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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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Non-Intrusive Inference Reduced Order Model for Fluids Using Deep Multistep Neural Network
Xie, X., Zhang, G., & Webster, C. (2019) · Mathematics · doi:10.3390/math7080757 · Cited by 33
“In this work, we only consider the implicit multistep method, Adams–Moulton (AM) scheme [35] as it has better stability property.”
MethodEfficient Preconditioner Updates for Shifted Linear Systems
Bellavia, S., Simone, V. D., di Serafino, D., et al. (2011) · SIAM Journal on Scientific Computing · doi:10.1137/100803419 · Cited by 34
“, BDF and SDIRK are applied [1].”
Masciola, M. D., Jonkman, J. M., & Robertson, A. N. (2014) · doi:10.1115/omae2014-23508 · Cited by 34
Modelling the role of the basement membrane beneath a growing epithelial monolayer
Dunn, S.-J., Fletcher, A. G., Chapman, S. J., Gavaghan, D. J., & Osborne, J. M. (2011) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2011.12.013 · Cited by 34
Stabilization of non-linear differential-algebraic equation systems
Liu, X., & Ho, D. W. C. (2004) · International Journal of Control · doi:10.1080/00207170410001715013 · Cited by 34
Automatic domain partitioning for quadrilateral meshing with line constraints
Kowalski, N., Ledoux, F., & Frey, P. (2014) · Engineering With Computers · doi:10.1007/s00366-014-0387-5 · Cited by 33
“Then we use the Heun’s method [29], an easy-to-use variation of a Runge–Kutta method, to integrate Y = f ( X ) over T , starting from X i .”
MethodDing, Z., Zhou, Y. H., & Zhou, M. (2014) · IEEE Transactions on Automation Science and Engineering · doi:10.1109/tase.2013.2285574 · Cited by 33
“For the latter, since implicit Runge–Kutta method has a stable area [17], and the algorithm is guaranteed to converge in the stable area, the rounding error of the perturbation cannot increase and will decrease to 0 in the iteration process [2].”
MethodParametric mixed-integer 0–1 linear programming: The general case for a single parameter
Mitsos, A., & Barton, P. I. (2008) · European Journal of Operational Research · doi:10.1016/j.ejor.2008.01.007 · Cited by 33
“Note that similar error estimates are successfully applied to the solution of differential-algebraic equation systems [46], which suggests that our proposal will be successful in practice.”
BackgroundStructural Fault Detection and Isolation in Hybrid Systems
Khorasgani, H., & Biswas, G. (2017) · IEEE Transactions on Automation Science and Engineering · doi:10.1109/tase.2017.2749447 · Cited by 33
“In previous work [58], [59], we used the sensitivity analysis [60] to define a detectability ratio measure and an isolability ratio measure to quantify the performance of a residual in FDI.”
MethodQiu, C., Güttel, S., Ren, X., et al. (2019) · Geophysical Journal International · doi:10.1093/gji/ggz129 · Cited by 33
Computing Validated Solutions of Implicit Differential Equations
Hoefkens, J., Berz, M., & Makino, K. (2003) · Advances in Computational Mathematics · doi:10.1023/a:1022858921155 · Cited by 33
“DAE problems [ 1 ,13,18,19], they are usually based on multistep methods and generally do not provide the possibility of verification and validation.”
BackgroundMethodPseudo-Transient Continuation for Nonsmooth Nonlinear Equations
Fowler, K. R., & Kelley, C. T. (2005) · SIAM Journal on Numerical Analysis · doi:10.1137/s0036142903431298 · Cited by 33
“We use the known result for the scalar function max(0, v), 0 if v < 0, Omax(0, v)= [0, 1] if v = 0, 1 if v > 0, to obtain O -Lb 0 0 A 0 0 1F -1 - (1/p)max(0, v)(1-p)/P 0 1 0 0 max(0, v).”
BackgroundInfluentialDorn, O., Bertete‐Aguirre, H., Berryman, J. G., et al. (2002) · Inverse Problems · doi:10.1088/0266-5611/18/2/301 · Cited by 33
Zhou, Y., Gad, E., Nakhla, M. S., et al. (2011) · IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems · doi:10.1109/tcad.2011.2167326 · Cited by 33
“Digital Object Identifier 10.1109/TCAD.2011.2167326 The theoretical difficulty became known when Dahlquist established his so-called theoretical barriers, which showed that A-stable linear-multistep (LMS) methods cannot have orders higher than two [1].”
