
“The time evolution of a reactive system is commonly modeled using massaction kinetic equations [24–28], which, by applying Euler discretization method [29], can be written as”
MethodTooranjipour, P., Vatankhah, R., & Khosravifard, A. (2020) · Applied Mathematical Modelling · doi:10.1016/j.apm.2020.06.053 · Cited by 11
Lu, Q., & Rizwan-uddin, R.-U. (2020) · Annals of Nuclear Energy · doi:10.1016/j.anucene.2020.107747 · Cited by 11
Modeling the Steady-State and Dynamic Characteristics of Solid-Oxide Fuel Cells
Kee, R. J., Zhu, H., Braun, R. J., et al. (2012) · Advances in chemical engineering · doi:10.1016/b978-0-12-386874-9.00010-5 · Cited by 11
Zheng, H., & Chen, Q. (2022) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2022.114833 · Cited by 11
Parallel-in-Time Solution of Power Systems with Scheduled Events
Schroder, J. B., Falgout, R. D., Woodward, C. S., et al. (2018) · IEEE Power & Energy Society General Meeting (PESGM) · doi:10.1109/pesgm.2018.8586435 · Cited by 11
“If yn represents an approximation of y(tn), then the computation of yn from the previous step yn−1 is given by yn = yn−1 + h ∑s i=1 biKi, where the Ki satisfy the following nonlinear equations [18],”
An index 0 Differential-Algebraic equation formulation for multibody dynamics: Holonomic constraints
Haug, E. J. (2016) · Mechanics Based Design of Structures and Machines · doi:10.1080/15397734.2016.1246370 · Cited by 11
“While by the strict de nition of di erentiation index (Ascher and Petzold, 1998), this is an index 1 DAE, by de ning λ = µ̇, Eq.”
BackgroundMethodNote on the Convergence of the Implicit Euler Method
Faragó, I. (2013) · Lecture notes in computer science · doi:10.1007/978-3-642-41515-9_1 · Cited by 11
Mitchell, I. M., & Susuki, Y. (2008) · Lecture notes in computer science · doi:10.1007/978-3-540-78929-1_51 · Cited by 11
“One exception is [7], which adapts the d/dt tool [8] to a DAE setting by replacing the standard ODE integration schemes in d/dt with a standard DAE integration scheme: the equivalent ODE (3) and (5) is solved at each timestep, numerical errors in this process cause drift away from the constraint manifold, and so the r…”
MethodSpatial Aspects in Biological System Simulations
Resat, H., Costa, M. N., & Shankaran, H. (2010) · Methods in enzymology on CD-ROM/Methods in enzymology · doi:10.1016/b978-0-12-381270-4.00017-2 · Cited by 11
“The dynamical stiffness issue has been the subject of many studies in a multitude of scientific disciplines, and there are good ODE solvers which overcome the numerical difficulties (Ascher and Petzold, 1998).”
BackgroundMethodInfluentialNonlinear dispersion of a pollutant ejected into a channel flow
Gorder, R. A. V., & Vajravelu, K. (2011) · Open Physics · doi:10.2478/s11534-011-0025-4 · Cited by 11
“Figures 4 and 5 show the agreement between our results and numerical results obtained in Maple 12 using the numerical ordinary boundary value problem solver (see references [12, 13 ] for details of the numerical method).”
Optimization of space radiators with step fins
Arslantürk, C. (2010) · Proceedings of the Institution of Mechanical Engineers Part G Journal of Aerospace Engineering · doi:10.1243/09544100jaero670 · Cited by 11
Navarro, A. K. W., & Vassiliadis, V. S. (2013) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2013.11.004 · Cited by 11
Ibáñez, J., Hernández, V., Arias, E., et al. (2008) · Computer Physics Communications · doi:10.1016/j.cpc.2008.11.013 · Cited by 11
Analytical model and implementation by equations of three-phase diode bridge rectifiers operation
di Gerlando, A., Foglia, G. M., Iacchetti, M. F., et al. (2010) · doi:10.1109/icelmach.2010.5608272 · Cited by 11
“Strictly speaking, the model of the 12 pulse series rectifier is described by a Differential - Algebraic Equations (DAEs) system [ 11 ] consisting of three ODEs for iPa(t), iPb(t), vc(t) and of three algebraic constraint equations (33).”
