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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Karras et al. showed that Heun's 2nd order method provides an excellent tradeoff between sample quality and sampling speed.
“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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Reversed graph embedding resolves complex single-cell trajectories
Qiu, X., Mao, Q., Tang, Y., Wang, L., Chawla, R., Pliner, H. A., & Trapnell, C. (2017) · Nature Methods · doi:10.1038/nmeth.4402 · Cited by 5,070 · see citing works
Stochastic Simulation of Chemical Kinetics
Gillespie, D. T. (2006) · Annual Review of Physical Chemistry · doi:10.1146/annurev.physchem.58.032806.104637 · Cited by 2,072 · see citing works
“A well-known strategy for numerically solving a stiff ODE dx / dt = f (x) is to replace the explicit updating formula xt+� t = xt + f (xt)� t with an implicit formula, such as xt+� t = xt + f (xt + � t)� t ( 36 ).”
BackgroundMethodInfluentialDiffusion Models: A Comprehensive Survey of Methods and Applications
Yang, L., Zhang, Z., Song, Y., Hong, S., Xu, R., Zhao, Y., Zhang, W., Cui, B., & Yang, M.-H. (2024) · ACM Computing Surveys · doi:10.1145/3626235 · Cited by 1,704 · see citing works
“In the following, we will discuss them in detail.”
BackgroundSolving Differential Equations in R: Package deSolve
Soetaert, K., Petzoldt, T., & Setzer, R. (2010) · Journal of Statistical Software · doi:10.18637/jss.v033.i09 · Cited by 1,694 · see citing works
“Except for the Runge-Kutta methods, all solvers implemented in deS-olve are variable order, variable step methods, that use the backward differentiation formulas and Adams methods, two important families of multistep methods (Asher and Petzold 1998).”
BackgroundMethodInfluentialDiscrete mechanics and variational integrators
Marsden, J. E., & West, M. (2001) · Acta Numerica · doi:10.1017/s096249290100006x · Cited by 1,553 · see citing works
“Constrained mechanical systems are particular examples of differential algebraic systems, and many of the techniques for the numerical integration of such systems can also be applied in the mechanical setting (see Hairer and Wanner (1996) and Ascher and Petzold (1998)).”
MethodACADO toolkit—An open‐source framework for automatic control and dynamic optimization
Houska, B., Ferreau, H. J., & Diehl, M. (2010) · Optimal Control Applications and Methods · doi:10.1002/oca.939 · Cited by 996 · see citing works
“Note that the ACADO BDF integrator, which is based on the algorithmic ideas in [5, 6, 27], can also deal with fully implicit differential algebraic equations of index 1, which have the form”
BackgroundMOOSE: A parallel computational framework for coupled systems of nonlinear equations
Gaston, D. R., Newman, C., Hansen, G. A., et al. (2009) · Nuclear Engineering and Design · doi:10.1016/j.nucengdes.2009.05.021 · Cited by 958 · see citing works
Reinforcement Learning and Dynamic Programming Using Function Approximators
Buşoniu, L., Babuška, R., Schutter, B. D., & Ernst, D. (2010) · CRC Press eBooks · doi:10.1201/9781439821091 · Cited by 919 · see citing works
Physically Based Deformable Models in Computer Graphics
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Stefanini, L., & Bede, B. (2008) · Nonlinear Analysis · doi:10.1016/j.na.2008.12.005 · Cited by 694 · see citing works
Learning Deep Transformer Models for Machine Translation
Wang, Q., Li, B., Xiao, T., Zhu, J., Li, C., Wong, D. F., & Chao, L. S. (2019) · Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics · doi:10.18653/v1/p19-1176 · Cited by 673 · see citing works
“• Inspired by the linear multi-step method in numerical analysis (Ascher and Petzold, 1998), we propose an approach based on dynamic linear combination of layers (D LCL ) to memorizing the features extracted from all preceding layers.”