BackgroundInfluentialFrostig, Y., & Thomsen, O. T. (2007) · Journal of mechanics of materials and structures · doi:10.2140/jomms.2007.2.1355 · Cited by 33
“(24) The nonlinear response is determined numerically using trapezoid or mid-point methods with Richardson extrapolation or deferred corrections (see [Ascher and Petzold 1998]), as implemented in Maple.”
Numerical methods for simulation in applied dynamics
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“However, there are other types of indices for descriptor systems, such as the differentiation index [9], the tractability index [86], and so on.”
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Hoang, M. T., & Ehrhardt, M. (2024) · Journal of Computational and Applied Mathematics · doi:10.1016/j.cam.2024.115752 · Cited by 32
Scene-Dependent Prediction in Latent Space for Video Anomaly Detection and Anticipation
Cao, C., Zhang, H., Lu, Y., Wang, P., & Zhang, Y. (2024) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2024.3461718 · Cited by 46
Analytical and numerical solutions for shapes of quiescent two-dimensional vesicles
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Design of Dynamic Systems Using Surrogate Models of Derivative Functions
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Simulation of the Growth Kinetics of FeB and Fe2B Layers on AISI D2 Steel by the Integral Method
Keddam, M., & Kulka, M. (2018) · The Physics of Metals and Metallography · doi:10.1134/s0031918x18090065 · Cited by 32
Clément, J.-B., Golay, F., Ersoy, M., et al. (2021) · Advances in Water Resources · doi:10.1016/j.advwatres.2021.103897 · Cited by 32
Block implicit methods for Odes
Brugnano, L., & Trigiante, D. (2000) · Florence Research (University of Florence) · Cited by 32
Use of Virtual Cell in Studies of Cellular Dynamics
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“Two of the solvers with variable time step, IDA and CVODE, are unconditionally stable (50).”
Theoretical basis of parametric-resonance-based atomic force microscopy
Prakash, G., Hu, S., Raman, A., et al. (2009) · Physical Review B · doi:10.1103/physrevb.79.094304 · Cited by 32
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Fabozzi, D., & Cutsem, T. V. (2009) · IEEE Power & Energy Society General Meeting (PESGM) · doi:10.1109/pes.2009.5275463 · Cited by 32
“This property is called A-Stability [5] and holds when the region of absolute stability of the method contains the entire left half plane (h (λ) 0).”
BackgroundMethodInfluentialAfanas’ev, V. P., Efremenko, D., & Kaplya, P. S. (2016) · Journal of Electron Spectroscopy and Related Phenomena · doi:10.1016/j.elspec.2016.04.006 · Cited by 32
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“We chose the LSODA solver [63] based on the backward differentiation formulas [6] with adaptive step size determination, since we did not know the time constants of the various binding steps in advance and since the stopped-flow data achieved measurement rates as high as 80 kHz.”
MethodThe Newmark Integration Method for Simulation of Multibody Systems: Analytical Considerations
Gavrea, B., Negruţ, D., & Potra, F. A. (2005) · doi:10.1115/imece2005-81770 · Cited by 31
“This method is implicit and A-stable (stable in the whole left-hand plane) [2] provided [3]”
MethodCosta, M. R. P. F. N., & Dias, R. C. S. (2003) · Macromolecular Theory and Simulations · doi:10.1002/mats.200350011 · Cited by 31
“A widely used measure of numerical stiffness([7]) is found by multiplying the end value of time tf by the lowest real part of the eigenvalues of the Jacobian of the system right hand sides.”
MethodA Versatile Time-Domain Approach to Simulate Oscillators in RF Circuits
Maffezzoni, P. (2009) · IEEE Transactions on Circuits and Systems I Regular Papers · doi:10.1109/tcsi.2008.2002659 · Cited by 31
A fast lattice Green's function method for solving viscous incompressible flows on unbounded domains
Liska, S., & Colonius, T. (2016) · Journal of Computational Physics · doi:10.1016/j.jcp.2016.04.023 · Cited by 31
“(9), as a system of differential algebraic equations (DAEs) of index 2 prior to choosing a time integration scheme can have undesirable consequences on the quality of the numerical solution [34,35].”