BackgroundMethodInfluentialGenerating optimal dynamic motions for closed-chain robotic systems
Chettibi, T., Haddad, M., Labed, A., et al. (2005) · European Journal of Mechanics - A/Solids · doi:10.1016/j.euromechsol.2005.01.005 · Cited by 11
Talwar, J., Mohanty, R. K., & Singh, S. (2015) · Applied Mathematics and Computation · doi:10.1016/j.amc.2015.03.057 · Cited by 11
Direct optimization of dynamic systems described by differential‐algebraic equations
Fabien, B. C. (2008) · Optimal Control Applications and Methods · doi:10.1002/oca.838 · Cited by 11
“First, the norm, ‖̄ j,l+1‖, can be a weighted or scaled norm [1, 2].”
MethodIn virtuo Experiments Based on the Multi-Interaction System Framework: the RéISCOP Meta-Model.
Desmeulles, G., Bonneaud, S., Redou, P., Rodin, V., & Tisseau, J. (2009) · doi:10.3970/cmes.2009.047.299 · Cited by 11
“But, these systems are routinely processed using the traditional tools of ODE numerical resolution ( Asher and Petzold, 1997 ) which adapt well to synchronous resolutions.”
MethodThe method of a floating frame of reference for non-smooth contact dynamics
Lozovskiy, A. V., & Dubois, F. (2016) · European Journal of Mechanics - A/Solids · doi:10.1016/j.euromechsol.2016.01.007 · Cited by 11
“The scheme used for temporal discretization is a Newmark symmetric one (also called trapezoidal), known for being a second-order accurate, unconditionally stable implicit scheme, [23], [24].”
MethodBetsch, P., Altmann, R., & Yang, Y. (2016) · Computational methods in applied sciences · doi:10.1007/978-3-319-30614-8_1 · Cited by 11
Benchmarking of adjoint sensitivity-based optimization techniques using a vehicle ride case study
Zhu, Y., Sandu, C., Dopico, D., et al. (2017) · Mechanics Based Design of Structures and Machines · doi:10.1080/15397734.2017.1338576 · Cited by 11
“However, the current adjoint sensitivity approaches are coupled with index-3 and index-1 DAE formulations, which are not the best options because of the numerical di culties to solve the DAE systems (Brenan et al., 1989; Ascher and Petzold, 1998).”
MethodHernández-Díaz, A. M., Bueno-Crespo, A., Pérez‐Aracil, J., et al. (2018) · Applied Soft Computing · doi:10.1016/j.asoc.2018.07.009 · Cited by 11
A new modified nodal analysis for nano-scale memristor circuit simulation
Yu, H., & Fei, W. (2010) · doi:10.1109/iscas.2010.5537950 · Cited by 11
“, by a backward-differential-formula (BDF) [6], [8].”
Numerical Analysis of Dynamical Systems
Guckenheimer, J. M. (1999) · Cited by 11
Stochastic DAEs in Transient Noise Simulation
Winkler, R. (2004) · Mathematics in industry · doi:10.1007/978-3-642-55872-6_45 · Cited by 11
An Efficient Non-standard Finite Difference Scheme for an Ionic Model of Cardiac Action Potentials
Spiteri, R. J., & MacLachlan, M. C. (2003) · The Journal of Difference Equations and Applications · doi:10.1080/1023619031000146878 · Cited by 11
“In this paper, we define a problem to be stiff if the time step used by an explicit numerical method such as forward Euler [1,10] is based on considerations of stability and not of accuracy.”
MethodCombined DAE and Sliding Mode Control Methods for Simulation of Constrained Mechanical Systems
Compere, M. D., & Longoria, R. G. (2000) · Journal of Dynamic Systems Measurement and Control · doi:10.1115/1.1320450 · Cited by 11
Lee, L.-F., Bhatt, R., & Krovi, V. (2006) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/robot.2005.1570103 · Cited by 11
“By doing so, we would like to link (and leverage) the extensive literature on formulation and implementation of computational simulation of multibody systems [13-17] to the problem of motion planning of robot collectives.”
