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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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Meliopoulos, A. P. S., Cokkinides, G. J., & Stefopoulos, G. K. (2006) · Proceedings of the IEEE Conference on Decision & Control, including the Symposium on Adaptive Processes · doi:10.1109/cdc.2005.1583252 · Cited by 17
“Any Lobatto method with s collocation points has an order of accuracy of 2 2s , and therefore the method is order-four accurate [22-24].”
BackgroundMethodInfluentialOn the numerical solution of differential–algebraic equations with index-2
Çeli̇k, E. (2003) · Applied Mathematics and Computation · doi:10.1016/j.amc.2003.08.018 · Cited by 17
Parameter optimization for differential equations in asset price forecasting
Duran, A., & Caginalp, G. (2008) · Optimization methods & software · doi:10.1080/10556780801996178 · Cited by 17
“• We have methods that are efficient for certain stiff and non-stiff applications [3].”
MethodA stabilized explicit Lagrange multiplier based domain decomposition method for parabolic problems
Zheng, Z., Simeon, B., & Petzold, L. R. (2008) · Journal of Computational Physics · doi:10.1016/j.jcp.2008.01.057 · Cited by 17
“following the standard theory for numerical ODEs (e.g., [ 44 ]).”
Self-citationTree‐based characterization of low index circuit configurations without passivity restrictions
Encinas, A. J., & Riaza, R. (2007) · International Journal of Circuit Theory and Applications · doi:10.1002/cta.417 · Cited by 17
Supplementary techniques for 2S-DIRK-based EMT simulations
Noda, T., Kikuma, T., & Yonezawa, R. (2014) · Electric Power Systems Research · doi:10.1016/j.epsr.2014.04.011 · Cited by 17
Coupled Electromagnetic Field/Circuit Simulation. Modeling and Numerical Analysis
Baumanns, S. (2012) · Kölner Universitäts PublikationsServer (Universität zu Köln) · Cited by 17
Partial Slip Effects on a Rotating Flow of Two Phase Nano Fluid Over a Stretching Surface
Nadeem, S., Rehman, A. U., Mehmood, R., et al. (2014) · Current Nanoscience · doi:10.2174/1573413710666140604221338 · Cited by 16
A massless boundary component mode synthesis method for elastodynamic contact problems
Tec, C. D. M., Gross, J., & Krack, M. (2021) · Computers & Structures · doi:10.1016/j.compstruc.2021.106698 · Cited by 16
“For the theoretical background and alternative techniques we refer to [9, 16, 8, 43].”
BackgroundMethodAn Efficient Third-Derivative Hybrid Block Method for the Solution of Second-Order BVPs
Rufai, M. A. (2022) · Mathematics · doi:10.3390/math10193692 · Cited by 16
“For more explanation on shooting methods theory, see Ascher et al. [3], Ascher and Petzold [4], Atkinson et al. [5], Keskin [6], and Hoffman [7].”
de Castro, A., Bochev, P., Kuberry, P., et al. (2023) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2023.116398 · Cited by 16
Symbolic Numeric Index Analysis Algorithm for Differential Algebraic Equations
Chowdhry, S., Krendl, H., & Linninger, A. A. (2004) · Industrial & Engineering Chemistry Research · doi:10.1021/ie0341754 · Cited by 16
On Exact Conservation for the Euler Equations with Complex Equations of State
Banks, J. W. (2010) · Communications in Computational Physics · doi:10.4208/cicp.090909.100310a · Cited by 16
“3) Starting from the state [ρ0,u0,p0], the solution is determined in a Runge-Kutta like strategy [17] pk+1≈ pk− ∆ρ 6 [”
MethodAnalysis of Linear Variable Coefficient Delay Differential-Algebraic Equations
Ha, P., Mehrmann, V., & Steinbrecher, A. (2014) · Journal of Dynamics and Differential Equations · doi:10.1007/s10884-014-9386-x · Cited by 16
Partitioned time discretization for parallel solution of coupled ODE systems
Connors, J. M., & Miloua, A. A. (2010) · BIT Numerical Mathematics · doi:10.1007/s10543-010-0295-z · Cited by 16
“Convergence of the single step algorithms now follows from the Lax equivalence theorem (see also [ 1 ]).”