BackgroundA Survey of Dimension Reduction Methods for High-dimensional Data Analysis and Visualization
Engel, D., Hüttenberger, L., & Hamann, B. (2012) · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · doi:10.4230/oasics.vluds.2011.135 · Cited by 31
Lauper, B. B., Anthamatten, E., Raths, J. O., et al. (2021) · ACS Environmental Au · doi:10.1021/acsenvironau.1c00023 · Cited by 30
Cross, E. A., & Mitchell, I. M. (2008) · Proceedings of the ... American Control Conference/Proceedings of the American Control Conference · doi:10.1109/acc.2008.4586828 · Cited by 30
“Fortunately, a number of numerical schemes are available for accurately approximating the solution of DAEs with index one [5]; for example, the sample trajectories used in this paper were generated with MATLAB’s ode15s [6].”
MethodFluid Rewards for a Stochastic Process Algebra
Tribastone, M., Ding, J., Gilmore, S. T., et al. (2011) · IEEE Transactions on Software Engineering · doi:10.1109/tse.2011.81 · Cited by 30
“, [29]), one criterion checks if the norm of the derivative of the”
BackgroundMehmood, R., Nadeem, S., & Akbar, N. S. (2016) · Journal of Applied Fluid Mechanics · doi:10.18869/acadpub.jafm.68.228.24458 · Cited by 30
A process simulator interface for multiobjective optimization of chemical processes
López, C. A. M., Telen, D., Nimmegeers, P., et al. (2017) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2017.09.014 · Cited by 30
Chen, R., Wu, B., Cheng, Z., et al. (2020) · International Journal for Numerical Methods in Biomedical Engineering · doi:10.1002/cnm.3392 · Cited by 30
Embedded pairs for optimal explicit strong stability preserving Runge–Kutta methods
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Zhu, H., Ricote, S., Coors, W. G., et al. (2014) · Solid State Ionics · doi:10.1016/j.ssi.2014.07.009 · Cited by 30
Projected Pseudotransient Continuation
Kelley, C. T., Liao, L., Qi, L., et al. (2008) · SIAM Journal on Numerical Analysis · doi:10.1137/07069866x · Cited by 30
“This is different from the standard approach in an algorithm for initial value problems [2], where the time step is controlled with stability and accuracy in mind.”
BackgroundMethodA Comparison of Stiff ODE Solvers for Astrochemical Kinetics Problems
Nejad, L. A. M. (2005) · Astrophysics and Space Science · doi:10.1007/s10509-005-2100-z · Cited by 30
Arbitrary order numerical method for solving differential-algebraic equation by Padé series
Çeli̇k, E., & Bayram, M. (2002) · Applied Mathematics and Computation · doi:10.1016/s0096-3003(02)00080-2 · Cited by 30
“where F and y is a vector function for which we assumed sufficient differentiability [3,4,6], and the initial values to be consistent, i.”
BackgroundComparison of fully coupled approaches for approximating nonlinear transport and reaction problems
Kanney, J., Miller, C. T., & Barry, D. A. (2003) · Advances in Water Resources · doi:10.1016/s0309-1708(02)00188-4 · Cited by 30
Nonconforming Immersed Finite Element Methods for Interface Problems
Zhang, X. (2013) · VTechWorks (Virginia Tech) · Cited by 30
Berry, R. A., USDOE Office of Nuclear Energy (NE), Peterson, J. C., et al. (2015) · doi:10.2172/1239866 · Cited by 30
“77 3.3.2 HEM Constitutive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.1 Spatial Discretization Algorithm . . . . . . . . . . . . . . . . . . . . . . . .…”
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Mixed finite element formulation for dynamics of porous media
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An efficient optimized adaptive step-size hybrid block method for integrating differential systems
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“The numerical solution of the non-linear governing set of differential equations can be achieved using numerical schemes such as the multiple-point shooting method, see Stoer and Bulirsch (1980), or the finite-difference (FD) approach using trapezoid or mid-point methods with Richardson extrapolation or deferred corre…”
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Computing periodic solutions of linear differential-algebraic equations by waveform relaxation
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“The mstep constant BDF method converges to O(τ) for m < 7 [1].”
MethodA chemical kinetics algorithm for geochemical modelling
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“Although the computation of these derivatives requires some computational effort, it allows the use of larger time steps and increases the stability of the integration (Ascher and Petzold, 1998; Hairer and Wanner, 2010).”