BackgroundOptimal off‐line trajectory planning of hybrid fuel cell/gas turbine power plants
Kameswaran, S., Biegler, L. T., Junker, S. T., et al. (2007) · AIChE Journal · doi:10.1002/aic.11089 · Cited by 11
Man-portable power generation devices : product design and supporting algorithms
Mitsos, A. (2006) · DSpace@MIT (Massachusetts Institute of Technology) · Cited by 11
1Approximation Algorithms for Optimization of Combinatorial Dynamical SystemsPreprint
Yang, I., Burden, S. A., Rajagopal, R., Sastry, S., & Tomlin, C. J. (2016) · Cited by 10
ODE Transformer: An Ordinary Differential Equation-Inspired Model for Neural Machine TranslationPreprint
Li, B., Du, Q., Zhou, T., Zhou, S., Zeng, X., Xiao, T., & Zhu, J. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2104.02308 · Cited by 23
Rufai, M. A., Carpentieri, B., & Ramos, H. (2023) · Fractal and Fractional · doi:10.3390/fractalfract7100703 · Cited by 10
Sarkar, T., Biswas, P., & Srivastava, P. K. (2024) · International Journal of Biomathematics · doi:10.1142/s1793524524500013 · Cited by 10
Application of the NSFD method in a Malaria model with nonlinear incidence and recovery rates
Sarkar, T. K., Srivastava, P. K., & Biswas, P. (2024) · The European Physical Journal Plus · doi:10.1140/epjp/s13360-024-05045-x · Cited by 10
Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review
Boscarino, S., & Russo, G. (2024) · SeMA Journal · doi:10.1007/s40324-024-00351-x · Cited by 10
“and differential algebraic equations (DAE) [5, 14].”
Yang, Y., Lin, S., Chen, S., et al. (2024) · IEEE Transactions on Power Systems · doi:10.1109/tpwrs.2024.3416150 · Cited by 10
“The conventional explicit Euler and Milstein iteration schemes are 0.5-order convergence and first-order convergence, respectively [15], [16], indicating that the MEPC iteration scheme exhibits higher convergence precision.”
InfluentialConsistency-based diagnosis using data-driven residuals and limited training data
Mohammadi, A., Krysander, M., & Jung, D. (2025) · Control Engineering Practice · doi:10.1016/j.conengprac.2025.106283 · Cited by 12
Localization Theorems for Nonlinear Eigenvalue Problems
Bindel, D., & Hood, A. (2015) · SIAM Review · doi:10.1137/15m1026511 · Cited by 8
Polyanin, A. D., & Shingareva, I. K. (2017) · International Journal of Non-Linear Mechanics · doi:10.1016/j.ijnonlinmec.2017.05.009 · Cited by 10
Parallel-in-Time Solution of Power Systems with Unscheduled Events
Günther, S., Falgout, R. D., Top, P., et al. (2020) · IEEE Power & Energy Society General Meeting (PESGM) · doi:10.1109/pesgm41954.2020.9281595 · Cited by 10
N’Guessan, M.-A., Massot, M., Séries, L., et al. (2019) · Journal of Computational and Applied Mathematics · doi:10.1016/j.cam.2019.112542 · Cited by 10
“The integration of the DAE needs special care due to its stiffness [25, 26, 27]; in addition the satisfaction of the divergence-free constraint cannot be achieved with an explicit method.”
BackgroundKuwabara-Kono numerical dissipation: a new method to simulate granular matter
James, G., Vorotnikov, K., & Brogliato, B. (2019) · IMA Journal of Applied Mathematics · doi:10.1093/imamat/hxz034 · Cited by 10
Zimmerman, A. G., & Kowalski, J. (2021) · Computers & Mathematics with Applications · doi:10.1016/j.camwa.2020.11.008 · Cited by 10
“the second order method from the family of k order constant time step size (∆t) backward difference formulas (BDF-k) [22].”
