How to cite and citations


If you authored works where the IVP test set has been used, please send the releted references to francesca.mazzia at Please use the formats bibtex , ascii when giving credits to the IVP test set.

The following is a list of papers up to 2014 and phd theses where the IVP test set solvers and/or problems have been used. The list might be useful to researchers looking for papers related to their work. The list is not complete; 

Published Papers
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  • 2014
    1. González-Pinto, S., Hernández-Abreu, D., Simeon, B. Strongly A-stable first stage explicit collocation methods with stepsize control for stiff and differential-algebraic equations (2014) Journal of Computational and Applied Mathematics, 259 (PART A), pp. 138-152. DOI: 10.1016/
  • 2011
    1. González-Pinto, S., Hernández-Abreu, D. Global error estimates for a uniparametric family of stiffly accurate Runge-Kutta collocation methods on singularly perturbed problems (2011) BIT Numerical Mathematics, 51 (1), pp. 155-175. Cited 4 times. DOI: 10.1007/s10543-010-0304-2
  • 2009
    1. L. Aceto, C. Magherini. On the relations between B2 VMs and Runge-Kutta collocation methods, J. Comput. Appl. Math., 231 issue 1 (2009) 11-23.
    2. L. Brugnano, C. Magherini. Blended General Linear Methods based on Boundary Value Methods in the GBDF family, Journal of Numerical Analysis, Industrial and Applied Mathematics, 1-2 (2009) 23-40.
    3. S. González-Pinto, D. Hernández-Abreu, J. I.Montijano. An efficient family of strongly A-stable Runge-Kutta collocation methods for stiff systems and DAEs. Part I: Stability and order results, J. Comput. Appl. Math. (to appear), doi:10.1016/
    4. R. Lamour, F. Mazzia. Computation of consistent initial values for properly stated index 3 DAEs, BIT Numerical Mathematics, 49 issue 1 (2009) 161-175.


  • 2008
    1. R. Fazio. Numerical Scaling Invariance Applied to the van der Pol Model, Acta Applicandae Mathematicae: An International Survey Journal on Applying Mathematics and Mathematical Applications, 104 issue 1 (2008) 107-114.
    2. Gwenael Kervizic, Laurent Corcos. Dynamical modeling of the cholesterol regulatory pathway with Boolean networks, BMC Systems Biology 2:99 (2008).
    3. Nittka Robin, Sauter Manfred. Sobolev gradients for differential algebraic equations. Electron. J. Differential Equations, 42 (2008) 31 pp.


  • 2007
    1. M. Arnold, B. Burgermeister, A. Eichberger. Linearly implicit time integration methods in real-time applications: DAEs and stiff ODEs, Multibody System Dynamics, 17 issues 2-3 (2007) 99-117.
    2. Teijo Arponen, Samuli Piipponen, Jukka Tuomela. Analysing singularities of a benchmark problem, Journal Multibody System Dynamics, (2007) online.
    3. D. Guibert, D. Tromeur-Dervout. Parallel adaptive time domain decompositions for stiff systems of ODEs/DAEs, Computers and Structures, 85 issue 9 (2007) 553-562.
    4. Jingfang Huang, Jun Jia and Michael Minion.
      Arbitrary Order Krylov Deferred Correction Methods for Differential Algebraic Equations, Journal of Computational Physics, 221 issue 2 (2007) 739-760.
    5. Higinio Ramos, Jesas Vigo-Aguiar. A fourth-order Runge-Kutta method based on BDF-type Chebyshev approximations, J. Comput. Appl. Math., 204 (2007) 124-136.
    6. L. M. Skvortsov. Explicit multistep method for the numerical solution of stiff differential equations, Computational Mathematics and Mathematical Physics, 47 issue 6 (2007) 915-923.
    7. L. F. Shampine, S. Thompson. Stiff systems, Scholarpedia (2007) 2(3):2855.
    8. S. Hamdi, W. E. Schiesser, G. W. Griffiths. Method of lines, Scholarpedia (2007) 2(7):2859.