MethodInfluentialAnalytical and numerical study of photocurrent transients in organic polymer solar cells
de Falco, C., Sacco, R., & Verri, M. (2010) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2010.01.018 · Cited by 29
“4); on the other hand, as the emphasis of the present paper is on accurately estimating photocurrent transient times, it is necessary to apply advanced time-step control techniques [24, 25].”
On the nature and stability of differential-algebraic systems
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EFFICIENT SIMULATION OF BUSH AND ROLLER CHAIN DRIVES
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“On the other hand, for most backward differentiation formula (BDF) type methods [1,11,55], the number of operations is only O(N) for each time step, with α ≤ 3 (α = 3 when Newton’s method and direct Gauss elimination are applied to solve the spatial nonlinear equations, and α = 1 when fast algorithms are applied to sp…”
MethodReliable approximations for a hepatitis B virus model by nonstandard numerical schemes
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“For a definition of the absolute stability domain we refer to e.g. Hairer andWanner (1991) and Ascher and Petzold (1998).”
BackgroundMethodInfluential3D forward modelling of time domain electromagnetic data
Haber, E., Ascher, U. M., & Oldenburg, D. W. (2002) · Second International Meeting for Applied Geoscience & Energy · doi:10.1190/1.1817334 · Cited by 28
Self-citationOn multi-time-step monolithic coupling algorithms for elastodynamics
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“(b) BDF and IRK schemes are developed for first-order DAEs.”
InfluentialGeneralized Derivatives of Differential–Algebraic Equations
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“However, the situation is much more complicated with nonlinear ODEs which are common in physics-based animation.”
BackgroundInfluentialFrostig, Y., Birman, V., & KARDOMATEAS, G. A. (2018) · International Journal of Solids and Structures · doi:10.1016/j.ijsolstr.2018.02.023 · Cited by 28
“It uses trapezoid or mid-point methods with Richardson extrapolation or deferred corrections, ( Ascher and Petzold, 1998 ), along with parametric or arc-length continuation methods ( Keller, 1992 ).”
MethodIMEXnet: A Forward Stable Deep Neural NetworkPreprint
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Languages and Tools for Optimization of Large-Scale Systems
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Simulation of Motion of Constrained Multibody Systems Based on Projection Operator
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“Surveys of the existing techniques for solving DAE may be found in [14, 13, 1, 15, 2, 6].”
MethodHamdi, I., Denis, M., Zouari, A. B., et al. (2014) · Continental Shelf Research · doi:10.1016/j.csr.2014.10.002 · Cited by 27
Solving time‐dependent PDEs using the material point method, a case study from gas dynamics
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“In both these cases the discontinuity in the first time derivative of the right hand side of equation (43)(after noting that xp is time dependent) means that the time integration method accuracy is restricted to first order unless special action is taken, [1], p.”
BackgroundZhu, Y., Dopico, D., Sandu, C., et al. (2015) · Journal of Computational and Nonlinear Dynamics · doi:10.1115/1.4029601 · Cited by 27
MMC-HVDC: Simulation and control system
Debnath, S., & Madhusudhan, C. (2016) · IEEE Energy Conversion Congress and Exposition (ECCE) · doi:10.1109/ecce.2016.7855248 · Cited by 27
Continuum Gating Current Models Computed with Consistent Interactions
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Analysis of the variable step method of Dahlquist, Liniger and Nevanlinna for fluid flow
Layton, W., Pei, W., Qin, Y., et al. (2021) · Numerical Methods for Partial Differential Equations · doi:10.1002/num.22831 · Cited by 27
“1) the remainder use the constant time step implicit midpoint or the trapezoidal schemes, for example, [2, 3, 8, 31, 39], (often combined with fractional steps, or with ad hoc fixes to correct for oscillations due to lack of L-stability [6, 38, 44]), or the backward differentiation formula 2 (BDF2) method [1, 18, 22,…”
MethodRamos, H., & Rufai, M. A. (2021) · Mathematics and Computers in Simulation · doi:10.1016/j.matcom.2021.10.023 · Cited by 27
An initialization subroutine for DAEs solvers: DAEIS
Wu, B., & White, R. E. (2001) · Computers & Chemical Engineering · doi:10.1016/s0098-1354(00)00655-4 · Cited by 27
“Solving nonlinear equations is a fundamental problem in numerical simulations (Rice & Do, 1995; Ascher & Petzold, 1998; Shacham, Brauner, & Pozin, 1998).”
