EPODE

References

Available informations:

References for stiff problems

References for parallelism

Internet resources of information

Author experience

References for stiff IVP of ODEs

Aiken,C(editor), Stiff Computation, Oxford University Press, 1985.
Albrecht,P Numerical Treatement of ODEs: The Theory of A-Methods, Numerische Mathematik 47 (1985) 59-87.
Alfeld,P., Lambert,J.D Correction in the Dominant Space: A Numerical Techniqe for a Certain Class of Stiff Initial Value Problems, Mathematics of Computation 31 (1977) 922-938.
Bader,G., Deuflhard,P A Semi-Implicit Mid-Point Rule for Stiff Systems of ODE, Numerische Mathematik 41 (1983) 373-398.
Beaudet,P.R Multi-off-grid Methods in Multi-step-integration of Ordinary Differential Equation, in cite.
Bettis,D.G(editor) Proceedings of the Conference on the Numerical Solution of Ordinary Differential Equations, 19-20 October 1972, The University of Texas at Austin, Springer, LNM 362, 1974.
Bickart,T.A., Jury,E.I Arithmetic Test for A-stability, A(a)-stability and Stiff-stability, BIT 18 (1978) 9-21.
Bickart,T.A., Picel,Z High Order Stiffly Stable Composite Methods for Numerical Integration of Stiff Differential Equations, BIT 13 (1973) 271-286.
Bickart,T.A., Rubin,W Composite Multistep Methods and Stiff Stability, in cite.
Bjorck,A Some Methods for Separating Stiff Components in Initial Value Problems, in cite
Bjurel,G Modified Linear Multistep Methods for a Class of Stiff Ordinary Differential Equations, BIT 12 (1972) 142-160.
Brown,R.L Some Characteristics of Implicit Multistep Multi-Derivative Integration Formulas, SIAM Journal of Numerical Analysis 14 (1977) 982-993.
Bulirsch,R.,Grigorieff,R.D (editors) Numerical Treatement of Differential Equations, Proceedings of a Conference Held at Oberwolfach, july 4-10, 1976, Springer, LNM 631, 1978.
Burrage,K., Chipman,F,H The Stability Properties of Singly-Implicit General Linear Methods, IMA Journal of Numerical Analysis 5 (1985) 287-295.
Butcher,J.C Order Conditions for a General Class of Numerical Methods for Ordinary Differential Equations, in cite.
Butcher,J.C The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, John Willey and Sons Inc., Chichester, 1987.
Carroll,J A Composite Integration Scheme for the Numerical Solution of Systems of Ordinary Differential Equations, Journal of Computational and Applied Mathematics 25 (1989) 1-13
Cash,J.R A Note on the Exponential Fitting of Blended, Extended Linear Multistep Methods, BIT 21 (1981) 450-454.
Cash,J.R On Improving the Absolute Stability of Local Extrapolation, Numerische Mathematik 40 (1982) 329-337.
Cash,J.R On the Design of High Order Exponentially Fitted Formulae for the Numerical Integration of Stiff Systems, Numerische Mathematik 36 (1981) 253-266.
Cash,J.R On the Exponential Fitting of Composite, Multiderivative Linear Multistep Methods, SIAM Journal of Numerical Analysis 18 (1981) 808-821.
Cash,J.R On the Integration of Stiff Systems of ODEs Using Extended Backward Differentiation Formulae, Numerische Mathematik 34 (1980) 235-246.