Method3D Airborne EM Forward Modeling Based on Time-Domain Spectral Element Method
Yin, C., Gao, Z., Su, Y., et al. (2021) · Remote Sensing · doi:10.3390/rs13040601 · Cited by 10
“The most frequently used method is the backward difference that is unconditionally stable without much limitations on the time-steps, so that the numerical accuracy in the forward modeling can be assured [44,45].”
MethodModeling Languages for Biochemical Network Simulation: Reaction vs Equation Based Approaches
Wiechert, W., Noack, S., & Elsheikh, A. (2009) · Springer eBooks · doi:10.1007/10_2009_64 · Cited by 10
“As a consequence several powerful numerical DAE solvers are available today [27].”
BackgroundSimulation of nonlinear random finite depth waves coupled with an elastic structure
Srisupattarawanit, T., Niekamp, R., & Matthies, H. G. (2005) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2005.02.027 · Cited by 10
An optimization method for solving some differential algebraic equations
El-Khateb, M. A., & Hussien, H. S. (2008) · Communications in Nonlinear Science and Numerical Simulation · doi:10.1016/j.cnsns.2008.07.003 · Cited by 10
Stiffness detection and reduction in discrete stochastic simulation of biochemical systems
Pu, Y., Watson, L. T., & Cao, Y. (2011) · The Journal of Chemical Physics · doi:10.1063/1.3548838 · Cited by 10
A dynamic step-size adaptation roach infestation optimization
Obagbuwa, I. C., Adewumi, A. O., & Adebiyi, A. A. (2014) · IEEE International Advance Computing Conference · doi:10.1109/iadcc.2014.6779498 · Cited by 10
“The simple Euler explicit method is given [9], [10] as: xn+1 = xn + hfxn, tn.”
Modular Multibody Formulation for Simulating Off-Road Tracked Vehicles
Omar, M. A. (2014) · Studies in Engineering and Technology · doi:10.11114/set.v1i2.462 · Cited by 10
“The proposed solver algorithm utilizes a predictor-corrector integrator with variable-order interpolation polynomials and variable time step (Bernan, Campbell, & Petzold, 1996) and (Ascher & Petzold, 1998).”
MethodRamirez-Chavez, L. E., Coello, C. A. C., & Rodríguez-Tello, E. (2012) · Cited by 10
Viscous Regularization for the Non-equilibrium Seven-Equation Two-Phase Flow Model
Delchini, M., Ragusa, J. C., & Berry, R. A. (2016) · Journal of Scientific Computing · doi:10.1007/s10915-016-0217-6 · Cited by 10
Knab, T. D., Clermont, G., & Parker, R. S. (2015) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2015.09.099 · Cited by 10
Walz, O., Marks, C., Viell, J., et al. (2017) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2016.12.014 · Cited by 10
Three-Dimensional Transient Electromagnetic Modeling Based on Fictitious Wave Domain Methods
Ji, Y., Hu, Y., & Imamura, N. (2017) · Pure and Applied Geophysics · doi:10.1007/s00024-017-1528-8 · Cited by 10
Modeling and Risk Analysis of Hardware-Software Systems
Zelenova, S. A., & Zelenov, S. V. (2017) · Proceedings of the Institute for System Programming of RAS · doi:10.15514/ispras-2017-29(5)-13 · Cited by 10
Polyanin, A. D., & Shingareva, I. K. (2018) · Applied Mathematics and Computation · doi:10.1016/j.amc.2018.04.071 · Cited by 10
Sensitivity Analysis for Hybrid Systems and Systems With Memory
Serban, R., & Recuero, A. M. (2019) · Journal of Computational and Nonlinear Dynamics · doi:10.1115/1.4044028 · Cited by 10
“The main reason for including this type of problems in our analysis is that this is the form of DAEs obtained when modeling incompressible flow by discretized Navier-Stokes equations in CFD [15] or with a stabilized index reduction of the index-3 equations of motion in multibody system problems [16,17].”