MethodInfluentialParameter estimation for differential equations: a generalized smoothing approach
Ramsay, J. O., Hooker, G., Campbell, D., & Cao, J. (2007) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.1467-9868.2007.00610.x · Cited by 613 · see citing works
Dynamical Systems, the Three-Body Problem and Space Mission Design
Koon, W. S., Lo, M. W., Marsden, J. E., & Ross, S. D. (2000) · WORLD SCIENTIFIC eBooks · doi:10.1142/9789812792617_0222 · Cited by 557 · see citing works
Implicit–Explicit Runge–Kutta Schemes and Applications to Hyperbolic Systems with Relaxation
Pareschi, L., & Russo, G. (2005) · Journal of Scientific Computing · doi:10.1007/s10915-004-4636-4 · Cited by 549 · see citing works
“(i) There is a close analogy between hyperbolic systems with stiff relaxation and differential algebraic equations (DAE) [ 2 ].”
BackgroundModeling Elementary Heterogeneous Chemistry and Electrochemistry in Solid-Oxide Fuel Cells
Zhu, H., Kee, R. J., Janardhanan, V. M., et al. (2005) · Journal of The Electrochemical Society · doi:10.1149/1.2116607 · Cited by 524 · see citing works
“equations form a differential-algebraic system, which can be solved numerically.(40) To solve these equations, they are first cast in finitevolume form.”
BackgroundAn overview of simultaneous strategies for dynamic optimization
Biegler, L. T. (2007) · Chemical Engineering and Processing - Process Intensification · doi:10.1016/j.cep.2006.06.021 · Cited by 520 · see citing works
“Moreover, it is well known hat sequential approaches have properties of single shooting ethods and cannot handle open loop instability [7,8].”
BackgroundModeling and Control of Complex Physical Systems
Duindam, V., Macchelli, A., Stramigioli, S., et al. (2009) · Springer eBooks · doi:10.1007/978-3-642-03196-0 · Cited by 506 · see citing works
Consolidation of unorganized point clouds for surface reconstruction
Huang, H. F., Li, D., Zhang, H., Ascher, U. M., & Cohen‐Or, D. (2009) · ACM Transactions on Graphics · doi:10.1145/1618452.1618522 · Cited by 491 · see citing works
Self-citationDifferential variational inequalities
Pang, J., & Stewart, D. E. (2007) · Mathematical Programming · doi:10.1007/s10107-006-0052-x · Cited by 443 · see citing works
“In the past couple decades, differential algebraic equations (DAEs) have become a very important generalization of ODEs and have been studied extensively; see the two excellent monographs [19,77] and the references therein.”
BackgroundStiffness in stochastic chemically reacting systems: The implicit tau-leaping method
Rathinam, M., Petzold, L. R., Cao, Y., & Gillespie, D. T. (2003) · The Journal of Chemical Physics · doi:10.1063/1.1627296 · Cited by 435 · see citing works
Self-citationCao, Y., Li, S., Petzold, L. R., et al. (2003) · SIAM Journal on Scientific Computing · doi:10.1137/s1064827501380630 · Cited by 410 · see citing works
“Asymptotic numerical stability, the stability for a fixed stepsize h and n → ∞, has been thoroughly studied for ODE and DAE systems [18, 2, 1, 16, 17].”
MethodSelf-citationStable architectures for deep neural networks
Haber, E., & Ruthotto, L. (2017) · Inverse Problems · doi:10.1088/1361-6420/aa9a90 · Cited by 611 · see citing works
“, [2, 4] for the proof of stability criteria for the forward Euler method.”
MethodKeyes, D. E., McInnes, L. C., Woodward, C. S., et al. (2013) · The International Journal of High Performance Computing Applications · doi:10.1177/1094342012468181 · Cited by 394 · see citing works
“, (Ascher and Petzold, 1998; Lambert, 1991)) diffusion solver in the diffusion solve (Ropp et al.”
BackgroundMethodInfluentialA semidiscrete finite volume formulation for multiprocess watershed simulation
Qu, Y., & Duffy, C. J. (2007) · Water Resources Research · doi:10.1029/2006wr005752 · Cited by 319 · see citing works
“However, the multiple timescales arising from W08419 QU AND DUFFY: MULTIPROCESS WATERSHED SIMULATION 9 of 18 W08419 watershed processes typically make (29) a highly stiff system [Ascher and Petzold, 1998].”