  • 2006
    1. L. Brugnano, C. Magherini, F. Mugnai. Blended implicit methods for the numerical solution of DAE problems, J. Comput. Appl. Math., 189 issue 1 (2006) 34-50.
    2. B. Burgermeister, M. Arnold, B. Esterl. DAE time integration for real-time applications in multi-body dynamics, ZAMM, 86, issue 10 (2006) 759-771.
    3. Jingfang Huang, Jun Jia, Michael Minion. Accelerating the convergence of spectral deferred correction methods, Journal of Computational Physics archive, 214 issue 2 (2006) 633-656.
    4. M. Korch, T. Rauber. Optimizing locality and scalability of embedded Runge-Kutta solvers using block based pipelining, J. Paral. Distr. Comput., 66 issue 3 (2006) 444-468.
    5. S. Schlenkrich, A. Walther, A. Griewank. Application of AD-based quasi-Newton methods to stiff ODEs. Automatic differentiation: applications, theory, and implementations, Lect. Notes Comput. Sci. Eng., 50, Springer Berlin 2006. 89-98.
    6. G. Soderlind, L. Wang. Adaptive time-stepping and computational stability, J. Comput. Appl. Math., 185 issue 2 (2006) 225-243.
    7. G. Soderlind, L. Wang. Evaluating numerical ODE/DAE methods, algorithms and software, J. Comput. Appl. Math., 185 issue 2 (2006) 244-260.


  • 2005
    1. J. R. Cash. Efficient time integrators in the numerical method of lines, J. Comput. Appl. Math., 183 issue 2 (2005) 259-274.
    2. S. Gonzalez-Pinto, R. Rojas-Bello. Speeding up Newton-type iterations for stiff problems, J. Comput. Appl. Math., 181 issue 2 (2005) 266-279.
    3. C. Lunk, B. Simeon. Runge-Kutta-Nystršom methods with maximized stability domain in structural dynamics, Appl. Numer. Math., 53 issue 2 (2005) 373-389.
    4. N. S. Nedialkov, J. D. Pryce. Solving Differential-Algebraic Equations by Taylor Series (I): Computing Taylor Coefficients, BIT Numerical Mathematics, 45 issue 3 (2005) 561-591.
    5. J. E. Onoda Pessanha, O. Saavedra, A. Paz, C. Portugal. Power System Stability Computer Simulation Using a Differential-Algebraic Equation Solver, International Journal of Emerging Electric Power Systems, 4 issue 2 (2005) article 3.
    6. S. Schlenkrich, A. Griewank, A. Walther. Efficient similarity factorization of rank-1 modifications. PAMM, 5 issue 1 (2005) 793-794.


  • 2004
    1. E. Arias, V. Hernández, J.J. Ibáñez. High performance algorithms for computing nonsingular Jacobian-free piecewise linearization of differential algebraic equations. Integral methods in science and engineering (Saint Étienne, 2002), 7–12, Birkhäuser Boston, Boston, MA, 2004.
    2. L. Brugnano, C. Magherini. The BiM code for the numerical solution of ODEs, J. Comput. Appl. Math., 164-165 issue 1 (2004) 145-158.
    3. M. P. Calvo, A. Portillo. Are high order variable step equistage initializers better than standard starting algorithms?, J. Comput. Appl. Math., 169 Issue 2 (2004), 333-344.
    4. C. Lunk, B. Simeon. Runge–Kutta–Nyström methods with maximized stability domain in structural dynamics, Appl. Numer. Math., 53 issue 2-4 (2004) 373-389.


  • 2003
    1. F. Aluffi-Pentini, V. De Fonzo, V. Parisi. A novel algorithm for the numerical integration of systems of ordinary differential equations arising in chemical problems, J. of Math. Chemistry, 33 issue 1 (2003) 1-15.
    2. Y. Cao, L. Petzold. A subspace error estimate for linear systems, SIAM J. Matrix. Anal. Appl., 24 issue 3 (2003) 787–801.
    3. J. R. Cash. Efficient Numerical Methods for the Solution of Stiff Initial-Value Problems and Differential Algebraic Equations, Proceedings: Mathematical, Physical and Engineering Sciences, 459 No. 2032 (Apr. 8, 2003) 797-815.
    4. S. Gonzalez-Pinto, J. I. Montijano, S. Perez-Rodriguez. Variable-order starting algorithms for implicit Runge-Kutta methods on stiff problems, Appl. Numer. Math., 44 issue 1-2 (2003) 77-94.