References for parallelism in solving IVP of ODEs

Amodio,P., Trigiante,D., A parallel direct method for solving IVPs for ODEs, Applied Numerical Mathematics 11 (1993), 85-93
Bellen,A., Introduction, International Conference on Parallel Methods for ODEs. The State of the Art, Grado (Italy), 10-13 September 1991, Applied Numerical Mathematics 11 (1993), 3-5.
Bellen,A., Parallelism across the steps for difference and differential equations, Lectures Notes in Mathematics 1386 (1989), 22-35
Bellen,A., Zennaro,M., Parallel algorithms for initial value problems for difference and differential equations, Journal of Computational and Applied Mathematics 25 (1989), 341-350
Bellen,A., Vemiglio,R., Zennaro,M., Parallel ODE solvers with stepsize control, Journal of Computational and Applied Mathematics 31 (1990), 277-293
Bellen,A., Zennaro,M., The use of RK formulae in waveform relaxation methods, Applied Numerical Mathematics 11 (1993), 95-114
Bendtsen, C., Highly stable parallel Runge-Kutta methods, Applied Numerical Mathematics 21 (1996), 1-8
Butcher,J.C., Diagonally implicit multi-stage integration methods, Applied Numerical Mathematics 11 (1993), 347-363
Burrage,K., The error behaviour of a general class of predictor-corrector methods, Applied Numerical Mathematics 8 (1991), 201-216
Burrage,K., Parallel methods for initial value problems, Applied Numerical Mathematics 11 (1993), 5-25
Burrage,K., The search for the Holy Grail or: Predictor-corrector methods for solving ODE IVPs, Applied Numerical Mathematics 11 (1993), 125-141
Burrage,K., Pohl,B., Implementing an ODE code on distributed memory computers, Computers \& Mathematics with Applications 28 (1994), no. 10-12, 235-252
Chartier,P., Phillipe,B., A parallel shooting technique for solving dissipative ODEs, Computing 51 (1993), 209-236
Chu,M.T., Hamilton,H., Parallel solutions of ODE's by multi-block methods, SIAM J. Sci. Statistic Comput. 27 (1978), 413-420
Cong,Nguyen huu, Parallel iteration of symmetric Runge-Kutta methods for nonstiff initial value problems, Journal of Computational and Applied Mathematics 51 (1994), 117-125
Cong,Nguyen huu, A parallel DIRK methods for stiff initial-value problems, Journal of Computational and Applied Mathematics 54 (1994), 121-127
Cooper,G.J., Vignesvaran,R., On the use of parallel processors for implicit Runge-Kutta methods, Computing 51 (1993), 135-150
Crisci,M., Parallel frontal methods for ODEs, BIT 34 (1994), 215-227
Enright,W.H., Hull,T.E., Lindberg, B., Comparing numerical methods for stiff systems of ODEs, BIT 15 (1975), 10-48
Evans,D.J., Megson,G.M., Construction of extrapolation tables by systolic arrays for solving ordinary differential equations, Parallel Computing 4 (1987), 33-48
Evans,D.J., Sanugi,B.B., A parallel Runge-Kutta integration method, Parallel Computing 11 (1989), 245-251
Gallopoulos,G. Houstis,E., Rice,J., Computer as Thinker/Doer: Problem-Solving Environments for Computational Science, IEEE Computational Science and Engineering, (1994)
Gear,C.W., Massive parallelism across space in ODEs, Apllied Numerical Mathematics 11 (1993), 27-43
Gear,C.W., Xuhai Xu, Parallelism across time in ODEs, Applied Numerical Mathematics 11 (1993), 45-68
Ghoshal,S.K., Gupta,M., Rajaraman,V., A parallel multistep predictor-corrector algorithm for solving ordinary differential equations, Journal of Parallel and Distributed Computing 6 (1989), 630-648