BackgroundSTRANDS: Interactive Simulation of Thin Solids using Cosserat Models
Pai, D. K. (2002) · Computer Graphics Forum · doi:10.1111/1467-8659.00594 · Cited by 319 · see citing works
“(3) As with the kinematic differential equation, we would like to compute this in relative coordinates, but there is the standard complication of differentiating a spatial force in “moving coordinates” (e.”
BackgroundModeling and simulation of wear in revolute clearance joints in multibody systems
Flores, P. (2008) · Mechanism and Machine Theory · doi:10.1016/j.mechmachtheory.2008.08.003 · Cited by 316 · see citing works
“Hence, numerical algorithms with variable step size and order are an important feature for the computational strategy [20,32,33].”
Method̊ström, K. J. A., & Kumar, P. R. (2014) · Automatica · doi:10.1016/j.automatica.2013.10.012 · Cited by 304 · see citing works
“In the 1980s there had been significant advances in numerical solution of such equations [244,110,24,278].”
BackgroundMethodSemi-implicit spectral deferred correction methods for ordinary differential equations
Minion, M. L. (2003) · Communications in Mathematical Sciences · doi:10.4310/cms.2003.v1.n3.a6 · Cited by 264 · see citing works
“The SISDC[3] method also inherits this instability in a range of ∆t.”
BackgroundMethodSimultaneous dynamic optimization strategies: Recent advances and challenges
Kameswaran, S., & Biegler, L. T. (2006) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2006.05.034 · Cited by 239 · see citing works
Hucka, M., Finney, A., Doyle, J. C., Kitano, H., Kovitz, B., Shapiro, B. E., Funahashi, A., Matthews, J., Bornstein, B., Keating, S., et al. (2004) · Systems Biology · doi:10.1049/sb:20045008 · Cited by 232 · see citing works
Santina, C. D., Bicchi, A., & Rus, D. L. (2020) · IEEE Robotics and Automation Letters · doi:10.1109/lra.2020.2967269 · Cited by 234 · see citing works
“Thus Bα loses rank in the straight configuration, making (18) a differential algebraic equation, which is of course much more difficult to study, control, and simulate [19] than a standard ordinary differential equation.”
BackgroundRaja, L. L., Kee, R. J., Deutschmann, O., et al. (2000) · Catalysis Today · doi:10.1016/s0920-5861(00)00271-6 · Cited by 229 · see citing works
“[4] The Boundary-Layer equations are cast as a system of differentialalgebraic equations (DAE) by the method-of-lines approach.[5] These are then solved using the DASSL software.”
MethodThree dimensional inversion of multisource time domain electromagnetic data
Oldenburg, D. W., Haber, E., & Shekhtman, R. (2012) · Geophysics · doi:10.1190/geo2012-0131.1 · Cited by 226 · see citing works
“(Ascher and Petzold, 1998).”
MethodAlthoff, M. (2018) · EPiC series in computing · doi:10.29007/zbkv · Cited by 223 · see citing works
Kameswaran, S., & Biegler, L. T. (2007) · Computational Optimization and Applications · doi:10.1007/s10589-007-9098-9 · Cited by 220 · see citing works
“Radau collocation is well suited for stiff ODEs (L-stable), initial value DAEs, and is algebraically stable [ 3 ].”
BackgroundAlgorithms for strong coupling procedures
Matthies, H. G., Niekamp, R., & Steindorf, J. (2005) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2004.11.032 · Cited by 220 · see citing works
Convergence of the forward-backward sweep method in optimal control
McAsey, M. J., Mou, L., & Han, W. (2012) · Computational Optimization and Applications · doi:10.1007/s10589-011-9454-7 · Cited by 214 · see citing works
“The books [1, 27, 30] have extensive treatments of differential-algebraic systems, concentrating more on initial value problems rather than boundary value problems.”