MethodArslantürk, C. (2010) · Heat and Mass Transfer · doi:10.1007/s00231-010-0673-8 · Cited by 16
“In order to investigate the accuracy of the HPM solution, the problem is also solved numerically by using MAPLE which uses a finite difference method with Richardson extrapolation [ 17 ], and the corresponding results are compared with the HPM solution and presented in Fig. 3. The results of the comparison show that t…”
MethodInfluentialPerformance Analysis of Service Composition Based on Fuzzy Differential Equations
Ding, Z., Shen, H., & Kandel, A. (2010) · IEEE Transactions on Fuzzy Systems · doi:10.1109/tfuzz.2010.2089633 · Cited by 16
“For the rounding error, since the implicit Runge–Kutta method has stable area [11], and the algorithm is guaranteed to converge in the stable area, the rounding error of the perturbation cannot increase and thus will decrease to 0 in the iteration process [2].”
MethodQuadratized model of nonlinear saturable-core inductor for time-domain simulation
Stefopoulos, G. K., Cokkinides, G. J., & Meliopoulos, A. P. S. (2009) · IEEE Power & Energy Society General Meeting (PESGM) · doi:10.1109/pes.2009.5275663 · Cited by 16
Ghaneai, H., & Hosseini, M. M. (2015) · Applied Mathematical Modelling · doi:10.1016/j.apm.2015.10.002 · Cited by 16
Static and Dynamic Debugging of Modelica Models
Pop, A., Sjölund, M., Asghar, A., et al. (2012) · Linköping electronic conference proceedings · doi:10.3384/ecp12076443 · Cited by 16
A method based on Rayleigh quotient gradient flow for extreme and interior eigenvalue problems
Liu, S., & Luo, X.-L. (2010) · Linear Algebra and its Applications · doi:10.1016/j.laa.2009.12.023 · Cited by 16
“On the definition and properties of DAEs, one can refer to [1,16] and references therein.”
BackgroundExact Finite Difference Schemes for Three-Dimensional Linear Systems with Constant Coefficients
Á, Đ. Q., & Tuan, H. M. (2017) · Vietnam Journal of Mathematics · doi:10.1007/s10013-017-0249-8 · Cited by 16
Leroux, N. R., Marsh, C. B., & Pomeroy, J. W. (2020) · Water Resources Research · doi:10.1029/2020wr027466 · Cited by 16
“The spatial derivatives of the NERE equation were discretized using the cell‐centered finite difference method, and the explicit Heun scheme, a modified Euler's method (e.g., Ascher & Petzold, 1998), was applied to solve for the ODE time component of the 2‐”
MethodChakkingal, A., Pirro, L., da Cruz, A. R. C., Barrios, A. J., Virginie, M., Khodakov, A. Y., & Thybaut, J. W. (2021) · Chemical Engineering Journal · doi:10.1016/j.cej.2021.129633 · Cited by 16
Modeling of Aircraft Prescribed Trajectory Flight as an Inverse Simulation Problem
Blajer, W., Graffstein, J., & Krawczyk, M. (2009) · Springer eBooks · doi:10.1007/978-1-4020-8778-3_14 · Cited by 16
“In order to improve numerical accuracy of solution to DAEs (18), the Euler method as above can possibly be replaced by a higher-order backward difference approximation method [ 11 ,12].”
MethodHoang, T. L., Marian, J., Bulatov, V. V., & Hosemann, P. (2015) · Journal of Computational Physics · doi:10.1016/j.jcp.2015.07.061 · Cited by 16
“The SCD method is based on the stochastic simulation algorithm (SSA) proposed originally by Gillespie for simulations of chemical kinetics in well-stirred systems [7,8].”