Hackbusch,W., (ed.) Parallel algorithms for partial differential equations, Proceedings of the Sixth GAMM-Seminar, Kiel, January 19-21, 1990, Notes on Numerical Fluid Mechanics (NMFM) vol. 31, Friedr. Vieweg \& Sohn (1991)
Hairer,E., P.P. Norsett, G. Wanner, Solving Ordinary Differential Equations, I. Nonstiff Problems (1987) , Springer Verlag
Hairer,E., P.P. Norsett, G. Wanner, Solving Ordinary Differential Equations, II. Stiff and Differential-Algebraic Problems (1991), Springer Verlag.
van der Houwen,P.J., Sommeijer,B.P., Parallel iteration of high-order Runge-Kutta methods with stepsize control, Journal of Computational and Applied Mathematics 29 (1990), 111-127
van der Houwen,P.J., Sommeijer,B.P., Iterated Runge-Kutta methods on parallel computers, SIAM J. Sci. Stat. Comput. 12 (1991), no. 5, 1000-1028
van der Houwen,P.J., Sommeijer,B.P., Couzy,W., Embedded diagonally implicit Runge-Kutta in parallel computers, Mathematics of Computation 58 (1992), no. 197, 135-159
van der Houwen,P.J., Parallel step-by-step methods, Applied Numerical Mathematics 11 (1993), 69-81
van der Houwen,P.J., Cong,Ngyuen huu, Parallel block predictor-corrector methods of Runge-Kutta type, Applied Numerical Mathematics 13 (1993), 109-123
van der Houwen,P.J., Sommeijer,B.P., Analysis of parallel diagonally implicit iteration of Runge-Kutta methods, Applied Numerical Mathematics 11 (1993), 169-188
van der Houwen,P.J., Sommeijer,B.P., Preconditioning in parallel Runge-Kutta methods for stiff initial value problems, Computers Math. Applic. 28 (1994), no. 10-12, 17-31
van der Houwen,P.J., Sommeijer,B.P., van der Veen,W.A., Parallel iteration across the steps of high-order Runge-Kutta methods for nonstiff initial value problems, Journal of Computational and Applied Mathematics 60 (1995), 309-329
Hutchinson,D., Khalaf,B.M.S., Parallel algorithms for solving initial value problems: front broadering and embedded parallelism, Parallel Computing 17 (1991), 957-968
Iserles,A., Norsett,S.P., On the Theory of Parallel Runge-Kutta Methods, IMA Journal of Numerical Analysis 10 (1990), 463-488
Jackiewicz,Z., Stability analysis of time-point relaxation RK methods with respect to bidiagonal systems of differential equations, Applied Numerical Mathematics 11 (1993), 189-209
Jackson,K.R., Norsett,S.P., The potential for parallelism in Runge-Kutta methods. Part I: Runge-Kutta formulas in standard form, SIAM J. Numer. Anal. 32 (1995), no. 1, 49-82
Kahaner,D.K., Ng,E., Schiesser,W.E., Thompson,S., Experiments with an ordinary differential solver in the parallel solution of method of lines on a shared-memory parallel computer, Journal of Computational and Applied Mathematics 25 (1991), 231-253
Kamel,M.S., Ma,K.S., Enright,W.H., ODEXPERT: An expert system to select numerical solvers for initial value ODE systems, ACM Transactions on Mathematical Software 19 (1993) no. 1, 10-21.
Karakashian,O.A., Rust,W., On the parallel implementation of implicit Runge-Kutta methods, Parallel Computing 5, 20-26
Katz,I.N., Franklin,M.A., Sen,A., Optimally stable parallel predictors for Adams-Moulton correctors, Comp. \& Maths. with Appls., Vol. 3 (1977), 217-233
Kerckhoffs,E.J.H., Multiprocessor algorithms for ODEs, in cite, 325-346
Leimkuhler,B., Ruehli,A., Rapid convergence of wavefom relaxation, Applied Numerical Mathematics 11 (1993), 211-224