BackgroundNumerical Solution of Ordinary Differential Equations
ATKINSON, K. E., Han, W., & Stewart, D. E. (2009) · Wiley eBooks · doi:10.1002/9781118164495 · Cited by 207 · see citing works
“A number of texts deal specifically with DAEs, such as Ascher and Petzold [10] and Brenan et al.”
BackgroundModeling and simulation: tools for metabolic engineering
Wiechert, W. (2002) · Journal of Biotechnology · doi:10.1016/s0168-1656(01)00418-7 · Cited by 206 · see citing works
“From a computational perspective so-called continuation methods are well suited to solve this problem (Allgower and Georg, 1990).”
Geometric multiscale modeling of the cardiovascular system, between theory and practice
Quarteroni, A., Veneziani, A., & Vergara, C. (2016) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2016.01.007 · Cited by 210 · see citing works
“In mathematical terms, 0D models are described by Differential Algebraic Systems (DAE) of equations and bridging region compatibility guarantees that the system is of type 1 [6], so it reduces to a standard system of ODE.”
Reversible Architectures for Arbitrarily Deep Residual Neural Networks
Chang, B., Meng, L., Haber, E., Ruthotto, L., Begert, D., & Holtham, E. (2018) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v32i1.11668 · Cited by 248 · see citing works
“(7) is the Verlet method (Ascher and Petzold 1998) that reads We choose Eq.”
MethodInfluentialOptimization-Based Control for Dynamic Legged Robots
Wensing, P. M., Posa, M., Hu, Y. H., Escande, A., Mansard, N., & Prete, A. D. (2023) · IEEE Transactions on Robotics · doi:10.1109/tro.2023.3324580 · Cited by 216 · see citing works
“However, to accurately represent the interaction be-tween rigid bodies, F contact is numerically stiff [81], inheriting the mechanical stiffness of the material properties in the robot.”
BackgroundTopology optimization of unsteady incompressible Navier–Stokes flows
Deng, Y., Liu, Z., Zhang, P., et al. (2011) · Journal of Computational Physics · doi:10.1016/j.jcp.2011.05.004 · Cited by 201 · see citing works
“The spatial integrations in the iterative procedure can be performed by the inner function postint of Comsol.”
MethodInfluentialCommer, M., & Newman, G. A. (2004) · Geophysics · doi:10.1190/1.1801936 · Cited by 200 · see citing works
Constrained process monitoring: Moving‐horizon approach
Rao, C. V., & Rawlings, J. B. (2002) · AIChE Journal · doi:10.1002/aic.690480111 · Cited by 198 · see citing works
Dynamics of Particles and Fields
Martin, B. (1999) · Advances in imaging and electron physics · doi:10.1016/s1076-5670(08)70227-1 · Cited by 199 · see citing works
“Other important current work relying on differential algebraic methods is the practical study of differential equations under algebraic constraints, so-called differential algebraic equations ((Ascher and Petzold 1998); (Griepentrog and März 1986); (Brenan, Campbell, and Petzold 1989; Matsuda 1980)).”
BackgroundA Review on Deep Learning in Medical Image Reconstruction
Zhang, H., & Dong, B. (2020) · Journal of the Operations Research Society of China · doi:10.1007/s40305-019-00287-4 · Cited by 192 · see citing works
“We first show that how ResNet is related to forward Euler scheme in numerical ODEs.”
MethodInfluentialClark, M., & Kavetski, D. (2010) · Water Resources Research · doi:10.1029/2009wr008894 · Cited by 190 · see citing works
“The vast literature on differential equations, including numerical solutions [e.g., Ascher and Petzold, 1998], as well as specialized techniques for parameter estimation [e.g., Cao et al., 2002], can be exploited to control numerical errors, reduce runtimes, etc. [29] 3.”
BackgroundMethodInfluentialContamination Source Determination for Water Networks
Laird, C. D., Biegler, L. T., Waanders, B. V. B., et al. (2005) · Journal of Water Resources Planning and Management · doi:10.1061/(asce)0733-9496(2005)131:2(125 · Cited by 189 · see citing works
Implicit methods for efficient musculoskeletal simulation and optimal control
van den Bogert, A. J., Blana, D., & Heinrich, D. (2011) · Procedia IUTAM · doi:10.1016/j.piutam.2011.04.027 · Cited by 187 · see citing works
“One possible remedy is to use implicit ODE solvers [23], but the computational cost of numerically estimating the Jacobian matrix of the ODE is often higher than the cost of using an explicit solver with smaller time steps.”