  • 2002
    1. E. Barth, B. Leimkuhler, S. Reich. A test set for molecular dynamics algorithms. Computational methods for macromolecules: challenges and applications (New York, 2000), Lect. Notes Comput. Sci. Eng., 24, Springer, Berlin (2002) 73-103.
    2. L. Brugnano, C. Magherini. Blended implementation of block implicit methods for ODEs, Appl. Numer. Math., 42 issue 1 (2002) 29-45.
    3. F. Cameron, M. Palmroth, R. Piché. Quasi stage order conditions for SDIRK methods, Appl. Numer. Math., 42 issue 1 (2002) 61-75.
    4. F. Iavernaro, F. Mazzia. Generalization of backward differentiation formulas for parallel computers, Numerical Algorithms 31 (2002) 139-155.
    5. F. Iavernaro, F. Mazzia. Parallel implicit predictor corrector methods, Appl. Numer. Math., 42 issue 1 (2002) 235-250.
    6. J. M. Mantas Ruiz, J. Ortega Lopera, J. A. Carrillo De La Plata. Component-Based derivation of a parallel Stiff ODE solver implemented in a cluster of computers, Intern. J. Paral. Program., 30 issue 2 (2002) 99-148.
    7. Y. Cao, L. Petzold. A Subspace Error Estimate for Linear Systems, SIAM Journal on Matrix Analysis and Applications, 24 issue 3 (2002) 787-801.
    8. H. Podhaisky, B. A. Schmitt, R. Weiner. Design, analysis and testing of some parallel two-step W-methods for stiff systems, Appl. Numer. Math., 42 issue 1 (2002) 381-395.


  • 2001
    1. Frank and Van Der Houwen. Parallel iteration of the extended backward differentiation formulas, IMA J Numer Anal., 21 (2001) 367-385.


  • 2000
    1. K. Burrage, H. Suhartanto. Parallel iterated methods based on variable step-size multistep Runge-Kutta methods of Radau type for stiff problems, Adv. Comput. Math., 13 issue 3 (2000) 257-270.
    2. A. Frommer, D. B. Szyld. On asynchronous iterations. Numerical analysis 2000, Vol. III. Linear algebra, J. Comput. Appl. Math., 123 issue 1-2 (2000) 201-216.


  • 1999
    1. F. Iavernaro, F. Mazzia. On the extension of the code GAM for parallel computing, EURO-PAR’99 Parallel Processing, Lecture Notes in Computer Science, 1685, Springer, Berlin, (1999) 1136-1143.


  • 1998
    1. P. Amodio, F. Mazzia. An algorithm for the computation of consistent initial values for differential-algebraic equations. Differential algebraic equations (Grenoble, 1997), Numer. Algorithms, 19 issue 1-4 (1998) 13-23.
    2. F. Iavernaro, F. Mazzia. Solving ordinary differential equations by generalized Adams methods: properties and implementation techniques, Appl. Numer. Math., 28 (1998) 107-126.


  • 1997
    1. K. Burrage, H. Suhartanto. Parallel iterated method based on multistep Runge-Kutta of Radau type for stiff problems. Parallel methods for ODEs, Adv. Comput. Math., 7 issue 1-2 (1997) 59–77.
    2. P. J. van der Houwen, J. J. B. de Swart. Parallel linear system solvers for Runge-Kutta methods, Adv. Comput. Math., 7 issue 1-2 (1997) 157-181.
    3. P. J. van der Houwen, J. J. B. de Swart. Triangularly implicit iteration methods for ODE-IVP solvers, SIAM J. Sci. Comput., 18 issue 1 (1997), 41-55.