Nakshatrala, K. B., Hjelmstad, K. D., & Tortorelli, D. A. (2008) · International Journal for Numerical Methods in Engineering · doi:10.1002/nme.2303 · Cited by 27
“For a detailed discussion on domain decomposition methods see References [1–3].”
BackgroundInfluentialIntegrated Optimization Strategies for Dynamic Process Operations
Biegler, L. T. (2017) · Theoretical Foundations of Chemical Engineering · doi:10.1134/s004057951706001x · Cited by 27
“Moreover, sequential approaches are known to fail on unstable dynamic systems [3, 25].”
BackgroundSymbolic computation of differential equivalences
Cardelli, L., Tribastone, M., Tschaikowski, M., & Vandin, A. (2019) · Theoretical Computer Science · doi:10.1016/j.tcs.2019.03.018 · Cited by 26
Animating developable surfaces using nonconforming elements
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“…and Ko [2002] introduced the second order accurate BDF2 method to cloth simulation, which features much reduced damping yet still has stiff decay [Ascher and Petzold 1998], of crucial importance for dealing with stiff systems.2 They also fortuitously avoided locking with their implicit model of…”
BackgroundMethodInfluentialOn the optimal numerical time integration for Lagrangian DEM within implicit flow solvers
Tuley, R. J., Danby, M., Shrimpton, J. S., et al. (2009) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2009.10.003 · Cited by 27
“…= f (xn+1|h, un+1|h) un+1|h+1 = 1 3 ( 2Fn+1|h t m + 4un − un−1 ) xn+1|h+1 = 1 3 (2un+1|h t + 4xn − xn−1) (23) Similarly, the BD4 scheme we have used (Ascher & Petzold, 1998) is defined as: Fn+1|h = f (xn+1|h, un+1|h) un+1|h+1 = 1 25 ( 12Fn+1|h t m + 48un − 36un−1 + 16un−2 − 3un−3 ) xn+1|h+1…”
Ascher, U. M., Huang, H. F., & van den Doel, K. (2007) · BIT Numerical Mathematics · doi:10.1007/s10543-006-0112-x · Cited by 27
“In particular, if the exact solution of the ODE system satisfies such an invariant, how important is it to preserve this property exactly, or approximate it very well, while the numerical solution resulting from our discretization is far less accurate point-wise? In some instances, this is very important [6, 2].”
BackgroundSelf-citationRunge–Kutta–Chebyshev projection method
Zheng, Z., & Petzold, L. R. (2006) · Journal of Computational Physics · doi:10.1016/j.jcp.2006.07.005 · Cited by 27
“The mathematical structure of this DAE system is referred to as Hessenberg index 2 [3].”
MethodSelf-citationStochastic Modeling of Gene Regulatory Networks y
Samad, H. E., Khammash, M., Petzold, L. R., et al. (2005) · Cited by 27
Self-citationA Physiologically-Based Model for the Active Cardiac Muscle Contraction
Chapelle, D., Clément, F., Génot, F., et al. (2001) · Lecture notes in computer science · doi:10.1007/3-540-45572-8_18 · Cited by 27
“Performing a variational finite element space discretization with piecewiselinear continuous functions for the displacement and piecewise-constant functions for stress and deformation variables, we use a midpoint scheme for solving the resulting system of differential-algebraic equations (see [ 1 ]).”
MethodA variational principle for computing slow invariant manifolds in dissipative dynamical systemsPreprint
Lebiedz, D., Siehr, J., & Unger, J. (2011) · Cited by 27
Gharebaghi, S., Chaudhuri, N. R., He, T., et al. (2023) · IEEE Transactions on Smart Grid · doi:10.1109/tsg.2023.3338181 · Cited by 26
“…and optimal preventive control: This subprocess involves conducting cascade simulations using variable time-step BEM with a stiff decay property [21] that enables the use of large steps to accelerate dynamic simulations by disregarding fast oscillations in the exact time-domain trajectories.”
MethodThermal response prediction of a prototype Energy Micro-Pile
Ronchi, F., Salciarini, D., Cavalagli, N., et al. (2018) · Geomechanics for Energy and the Environment · doi:10.1016/j.gete.2018.07.001 · Cited by 26
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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