Geometric integrators for ODEs
McLachlan, R. I., & Quispel, G. R. W. (2006) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/39/19/s01 · Cited by 182 · see citing works
“A great deal is known about how to integrate DAEs in general ([51] chapters VI, VII, [4]).”
BackgroundNumerical Modeling in Open Channel Hydraulics
Szymkiewicz, R. (2010) · Water science and technology library · doi:10.1007/978-90-481-3674-2 · Cited by 183 · see citing works
Nicholson, B. L., Siirola, J. D., Watson, J., et al. (2017) · Mathematical Programming Computation · doi:10.1007/s12532-017-0127-0 · Cited by 179 · see citing works
“For more information on these techniques and details of their applicability, we defer to [3,4, 9,10].”
Aghili, F. (2005) · IEEE Transactions on Robotics · doi:10.1109/tro.2005.851380 · Cited by 172 · see citing works
Haber, E., Ascher, U. M., & Oldenburg, D. W. (2004) · Geophysics · doi:10.1190/1.1801938 · Cited by 175 · see citing works
“Common methods for very stiff equations are based on backward differentiation formulas (BDF) or on collocation at Radau points (Hairer and Wanner, 1991; Ascher and Petzold, 1998; Heling, 1998; Bastian, 1999; Turek, 1999).”
MethodSelf-citationVeerapaneni, S., Gueyffier, D., Zorin, D., et al. (2009) · Journal of Computational Physics · doi:10.1016/j.jcp.2008.11.036 · Cited by 174 · see citing works
“1 Spectral properties The choice of a computationally-efficient time-stepping scheme depends on whether (7) is stiff ([2], p.”
BackgroundDiscrete element modelling for process simulation in agriculture
Tijskens, E., Ramón, H., & Baerdemaeker, J. D. (2003) · Journal of Sound and Vibration · doi:10.1016/s0022-460x(03)00581-9 · Cited by 173 · see citing works
Hierarchical modeling and analysis of embedded systems
Alur, R., Dang, T., Esposito, J. M., et al. (2003) · Proceedings of the IEEE · doi:10.1109/jproc.2002.805817 · Cited by 172 · see citing works
“Linear Multistep Methods—see [54] for further details)”
Application of Centered Differentiation and Steepest Descent to Maximum Power Point Tracking
Xiao, W., Dunford, W. G., Palmer, P., et al. (2007) · IEEE Transactions on Industrial Electronics · doi:10.1109/tie.2007.899922 · Cited by 170 · see citing works
“The Euler method is simple in use, but shows local truncation errors [21], [22].”
Negruţ, D., Rampalli, R., Óttarsson, G., et al. (2006) · Journal of Computational and Nonlinear Dynamics · doi:10.1115/1.2389231 · Cited by 169 · see citing works
“For an ac count of relevant work in the area of numerical integration methods for the DAE of multibody dynamics the reader is re ferred to [2, 7, 4, 16, 1, 20, 28, 34] and references therein.”
MethodNumerical methods for mixed-integer optimal control problems
Säger, S. (2006) · Cited by 169 · see citing works
Modeling, simulation, sensitivity analysis, and optimization of hybrid systems
Barton, P. I., & Lee, C. K. (2002) · ACM Transactions on Modeling and Computer Simulation · doi:10.1145/643120.643122 · Cited by 162 · see citing works
Comparing Forward and Backward Reachability as Tools for Safety Analysis
Mitchell, I. M. (2007) · Springer eBooks · doi:10.1007/978-3-540-71493-4_34 · Cited by 164 · see citing works
Estimating cloth simulation parameters from video
Bhat, K. S., Twigg, C. D., Hodgins, J. K., et al. (2003) · KiltHub Repository · doi:10.5555/846276.846282 · Cited by 164 · see citing works
Mass-conserving subglacial hydrology in the Parallel Ice Sheet Model version 0.6
Bueler, E., & PELT, W. J. J. V. (2015) · Geoscientific model development · doi:10.5194/gmd-8-1613-2015 · Cited by 162 · see citing works
“(31) then the coupled hydrological equations system is differential algebraic (Ascher and Petzold, 1998), and harder to solve www.geosci-model-dev.net/8/1613/2015/ Geosci.”