MethodAn asynchronous integration and event detection algorithm for simulating multi-agent hybrid systems
Esposito, J. M., & Kumar, V. R. (2004) · ACM Transactions on Modeling and Computer Simulation · doi:10.1145/1029174.1029177 · Cited by 16
“Mirtich [2000] introduced a multibody Newtonian mechanics simulator for graphics applications which is built upon the Time Warp algorithm [Jefferson 1985]. In this framework, each rigid body is considered a process with its own local clock; collisions and other changes in contact states (rolling, sliding, etc.) play t…”
BackgroundMethodInfluentialDobbins, R. R., & Smooke, M. D. (2010) · Flow Turbulence and Combustion · doi:10.1007/s10494-010-9278-z · Cited by 16
Implementing state-space methods for multizone contaminant transport
Parker, S. T., Lorenzetti, D. M., & Sohn, M. D. (2013) · Building and Environment · doi:10.1016/j.buildenv.2013.09.021 · Cited by 16
The Optimal Control Approach to Dynamical Inverse Problems
Steiner, W., & Reichl, S. (2012) · Journal of Dynamic Systems Measurement and Control · doi:10.1115/1.4005365 · Cited by 16
Cognitive Polar Receiver Using Two Injection-Locked Oscillator Stages
Chen, C.-T., Hsiao, C.-H., Horng, T., et al. (2011) · IEEE Transactions on Microwave Theory and Techniques · doi:10.1109/tmtt.2011.2172811 · Cited by 16
“By starting with a numerical method known as the Euler method [17], the derivative can be replaced by the finite-difference approximation”
MethodTime Explicit Schemes and Spatial Finite Differences Splittings
Chehab, J., & da Costa, B. A. S. (2003) · Journal of Scientific Computing · doi:10.1023/b:jomp.0000008719.48134.4f · Cited by 16
“When using explicit schemes for such problems, either none or too many digits of accuracy are obtained, since the size of the time step is constrained not only by accuracy requirements, but must also obey absolute stability restrictions (see [1]).”
BackgroundArrieta–Camacho, J. J., & Biegler, L. T. (2005) · Annals of the New York Academy of Sciences · doi:10.1196/annals.1370.001 · Cited by 16
Delchini, M., Ragusa, J. C., & Berry, R. A. (2017) · Nuclear Engineering and Design · doi:10.1016/j.nucengdes.2017.04.034 · Cited by 16
Tan, P., Ni, M., Shao, Z., et al. (2017) · Applied Energy · doi:10.1016/j.apenergy.2017.05.185 · Cited by 16
Time Integration-Based IGD Methods for Eigen-Analysis of Large Delayed Cyber-Physical Power System
Ye, H., Li, T., & Liu, Y. (2019) · IEEE Transactions on Power Systems · doi:10.1109/tpwrs.2019.2936871 · Cited by 16
“The almost identical accuracy originates from the fact that PS collocation method is IRK method in nature [31].”
MethodDynamical System Inspired Adaptive Time Stepping Controller for Residual Network Families
Yang, Y., Wu, J., Li, H., Li, X., Shen, T., & Lin, Z. (2020) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v34i04.6141 · Cited by 17
“(2) is stable when the following condition holds (Ascher and Petzold 1998): max |1 + hλi(J(t))| ≤ 1, (4) where λi denotes the i-th eigenvalue of the Jacobian matrix defined as J(t) = ∇yF(y(t),W(t)).”
BackgroundMethodModel Based Diagnosis and Supervision of Industrial Gas Turbines
Larsson, E. (2014) · Linköping University Electronic Press eBooks · doi:10.3384/diss.diva-106256 · Cited by 16
“This means that the algebraic constraint g1 is invertible (at least locally), and can be solved for z2 (Hairer et al., 1991; Ascher and Petzold, 1998).”
BackgroundMethodInfluentialConstraint Reasoning for Differential Models
Cruz, J. (2005) · New University of Lisbon's Repository (New University of Lisbon) · Cited by 16
Linear and Nonlinear Model Order Reduction for Numerical Simulation of Electric Circuits
Mohaghegh, K. (2010) · Cited by 16
An Optimal Control Toolbox for MATLAB Based on CasADi
Leek, V. (2016) · KTH Publication Database DiVA (KTH Royal Institute of Technology) · Cited by 16
Belliveau, P., & Haber, E. (2017) · Geophysics · doi:10.1190/geo2017-0494.1 · Cited by 15
Şuşcă, M., Mihaly, V. M., Stănese, M., et al. (2021) · Mathematics · doi:10.3390/math9070731 · Cited by 15
Hoang, M. T. (2023) · International Journal of Computer Mathematics · doi:10.1080/00207160.2023.2248304 · Cited by 15
“is used to approximate the rate of convergence of the numerical schemes (see [8]).”