Lustman,L., Neta,B., Gragg,W., Solution of the ODE IVP on an Intel Hypercube, Computers \& Mathematics with applications 23 (1992), no. 10, 65-72
Lustman, C. Solution of linear systems of ODEs on an Intel hypercube, SIAM Journal on Scientific Computing 12 (1991), 1480-1485.
de Meyer,H., van Daele,M., vanden Berghe,G., On the inplementation of parallel iterated Runge-Kutta methods on a transputer network, Applied Numerical Mathematics 13 (1993), 155-163
Miranker,W.L., A survey of parallelism in numerical analysis, SIAM Rev. 13 (1971), 524-547
Miranker,W.L., Liniger,W., Parallel Methods for the Numerical Integration of Ordinary Differential Equations, Mathematics of Computation 21 (1967), 303-320
Norsett,S.P., Simonsen,H., Aspects of parallel Runge-Kutta methods, Lectures Notes in Mathematics 1386 (1989), 103-117
Orel,B., Parallel Runge-Kutta methods with real eigenvalues, Applied Numerical Mathematics 11 (1993), 241-250
Rauber, T., Runger, G., Load balancing schemes for extrapolation methods, Concurrency: practice and experience 9 (1997), no. 3
te Riele,H.J.J., Dekker,T.J., van der Vorst,H.A., (eds.) Algorithms and Applications on Vector and Parallel Computers, Special Topics in Supercomputing vol. 3, North-Holland, Elsevier Science Publishers, Amsterdam (1987)
Sommeijer,B.P., Couzy,W., van der Houwen,P.J., A-stable parallel block methods for ordinary and integro-differential equations, Applied Numerical Mathematics 9 (1992), 267-281
Sommeijer,B.P., Parallel iterated Runge-Kutta methods for stiff ordinary differential equations, Journal of Computational and Applied Mathematics 45 (1993), 151-168
Skeel,R.D., Tam,H.W., Potential for Parallelism in Explicit Linear Methods, 377-383, Parallel Computing 15 (1992)
Skelboe,S., Methods for parallel integration of stiff systems of ODEs, BIT 32 (1992), 689-701
Tam,H.W., Parallel Methods for the Numerical Solution of Ordianry Differential Equations, 393-404, Parallel Computing 17 (1993)
Tam,H.W., One-stage parallel methods for the numerical solution of ordinary differential equations, SIAM J. Sci. Stat. Comput. 13 (1992), no. 5, 1039-1061
Tam,H.W., Two-stage parallel methods for the numerical solution of ordinary differential equations, SIAM J. Sci. Stat. Comput. 13 (1992), no. 5, 1062-1084
Tam,H.W., Skeel,R.D., Stability of parallel explicit ODE methods, SIAM J. Numer. Anal. 30 (1995), no. 4, 1179-1192
Tirani,R., Romano,T., Parallel Interpolation of High-Order Runge-Kutta Methods, Computing 50 (1993), 175-184
Thomas,G., Ueberhuber,C.W., Visualisation of Scientific Parallel Programs, Lecture Notes in Computer Science no. 771, Springer Verlag, Berlin (1994)
Ueberhuber,C.W., Numerical Computation. Mtehods, Software and Analysis, Springer Verlag, Berlin (1997)
Vandewalle,S., Piessens,R., A parallel and vectorizable algorithm for solving parabolic partial differential equations, in cite, 216-227
Vanderwalle,St., Piessens,R., Numerical experiments with nonlinear multigrid waveform relaxation on a parallel processor, Applied Numerical Mathematics 8 (1991), 149-161
Worland,P.B., Parallel Methods for ODEs with Improved Absolute Stability Boundaries, Journal of Parallel and Distributed Computing 18 (1993), 25-32
Xiao-qiu,Song, De-gui,Liu, Zhao-ding,Yuan, Some kinds of parallel Runge-Kutta methods, Journal of Computational Mathematics, Supplementary Issue, 1992, 79-85