BackgroundInfluentialHsiao, S., Chiang, W.-S., Jang, J., Wu, H.-L., Lu, W.-S., Chen, W., & Wu, Y.-T. (2020) · The Science of The Total Environment · doi:10.1016/j.scitotenv.2020.144439 · Cited by 162 · see citing works
Application of implicit–explicit high order Runge–Kutta methods to discontinuous-Galerkin schemes
Kanevsky, A., Carpenter, M. H., Gottlieb, D. J., et al. (2007) · Journal of Computational Physics · doi:10.1016/j.jcp.2007.02.021 · Cited by 160 · see citing works
“For an s-stage ERK of order p, the amplification factor RðzÞ is given as [2] RðzÞ 1⁄4 1þ zþ z 2 2 þ þ z p p! þ Xs”
Allison, J. T., & Herber, D. R. (2014) · AIAA Journal · doi:10.2514/1.j052182 · Cited by 156 · see citing works
Modeling and Simulation Methods for Design of Engineering Systems
Sinha, R., Paredis, C. J. J., Liang, V.-C., et al. (2000) · Journal of Computing and Information Science in Engineering · doi:10.1115/1.1344877 · Cited by 156 · see citing works
“Declarative or equation-based languages, on the other hand, do not impose a fixed causality on the model.”
BackgroundA fast algorithm for simulating vesicle flows in three dimensions
Veerapaneni, S., Rahimian, A., Biros, G., et al. (2011) · Journal of Computational Physics · doi:10.1016/j.jcp.2011.03.045 · Cited by 155 · see citing works
“High-order variants of the explicit (28) and the semi-implicit (30) schemes are readily obtained using the backward difference formula [1].”
BackgroundMethodAn event-driven method to simulate Filippov systems with accurate computing of sliding motions
Piiroinen, P. T., & Kuznetsov, Y. A. (2008) · ACM Transactions on Mathematical Software · doi:10.1145/1356052.1356054 · Cited by 155 · see citing works
Optimizing process economics online using model predictive control
Amrit, R., Rawlings, J. B., & Biegler, L. T. (2013) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2013.07.015 · Cited by 154 · see citing works
High-order multi-implicit spectral deferred correction methods for problems of reactive flow
Bourlioux, A., Layton, A. T., & Minion, M. L. (2003) · Journal of Computational Physics · doi:10.1016/s0021-9991(03)00251-1 · Cited by 151 · see citing works
“13 displays the L1 errors of the solutions computed using the MISDC(4; 4; 2; 6), MISDC(5; 5; 2; 5), ARK4(3)6L[2]SA (hereafter ARK4), and ARK5(4)8LSA (hereafter ARK5) methods for N 1⁄4 32, 64, 128, 256, and 512.”
Modelling of anisotropic growth in biological tissues
Menzel, A. (2004) · Biomechanics and Modeling in Mechanobiology · doi:10.1007/s10237-004-0047-6 · Cited by 147 · see citing works
“Ascher and Petzold (1998) as well as Angeles (1988) and references cited therein.”
BackgroundLu, Y., Zhong, A., Li, Q., & Dong, B. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1710.10121 · Cited by 235 · see citing works
“More importantly, we demonstrate that the connection between deep networks and numerical ODEs enables us to design new and more effective deep networks.”
BackgroundMethodInfluentialA toolbox for simulation of hybrid systems in matlab/simulink
Sanfelice, R. G., Copp, D. A., & Ñañez, P. (2013) · International Conference on Hybrid Systems: computation and control · doi:10.1145/2461328.2461346 · Cited by 143 · see citing works
“, [17, 3]) and have permitted the development of integration schemes for accurate computation of solutions to differential equations.”