MethodInfluentialA dynamically consistent nonstandard finite difference scheme for a generalized SEIR epidemic model
Hoang, M. T., & Ehrhardt, M. (2023) · The Journal of Difference Equations and Applications · doi:10.1080/10236198.2023.2291151 · Cited by 15
“We now compare the NSFD scheme (16) with two wellknown standard Runge-Kutta schemes, namely the Euler scheme and the second-order Runge-Kutta (RK2) scheme (see [6]).”
MethodRobust Predictive Motion Planning by Learning Obstacle Uncertainty
Zhou, J. Z., Gao, Y., Johansson, O., et al. (2025) · IEEE Transactions on Control Systems Technology · doi:10.1109/tcst.2025.3533378 · Cited by 15
“This continuous-time model can be approximated to a discrete-time model in (1) using a sampling interval T with different discretization methods [32].”
A dynamical method of DAEs for the smallest eigenvalue problem
Luo, X.-L. (2012) · Journal of Computational Science · doi:10.1016/j.jocs.2012.01.002 · Cited by 15
Chaotic cryptographic scheme and its randomness evaluation
Stoyanov, B. P. (2012) · AIP conference proceedings · doi:10.1063/1.4758983 · Cited by 15
Non‐parametric estimation of exact consumer surplus with endogeneity in price
Vanhems, A. (2010) · Econometrics Journal · doi:10.1111/j.1368-423x.2010.00311.x · Cited by 15
“Various classical algorithms can be used such as Euler-Cauchy algorithm, Heun’s method or Runge Kutta method (see Ascher and Petzold (1998) or Collatz (1960) for a general overview of these numerical methods). For example, Hausman and Newey (1995) use a BuerlischStoer algorithm from Numerical recipes and Vartia (1983)…”
Stable, high order accurate adaptive schemes for long time, highly intermittent geophysics problems
Erickson, B. A., & Nordström, J. (2014) · Journal of Computational and Applied Mathematics · doi:10.1016/j.cam.2014.04.019 · Cited by 15
A new stable splitting for singularly perturbed ODEs
Schütz, J., & Kaiser, K. (2016) · Applied Numerical Mathematics · doi:10.1016/j.apnum.2016.04.004 · Cited by 15
Finite Difference Computing with Exponential Decay Models
Langtangen, H. P. (2016) · Lecture notes in computational science and engineering · doi:10.1007/978-3-319-29439-1 · Cited by 15
“The methods are just briefly listed, and we refer to the rich literature for more detailed descriptions and analysis - the books [12, 2, 3, 4] are all excellent resources on numerical methods for ODEs.”
MethodNumerical solution of the finite horizon stochastic linear quadratic control problem
Damm, T., Mena, H., & Stillfjord, T. (2017) · Numerical Linear Algebra with Applications · doi:10.1002/nla.2091 · Cited by 15
“where αj , β are the coefficients of the p-step BDF formula, see [3].”
MethodParallel cyclic reduction decomposition for dynamic optimization problems
Wan, W., Eason, J. P., Nicholson, B. L., et al. (2017) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2017.09.023 · Cited by 15
Nonstandard finite difference schemes for solving an SIS epidemic model with standard incidence
Hoang, M. T., & Egbelowo, O. F. (2019) · Rendiconti del Circolo Matematico di Palermo Series 2 · doi:10.1007/s12215-019-00436-x · Cited by 15
The mathematical foundations of physical systems modeling languages
Benveniste, A., Caillaud, B., & Malandain, M. (2020) · Annual Reviews in Control · doi:10.1016/j.arcontrol.2020.08.001 · Cited by 15
“Developing existence/uniqueness results and discretization schemes, for a given class of (single-mode or multimode) DAE systems [3];”
BackgroundRest-to-Rest Trajectory Planning for Planar Underactuated Cable-Driven Parallel Robots
Idà, E., Berti, A., Brückmann, T., et al. (2017) · Mechanisms and machine science · doi:10.1007/978-3-319-61431-1_18 · Cited by 15
Hybrid state machine model for fast model predictive control: Application to path tracking
Amir, M., & Givargis, T. D. (2017) · doi:10.1109/iccad.2017.8203777 · Cited by 15
“The framework contains efficient implementations for numerical integrators, Runge-Kutta [29] and BDF [30] to solve ODEs and differential algebraic equations(DAEs).”