Information on web sites

W.M. Lioen, J.J.B. de Swart, W.A. van der Veen,
Test set for IVP solvers, Internet document, 1996.
H. Olsson,
Short presentation of Godess project, Internet document, 1996.

Author experience

Petcu,D New Methods for Solving Stiff Differential Equation Systems, Analele Universitatii Timisoara, Seria stiinte Matematice, XXVI-3 (1988), 67-72.
Petcu,D Noi metode numerice de rezolvare a sistemelor de ecuatii diferentiale stiff, SNIC Proceedings of the IVth National Symposium of Informatics in Constructions, Timisora, 26-27 mai 1988, 2 (1989), 287-294.
Petcu,D On the Error Estimate of Newton's Method for a Holder Continuous F-derivate, Analele Universitatii din Timisoara, Seria stiinte Matematice, XXVIII-2 (1990) 149-164.
Petcu,D Search Method For One-Dimensional Minimization, Analele Universitatii din Timisoara, Seria stiinte Matematice, XXVIII-2 (1990) 165-175.
Petcu,D On the Hypothesis for the Error Estimate of Newton's Method, Proceedings of the IVth Symposium on Mathematics and Appl., Universitatea Tehnica Timisoara (1991) 130-140.
Petcu,D On the Kantorovich Hypothesis for Newton's Method, IWR Preprint 92-17, Universitat Heidelberg (1992).
Petcu,D Hybrid Methods for Stiff Differential Equations, IWR Preprint 92-20, Universitat Heidelberg (1992).
Petcu,D Higher Order Implicit One-Step Methods for Integration of Stiff Differential Systems, Analele Universitatii din Timisoara, Seria stiinte Matematice XXX-1 (1992) 59-87.
Petcu,D A Family of Multistep Linear Methods for Ordinary Differential Equations, Analele Universitatii Timisoara, Seria stiinte Matematice, XXX-2 (1992), 257-282.
Petcu,D.,Von Schwerin,R On the Integration of Stiff Systems Using Split Adams-Moulton Methods, IWR Preprint 93-48, Universitat Heidelberg (1993).
Petcu,D Modified Kantorovich Hyphothesis for Newton's Method, Informatica 4 (1993) 188-198.
Petcu,D On the Exponential Fitting of Some Linear Multistep Methods, IWR Preprint 93-43, Universitat Heidelberg (1993).
Petcu,D Some A-stable and L-stable One-step Methods for Stiff Differential Equations, Matarom 3 (1993) 68-86.
Petcu,D Some Stiff Stable Methods for the Numerical Integration of Ordinary Diffrential Equations, Proceedings of the 9th ROmanian SYmposium on Computer Science, ROSYCS '93, 12-13 November 1993, V.Felea, G.Ciobanu(editors), Universitatea A.I.Cuza Iasi (1994) 447-454.
Petcu,D Modified Adams-Moulton Schemes for Solving Stiff Differential Equations, Analele Universitatii Timisoara, Seria stiinte Matematice, XXX-2 (1993), 101-118.
Petcu,D., Efficient Methods for Numerical Solution of Ordinary Differential Equations, Proceedings of the fifth Symposium of Mathematics and its Applications, 29-30 October 1993, Ed. Mirton, Timisoara (1993), 234-240
Petcu,D Numerical methods for solving the stiff differential systems (in romanian), Ph.D. Thesis, University of Timisoara, 1994.
Petcu,D., Parallel algorithms for the numerical integration of stiff ODE systems, Analele Universitatii din Timisoara, Seria Matematica-Informatica, vol. XXXII, fasc. 1 (1994)
Petcu,D., A parallel algorithm for stiff ordinary differential equations, Informatica, Vilnius 5 (1994), no. 3-4, 373-384
Petcu,D., Parallel algorithms (in Romanian), Lectures Notes, Tipografia Universitatii Timisoara (1994)
Petcu,D., Multistep methods for stiff initial value problems, Mathemathical monographs 50, West University of Timisoara, Tipografia Universitatii din Timisoara (1995)
Petcu,D Second Derivative Multistep Methods for Stiff Ordinary Differential Equations, Analele Universitatii Timisoara, Seria stiinte Matematice si Informatica, XXXI-1 (1994), 110-134.
Petcu,D., One-step Methods for the Numerical Solution of Stiff Ordinary Differential Systems, Revue D'Analyse Numerique et de Theorie de L'Approximation, Tome 23, no. 2 (1994), 197-216
Petcu,D., Implicit Runge-Kutta methods on a distributed computational system, Analele Universitatii din Timisoara, Seria Matematica-Informatica, vol. XXXIII, fasc. 2 (1995), 209-220
Petcu,D., Solving differential equations on a multiprocessor machine, The sixth Symposium of Mathematics and its Applications, Technical University of Timisoara, Timisora, 2-5 november 1995, Ed. Mirton (1996), 289-297.
Petcu, D., A distributed implementation of Runge-Kutta methods, Fifth International Colloquim on Numerical Analysis, Plovdiv, Bulgaria, 13-17 august 1996, to appear.
Petcu,D. , Distributed and Parallel Implementations of Runge-Kutta methods, Proceedings of ICAOR: International Conference on Approximation and Optimization, Cluj-Napoca 29 July - 1 August 1996, ed. D.D. Stancu, Gh. Coman, W. Breckner, P. Blaga, vol. II, Transilvania Press (1997), 193-202
Petcu,D., Parallel numerical algorithms, Part I: Solving systems of linear, nonlinear or differential equations, Mathematical Monographies 60, Timisoara, Tipografia Universitatii din Timisora (1996)
Petcu,D., Parallel numerical algorithms, Part II: Solving partial differential equations, Mathematical Monographies 61, Timisoara, Tipografia Universitatii din Timisora (1996)
Petcu,D., Implementation of some multiprocessor algorithms for ODEs using PVM, Lectures Notes in Computer Science 1332: Recent Advances in Parallel Virtual Machine and Message Passing Interface, 4th European PVM/MPI Users' Group Meeting, Cracow, Poland, 03-05 November 1997, Proceedings, eds. Marian Bubak, Jack Dongarra, Jerzy Wasniewski, Springer-Verlag Berlin, 375-383
Petcu,D., A numerical software product for solving ordinary differential equations, Proceedings of the Seventh Symposium of Mathematics and its Applications (7th SMA), 6-9 November 1997, Tipografia Universitatii Politehnica Timisoara, 217-222.