BackgroundWhen Polymer Reaction Engineers Play Dice: Applications of Monte Carlo Models in PRE
Brandão, A. L. T., Soares, J. B. P., Pinto, J. C., & Alberton, A. L. (2015) · Macromolecular Reaction Engineering · doi:10.1002/mren.201500020 · Cited by 139 · see citing works
Cheny, Y., & Botella, O. (2009) · Journal of Computational Physics · doi:10.1016/j.jcp.2009.10.007 · Cited by 141 · see citing works
Implicit-explicit runge-kutta schemes and applications to hyperbolic systems with relaxation
Pareschi, L., & Russo, G. (2005) · Journal of Scientific Computing · doi:10.1007/bf02728986 · Cited by 141 · see citing works
“i) There is a close analogy between hyperbolic systems with stiff relaxation and differential algebraic equations (DAE) [3].”
InfluentialPETSc/TS: A Modern Scalable ODE/DAE Solver LibraryPreprint
Abhyankar, S. G., Brown, J., Constantinescu, E. M., et al. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1806.01437 · Cited by 139 · see citing works
“The theory of numerical algorithms for ODEs and DAEs is covered in several monographs, including [Hairer et al. 2008; Hairer and Wanner 2002; Ascher and Petzold 1998; Brenan et al. 1996; Butcher 2008].”
MethodInversion of time domain three-dimensional electromagnetic data
Haber, E., Oldenburg, D. W., & Shekhtman, R. (2007) · Geophysical Journal International · doi:10.1111/j.1365-246x.2007.03365.x · Cited by 138 · see citing works
Arqub, O. A., & Rashaideh, H. (2017) · Neural Computing and Applications · doi:10.1007/s00521-017-2845-7 · Cited by 138 · see citing works
A Geometric Approach to Solving Problems of Control Constraints: Theory and a DAE Framework
Blajer, W., & Kołodziejczyk, K. (2004) · Multibody System Dynamics · doi:10.1023/b:mubo.0000040800.40045.51 · Cited by 135 · see citing works
“Of course, by differentiating a number of constants of integration is introduced, which means that the initial values of x must satisfy the initial constraint conditions and all its differentiated forms (the solution to DAEs is very sensitive to the inconsistency of initial vales [ 7 , 8]).”
BackgroundMethodInfluentialStochastic modelling of gene regulatory networks
Samad, H. E., Khammash, M., Petzold, L. R., et al. (2005) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.1018 · Cited by 135 · see citing works
“Although implicit methods are clearly more expensive per step than explicit methods, this cost is more than compensated for in solving stiff systems by the fact that the implicit methods can take much larger timesteps [33].”
BackgroundSelf-citationThe manifold structure of limb coordination in walking Drosophila
DeAngelis, B. D., Zavatone-Veth, J. A., & Clark, D. A. (2019) · eLife · doi:10.7554/elife.46409 · Cited by 104 · see citing works
“…fstance 5ð Þ d 6 dt ¼ p t swing fswing 6ð Þ 1þa sin 3 6ð Þ þ p t stance fstance 6ð Þ (13) We integrated this system of ODEs with the parameters given in Table 3 using Heun’s method, a second-order explicit Runge-Kutta integrator (Ascher and Petzold, 1998) with a timestep of 0.025 ms, as it…”
InfluentialGarcía-Vallejo, D., Mayo, J. M., Escalona, J. L., et al. (2004) · Nonlinear Dynamics · doi:10.1023/b:nody.0000027747.41604.20 · Cited by 128 · see citing works
“An implicit method requires solving a non-linear system of equations in each time step [ 8 ] and the jacobian of this system has to be calculated.”
MethodA Review of Damping Models for Structures With Mechanical Joints1
Mathis, A. T., Balaji, N. N., Kuether, R. J., Brink, A. R., Brake, M. R. W., & Quinn, D. D. (2020) · Applied Mechanics Reviews · doi:10.1115/1.4047707 · Cited by 128 · see citing works
“The latter case can be reformulated as a Differential Algebraic Equation (DAE), enabling the use of relevant methods such as ǫ-embedding and singular perturbations that are developed primarily for solving DAEs [202, 203].”