MethodStripling, H. F., Anitescu, M., & Adams, M. L. (2012) · Annals of Nuclear Energy · doi:10.1016/j.anucene.2012.08.019 · Cited by 15
“For ODEs, Cao and Petzold showed that integrating the adjoint against local truncation error estimates provides an estimate for the global discretization error.”
MethodA Nonlinear Hybrid Life Support System: Dynamic Modeling, Control Design, and Safety Verification
Glavaški, S., Subramanian, D., Ariyur, K. B., et al. (2007) · IEEE Transactions on Control Systems Technology · doi:10.1109/tcst.2007.899649 · Cited by 15
“We use collocation on finite elements ([6], [7]) to convert the infinite dimensional, continuous, differential equations into a finite dimensional, discretized set of algebraic equations within each quarter cycle (see Fig.”
MethodOn the convergence of finite difference methods for PDE under temporal refinement
Love, E., & Rider, W. J. (2013) · Computers & Mathematics with Applications · doi:10.1016/j.camwa.2013.04.019 · Cited by 15
Szymkiewicz, R., & Gąsiorowski, D. (2012) · Journal of Hydrology · doi:10.1016/j.jhydrol.2012.07.009 · Cited by 15
Implicit integration methods for dislocation dynamics
Gardner, D. J., Woodward, C. S., Reynolds, D. R., et al. (2015) · Modelling and Simulation in Materials Science and Engineering · doi:10.1088/0965-0393/23/2/025006 · Cited by 15
“Instead, use of multi-stage methods [9, 10], where only one prior solution is required to achieve higher orders, can be a better fit.”
MethodModel order reduction for object-oriented models: a control systems perspective
Donida, F., Casella, F., & Ferretti, G. (2010) · Mathematical and Computer Modelling of Dynamical Systems · doi:10.1080/13873954.2010.506760 · Cited by 15
“One important point to remark upon is the fact that simplifications may increase the index of the DAE system (2), thus preventing the numerical solution with standard solvers [ 37 ].”
BackgroundA Multi-Resolution Discontinuous Galerkin Method for Unsteady Compressible Flows
Shelton, A., Smith, M., & Zhou, H. M. (2008) · AIAA Fluid Dynamics Conference and Exhibit · doi:10.2514/6.2008-4140 · Cited by 15
“explicit Runge-Kutta multistage schemes may be written [7] as w = u + ∆t s ∑”
BackgroundConstraint-consistent Runge–Kutta methods for one-dimensional incompressible multiphase flow
Sanderse, B., & Veldman, A. E. P. (2019) · Journal of Computational Physics · doi:10.1016/j.jcp.2019.02.001 · Cited by 15
“2Hessenberg form: ODEs coupled with constraint, with explicit identification of algebraic and differential variables, where the algebraic variables may all be eliminated using the same number of differentiations [5].”
BackgroundSpontaneous capillary propulsion of liquid droplets on substrates with nonuniform curvature
Galatola, P. (2018) · Physical Review Fluids · doi:10.1103/physrevfluids.3.103601 · Cited by 15
Effects of Nonlinear Propagation of Focused Ultrasound on the Stable Cavitation of a Single Bubble
Bakhtiari-Nejad, M., & Shahab, S. (2018) · Acoustics · doi:10.3390/acoustics1010003 · Cited by 15
“To solve in frequency-domain, sec nd-o der, diagonally-implicit Rung -Kut a (DIRK), and the CrankNicolson (CN) method, which is based on trapezoidal rule [48], are used.”
MethodInfluentialHoang, M. T., & Egbelowo, O. F. (2020) · Boletín de la Sociedad Matemática Mexicana · doi:10.1007/s40590-020-00275-2 · Cited by 15
Stewart, J. M., & Mommert, M. (2014) · Cambridge University Press eBooks · doi:10.1017/9781009029728 · Cited by 15
Characterizing differential algebraic equationswithout the use of derivative arrays
März, R. (2005) · Computers & Mathematics with Applications · doi:10.1016/j.camwa.2005.08.015 · Cited by 15
Shiraishi, F., Tomita, T., Iwata, M., et al. (2009) · Mathematical Biosciences · doi:10.1016/j.mbs.2009.09.001 · Cited by 15
High gain observer for a class of implicit systems
Hammouri, H., & Marchand, N. (2002) · Proceedings of the IEEE Conference on Decision & Control, including the Symposium on Adaptive Processes · doi:10.1109/cdc.2000.912868 · Cited by 15
“It would be interesting for further investigations to see if the proposed scheme can be generalized to higher implicity degrees as it can be done for some numerical methods [1].”