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  • E. Arge, A.M. Bruaset, H. P. Langtangen. Modern software tools for scientific computing, Birkhauser Boston, 1997.
  • C. Constanda, M. Ahués, A. Largillier. Integral methods in science and engineering: analytic and numerical techniques, Birkhauser Boston, 2004.
  • D.J. Higham, N.J. Higham. Matlab guide, second edition. SIAM, 2005.
  • P. Kunkel, V. Mehrmann. Differential-Algebraic Equations: analysis and numerical solution. EMS Textbook in Mathematics, 2006
  • Soetaert, Karline, Cash, Jeff, Mazzia, Francesca, Solving Differential Equations in R,  Springer-Verlag, Berlin Heidelberg, 2012

Technical Reports / Conference Proceedings
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  • M. Arnold. Half-explicit Runge-Kutta methods with explicit stages for differential-algebraic systems of index 2, University of Rostock, 1995.
  • Einar Mageroy. Numerical integration of systems arising from the method of lines, Department of Mathematical Sciences, The Norwegian University of Science and Technology (1997).
  • V. Hernandez, I. Blanquer, E. Arias, V. M. Garcia, L. Penalver and P. Ruiz. Nonlinear Control Systems Simulation Toolbox in SLICOT, SLICOT Working Note 2000-5, Working Group on Software- ESAT- Katholieke Universiteit Leuven-Belgium, 2000.
  • R. Lamour. Index determination for DAEs, Humboldt-University of Berlin, Institute of Mathematics, 2001.
  • L. Ordóñez Inda. Numerical test sets for EcosimPRO, 1st Meeting of EcosimPro Users, UNED, Madrid, 3-4 May 2001.
  • Sebastian Schlenkrich. Application of AD based Quasi-Newton-Methods on stiff ODEs, AD 2004 — Fourth International Workshop on Automatic Differentiation, July 19-23, 2004,
    Argonne National Laboratory Gleatcher Center, Chicago, IL, USA.
  • Michael Gunther, Uwe Feldmann, Jan ter Maten. Modelling and discretization of circuit problems, Technische Universiteit Eindhoven, 2005.
  • Rahul V. Kharche, Shaun A. Forth. Source Transformation for MATLAB Automatic Differentiation, AMOR REPORT 2005/01 December 2005, Applied Mathematics & Operational Research, Defence College of Management & Technology, ranfield University Shrivenham Swindon SN6 8LA UK.
  • Inmaculada Higueras, Teo Rolda. Positivity-Preserving and Entropy-Decaying IMEX Methods, Monografias del Seminario Matematico Garcia de Galdeano 33, 129-136 (2006).
  • Nedialko S. Nedialkov. Interval Tools for ODEs and DAEs, Technical report CAS 06-09-NN, Dept. of Computing and Software,McMaster University, Hamilton, ON, L8S 4K1, Canada, November 2006.
  • G. F. Corliss, R. B. Kearfott, N. Nedialkov, J. D. Pryce, S. Smith.Interval Subroutine Library Mission, Dagstuhl Seminar Proceedings 06021 Reliable Implementation of Real Number Algorithms: Theory and Practice (2006)
  • N. Nedialkov. Interval Tools for ODEs and DAEs, scan, pp.4, 12th GAMM – IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006), 2006.
  • Miguel Cecenas Falcon, Rina M. Campos Gonzalez. Pruebas de Acoplamiento de Canales Paralelos a Cinetica Neutronica Modal, 2007 LAS/ANS Symposium, XVIII SNM Annual Meeting, XXV SMSR Annual Meeting, Co-sponsored by AMEE, Cancun, Quintana Roo, Mexico, July 1-5, 2007, Proceedings IJM Cancun 2007 on CDROM.
  • B. Lilleberg, I. S. Ertesvag, K. E. Rian. Computational modeling of combustion instabilities in lean premixed turbulent combustors, MekIT’07- Fourth national conference on computational mechanics, 23-24 May 2007, Trondheim, Norway.
  • U. Nowak, S. Gebauer. A New Test Frame for Ordinary Differential Equation Solvers (2007), Konrad-Zuse-Zentrum fur Informationstechnik Berlin.
  • Eberhard H.A. Gerbrach. How to Derive Single-equation Descriptions for Output-quantities in Nonlinear Circuits using Differential Algebra, (2008 Re-Release). arXiv:0804.2992v1.