MethodShepherd, D., Miles, J., Heil, M., et al. (2019) · Journal of Scientific Computing · doi:10.1007/s10915-019-00965-8 · Cited by 10
A Large-Scale Assessment of Exact Model Reduction in the BioModels Repository
Pérez-Verona, I. C., Tribastone, M., & Vandin, A. (2019) · Lecture notes in computer science · doi:10.1007/978-3-030-31304-3_13 · Cited by 10
Application of integral method for investigating the boriding kinetics of AISI 316 steel
Zouzou, C., & Keddam, M. (2020) · Metallurgical Research & Technology · doi:10.1051/metal/2020011 · Cited by 10
On the application of shooting method for determining semicontinuous distillation limit cycles
Madabhushi, P. B., & Adams, T. A. (2020) · Process Safety and Environmental Protection · doi:10.1016/j.cherd.2020.05.019 · Cited by 10
Underactuated crane control for the automation of block erection in shipbuilding
Lee, H.-W., Roh, M.-I., & Ham, S.-H. (2021) · Automation in Construction · doi:10.1016/j.autcon.2021.103573 · Cited by 10
Application of Envelope-Following Techniques to the Simulation of Hybrid Power Systems
Linaro, D., del Giudice, D., Brambilla, A., et al. (2021) · IEEE Transactions on Circuits and Systems I Regular Papers · doi:10.1109/tcsi.2021.3136621 · Cited by 10
Hammoud, B., Olivieri, L., Righetti, L., Carpentier, J., & Prete, A. D. (2022) · Multibody System Dynamics · doi:10.1007/s11044-022-09818-z · Cited by 10
“To address all the above questions in an efficient way, we apply EIs only to the contact dynamics (which is stiff) while using an explicit Euler scheme for the remaining terms (which are not stiff).”
BackgroundMethodInfluentialEl-Ajou, A., Al-ghananeem, H., Saadeh, R., et al. (2023) · Frontiers in Physics · doi:10.3389/fphy.2023.1167797 · Cited by 10
“Many analytical, numerical, and numero-analytical techniques have been proposed previously and recently to provide solutions for differential equations with initial or boundary conditions, such as the Laplace and Fourier transforms method [2], the Adomian decomposition method [3–5], the variational iteration method [6…”
MethodNumerical Approximation of Rare Event Probabilities in Biochemically Reacting Systems
Mikeev, L., Sandmann, W., & Wolf, V. (2013) · Lecture notes in computer science · doi:10.1007/978-3-642-40708-6_2 · Cited by 10
“, [3, 8, 11]) applied to Eq.”
Dynamics and control of rotary cranes executing a load prescribed motion
Blajer, W., & Kołodziejczyk, K. (2006) · Cited by 10
Pseudo‐transient continuation for nonlinear transient elasticity
Gee, M. W., Kelley, C. T., & Lehoucq, R. B. (2008) · International Journal for Numerical Methods in Engineering · doi:10.1002/nme.2527 · Cited by 10
“3) to steady-state with a conventional initial value problem code [10, 1, 19] where the time step is controlled with stability and accuracy in mind.”
BackgroundUltrafast, Compact, and Energy Efficient All-Optical Switches Based on a Saturable Absorbing Cavity
Ma, P., Jäckel, H., Bona, G., et al. (2014) · IEEE Journal of Quantum Electronics · doi:10.1109/jqe.2014.2365995 · Cited by 10
“18, by separating the complex field amplitude into real and imaginary parts a = Re[a] + i Im[a], the coupled mode equation, together with the differential equation of α(t), became a set of three coupled real 1st-order differential equations, which were solved numerically by using the 4th-order Runge-Kutta method [22].”
MethodComment 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:10.3402/tellusa.v60i4.15298 · Cited by 10
Phase-field modeling of thermomechanical damage in tungsten under severe plasma transients
Crosby, T., & Ghoniem, N. M. (2012) · Computational Mechanics · doi:10.1007/s00466-012-0733-7 · Cited by 10
“The BDF takes the general form [1]:”
Computational issues of solving the 1D steady gradually varied flow equation
Artichowicz, W., & Szymkiewicz, R. (2014) · Journal of Hydrology and Hydromechanics · doi:10.2478/johh-2014-0031 · Cited by 10
“It is implicit, it is absolutely stable and additionally it is A – stable (Ascher and Petzold, 1998), which means that it can be applied to solve even stiff equations.”
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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