BackgroundInfluentialQuadratic integration method for transient simulation and harmonic analysis
Stefopoulos, G. K., Cokkinides, G. J., & Meliopoulos, A. P. S. (2008) · International Conference on Harmonics and Quality of Power · doi:10.1109/ichqp.2008.4668852 · Cited by 15
Efficient dynamic simulation of robotic systems with hierarchy
Esposito, J. M., & Kumar, V. (2002) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/robot.2001.933049 · Cited by 15
Augmented Lagrangian index-3 semi-recursive formulations with projections
Dopico, D., Varela, Á. L., & Luaces, A. (2020) · Multibody System Dynamics · doi:10.1007/s11044-020-09771-9 · Cited by 14
Magneto-Thermal Thin Shell Approximation for 3D Finite Element Analysis of No-Insulation Coils
Schnaubelt, E., Atalay, S., Woźniak, M., et al. (2023) · IEEE Transactions on Applied Superconductivity · doi:10.1109/tasc.2023.3340648 · Cited by 14
“An adaptive implicit Euler scheme is used for time integration [29].”
MethodLiang, Q., Jiang, K., & Zhang, P. (2013) · Mathematical Methods in the Applied Sciences · doi:10.1002/mma.2868 · Cited by 14
“However, if we set 1 D 1=2, 2 D 1=2, we can obtain the improved Euler scheme [14], which is a second-order scheme.”
MethodOn dual Schur domain decomposition method for linear first-order transient problems
Nakshatrala, K. B., Prakash, A. J., & Hjelmstad, K. D. (2009) · Journal of Computational Physics · doi:10.1016/j.jcp.2009.07.016 · Cited by 14
“Some of the common indices are – the differential index, local index, geometric index, tractability index, and perturbation index [1].”
BackgroundCrockatt, M., Christlieb, A. J., & Hauck, C. D. (2020) · Journal of Computational Physics · doi:10.1016/j.jcp.2020.109765 · Cited by 14
Makuch, N., Dziarski, P., Kulka, M., et al. (2021) · Materials · doi:10.3390/ma14185146 · Cited by 14
“The estimation of layers’ thicknesses by the integral diffusion model was carried out via a numerical solution of DAE system [27] by taking as initial conditions a0 = 44.”
MethodStröhle, T., & Betsch, P. (2022) · International Journal for Numerical Methods in Engineering · doi:10.1002/nme.6951 · Cited by 14
“Now, the first boundary condition in (14)2 yields f (t) = EA∂su(0, t), where (16) implies ∂su(0, t) = ( Φ′(t) − Ψ′(t) ) c−1.”
Zero Stability Well Predicts Performance of Convolutional Neural Networks
Chen, L., Jin, L., & Shang, M. (2022) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v36i6.20576 · Cited by 14
Semi-implicit Krylov deferred correction methods for differential algebraic equations
Bu, S., Huang, J., & Minion, M. L. (2012) · Mathematics of Computation · doi:10.1090/s0025-5718-2012-02564-6 · Cited by 14
“We first consider the system (see [5]) x1 = (α− 1 2−t )x1 + (2− t)αz + 3−t 2−t exp = f1(x1, z), x2 = 1−α t−2 x1 − 10(4)x2 + (α− 1)z + (10(4) + 1) exp = f2(x1, x2, z), 0 = (t + 2)x1 + (t(2) − 4)x2 − (t(2) + t− 2) exp = g(x1, x2) with α ∼ O(1).”
MethodGaston, D. R., Guo, L., Hansen, G. A., Huang, H., Johnson, R., Knoll, D. A., Newman, C., Park, H. K., Podgorney, R. K., Tonks, M., et al. (2012) · Communications in Computational Physics · doi:10.4208/cicp.091010.140711s · Cited by 13
“Time integration methods include implicit Euler, Crank-Nicolson and second order backward difference [11].”