MethodOptimization-based simulation of nonsmooth rigid multibody dynamics
Anitescu, M. (2005) · Mathematical Programming · doi:10.1007/s10107-005-0590-7 · Cited by 125 · see citing works
“Under the assumptions from this work, we have shown that the total energy of the system stays bounded on every finite time interval and that the geometrical constraints are satisfied with one order higher than the order of the method [2], which is one way to describe constraint stabilization [ 8 ].”
MethodThe use of particle swarm optimization for dynamical analysis in chemical processes
Ourique, C. O., Junior, E. C. B., & Pinto, J. C. (2002) · Computers & Chemical Engineering · doi:10.1016/s0098-1354(02)00153-9 · Cited by 124 · see citing works
Asynchronous contact mechanics
Harmon, D., Vouga, E., Smith, B., et al. (2009) · ACM Transactions on Graphics · doi:10.1145/1531326.1531393 · Cited by 123 · see citing works
Phase-field model of pitting corrosion kinetics in metallic materials
Ansari, T. Q., Xiao, Z., Hu, S., Li, Y., Luo, J., & Shi, S. (2018) · npj Computational Materials · doi:10.1038/s41524-018-0089-4 · Cited by 122 · see citing works
A Perspective on Methods for Trajectory Optimization
Ross, I. M., & Fahroo, F. (2002) · AIAA/AAS Astrodynamics Specialist Conference and Exhibit · doi:10.2514/6.2002-4727 · Cited by 122 · see citing works
Barber, J. O., Alberding, J. P., Restrepo, J. M., et al. (2008) · Annals of Biomedical Engineering · doi:10.1007/s10439-008-9546-4 · Cited by 122 · see citing works
Modeling Reaction and Transport of Multiple Species in Water Distribution Systems
Shang, F., Uber, J. G., & Rossman, L. A. (2007) · Environmental Science & Technology · doi:10.1021/es072011z · Cited by 120 · see citing works
Reachability Analysis of Nonlinear Differential-Algebraic Systems
Althoff, M., & Krogh, B. H. (2014) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2013.2285751 · Cited by 120 · see citing works
“DAEs occur in many practical applications, typically when there are constraints on the state variables [10], [14].”
BackgroundA User-Friendly Software Package for HIFU Simulation
Soneson, J. E., & Ebbini, E. S. (2009) · AIP conference proceedings · doi:10.1063/1.3131405 · Cited by 119 · see citing works
Anitescu, M., & Hart, G. D. (2004) · International Journal for Numerical Methods in Engineering · doi:10.1002/nme.1047 · Cited by 118 · see citing works
Nano boundary layers over stretching surfaces
Gorder, R. A. V., Sweet, E., & Vajravelu, K. (2009) · Communications in Nonlinear Science and Numerical Simulation · doi:10.1016/j.cnsns.2009.06.004 · Cited by 115 · see citing works
A computational strategy for multiscale systems with applications to Lorenz 96 model
Fatkullin, I., & Vanden‐Eijnden, E. (2004) · Journal of Computational Physics · doi:10.1016/j.jcp.2004.04.013 · Cited by 115 · see citing works
“[2,10–13,19]), the multiscale schemes also apply in the context of dynamical systems with stochastic”
BackgroundUnderstanding and Improving Transformer From a Multi-Particle Dynamic System Point of ViewPreprint
Lu, Y., Li, Z., He, D., Sun, Z., Dong, B., Qin, T., Wang, L., & Liu, T. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1906.02762 · Cited by 146 · see citing works
“In this way, the stack of self-attention sub-layers and position-wise FFN sub-layers with residual connections can be viewed as solving the ODE problem numerically using the Lie–Trotter splitting scheme [12] and the Euler’s method [13].”
MethodInfluential
- 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 methods for differential equations
- Physical Sciences › Mathematics › Modeling and Simulation › Mathematical and Computational Methods
- Physical Sciences › Computer Science › Computational Theory and Mathematics › 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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