Theses (back to top of page)

  • Erik Hamran Nilsen. The Implementation of SIRK Methods for Differential Algebraic Equations. Department of Mathematical Sciences The Norwegian University of Science and Technology, 1997.
  • Guangfu Sun. Berechnung von Gittermast-Fahrzeugkranen unter Berücksichtigung der Antriebs- und Regelungssysteme Lehrstuhl für F&ooml;rdertechnik Materialß Logistik der Technischen Universität München, 2001.
  • Enrique Arias Antúnez. Algoritmos de Altas Prestaciones para la Simulación, Estimación y Control de Sistemas No Lineales, Departamento de Sistemas Informáticos y Computación, Universidad Politécnica de Valencia, 2003.
  • Cecilia Magherini. Numerical solution of stiff ODE-IVPs via Blended Implicit methods: Theory and Numerics. Universita’ degli Studi di Firenze, 2004.
  • J. Niesen. On the Global Error of Discretization Methods for Ordinary Differential Equations. University of Cambridge, 2004.
  • Kirsten R. Meeker. Digital Filter Stepsize Control of DASPK and its Effect on Control Optimization Performance, University of California, Santa Barbara, December 2004.
  • Jesus Martin Vaquero. Metodos Exponential Fitting y Adaptados para Problemas Stiff Departamento de Matematica y Fisica Aplicadas y Ciencias de la Naturaleza,Universidad Rey Juan Carlos,Madrid, September 2005.
  • J. Bellmann. Sensitivitatsanalyse Differentiell-Algebraischer Systeme und adaptives Checkpointing, Istitut fur Wissenschaftliches Rechnen, Fachrichtung Mathematik, Technische Universitat Dresden, December 2006.
  • Joseba Makazaga Odria. Sobre los errores locales y globales de la integración de Ecuaciones Diferenciales Ordinarias mediante métodos de Runge-Kutta Explícitos. Departamento de Ciencias de la Computación e Inteligencia Artificial, Universidad del País Vasco, 2007.
  • Celso Freitas, Integração numérica de sistemas não lineares semi-implícitos via teoria de controle geométrico, Instituto de Matemática e Estatística da Universidade de São Paulo, 2011.

Seminars (back to top of page)

  • Centre for Analysis, Scientific computing and Applications (CASA), Department of Mathematics and Computer Science of Eindhoven University of Technology (TU/e), 10-03-2004, S.M.A. Allaart-Bruin Numerical Integration of DAE’s.
  • Supercomputer Education and Research Centre,Indial Institute of Science, Bangalore: SERC Research Seminar Day, Saturday, August 18, 2007 Dr. Soumyendu Raha Research in Scientific Computation

Software (back to top of page)

  • OdePkg: a package for solving ordinary differential equations and more.
  • CHAINSOLVER 2.20, transmutation simulation of samples during irradiation in nuclear reactors.
  • The control and systems library SLICOT
  • deSolve : General solvers for initial value problems of ordinary differential equations (ODE), partial differential equations (PDE) and differential algebraic equations (DAE). Authors: Karline Soetaert, Thomas Petzoldt, R. Woodrow Setzer.
  • deTestSet : Solvers and testset for stiff and nonstiff differential equations, and differential algebraic equations.
  • Example Programs for IDAS v.1.0.0.. Alan C. Hindmarsh and Radu Serban, Center for Applied Scientific Computing Lawrence Livermore National Laboratory.
  • XmdS: eXtensible multi-dimensional Simulator.
  • VODE_F90 (G.D Byrne, S. Thompson).
  • TEST_ODE Test problems for initial value solvers.

Teaching (back to top of page)