MethodGlobal and local methods for tracking the intersection curve between two surfaces
Borkowski, A., & Keller, W. (2005) · Journal of Geodesy · doi:10.1007/s00190-005-0437-2 · Cited by 14
“In this form, the problem has been treated frequently in literature ( Ascher and Petzold 1998; Hairer et.al.”
BackgroundFrederix, G., van Hasselt, J. G. C., Severens, J. L., et al. (2013) · Medical Decision Making · doi:10.1177/0272989x13476763 · Cited by 14
“Well-known LMS algorithms include the Adams algorithm(16) and the backward differentiation formula (BDF) method.(17) The Runge-Kutta and Adams methods are explicit methods that cannot adequately handle stiff problems, whereas the BDF method is an implicit method that can handle stiff ODEs.”
MethodDebugging symbolic transformations in equation systems
Sjölund, M., & Fritzson, P. (2011) · Equation-Based Object-Oriented Modeling Languages and Tools · Cited by 14
Reachability Analysis and Deterministic Global Optimization of DAE Models
Scott, J. K., & Barton, P. I. (2015) · Differential-algebraic equations forum · doi:10.1007/978-3-319-22428-2_2 · Cited by 13
“Nonetheless, this approach has become quite powerful due to the development of efficient and robust methods for numerical integration and sensitivity analysis of DAE systems [6, 24, 27, 49].”
MethodOzdemir, M. O., & Pasaogullari, U. (2016) · International Journal of Hydrogen Energy · doi:10.1016/j.ijhydene.2016.05.055 · Cited by 14
A real-time simulator framework for marine power plants with weak power grids
Skjong, S., & Pedersen, E. (2017) · Mechatronics · doi:10.1016/j.mechatronics.2017.09.001 · Cited by 14
“Also, when the system is proven stable with the Euler integration method, the results when using Runge-Kutta integration methods are also stable [23].”
MethodSide stream control in semicontinuous distillation
Madabhushi, P. B., & Adams, T. A. (2018) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2018.09.002 · Cited by 14
Delay-dependent stability of Runge–Kutta methods for linear delay differential–algebraic equations
Hu, X., Cong, Y., & Hu, G. (2019) · Journal of Computational and Applied Mathematics · doi:10.1016/j.cam.2019.06.016 · Cited by 14
Semi-Implicit Multistep Extrapolation ODE Solvers
Бутусов, Д., Вадимовна, Т. А., Fedoseev, P. S., et al. (2020) · Mathematics · doi:10.3390/math8060943 · Cited by 14
“stiff ODE systems excluding implicit methods, such as backward differentiation formulas (BDF) [3].”
MethodHrnet: Hamiltonian Rescaling Network for Image Downscaling
Chen, Y., Xiao, X., Dai, T., & Xia, S. (2020) · Proceedings - International Conference on Image Processing · doi:10.1109/icip40778.2020.9190729 · Cited by 13
Impact of Force Function Formulations on the Numerical Simulation of Centre-Based Models
Mathias, S., Coulier, A., Bouchnita, A., et al. (2020) · Bulletin of Mathematical Biology · doi:10.1007/s11538-020-00810-2 · Cited by 14
Luo, X.-L., Xiao, H., & Lv, J.-H. (2021) · Numerical Algorithms · doi:10.1007/s11075-021-01112-x · Cited by 14
“Firstly, we apply the implicit Euler method to the continuous Newton flow (9) [2, 5], then we obtain”
MethodHay, A., Étienne, S., Garon, A., et al. (2015) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2015.06.006 · Cited by 14
Tokić, A., Milardić, V., Uglešić, I., et al. (2015) · Electric Power Systems Research · doi:10.1016/j.epsr.2015.05.020 · Cited by 14
A stiff, variable time step transport solver for CONTAM
Lorenzetti, D. M., Dols, W. S., Persily, A. K., et al. (2013) · Building and Environment · doi:10.1016/j.buildenv.2013.05.008 · Cited by 14
“A stiff system forces a solver to take shorter steps than required to accurately represent the solution [2].”
Influential
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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