Further Reading

Numerical analysis (General)

• F. S. Acton. Numerical Methods That Work. Math. Assoc. Amer., Washington, DC, 1990. (reprint of 1970 edition).
• K. Atkinson. Elementary Numerical Analysis. John Wiley & Sons, New York, 2d edition, 1993.
• T. Gowers, J. Barrow-Green, I. Leader, eds. The Princeton Companion to Mathematics, Princeton University Press
• E. Isaacson and H. B. Keller. Analysis of Numerical Methods. Dover, New York, 1994. (reprint of 1966 edition).
• W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes. Cambridge University Press, New York, 2d edition, 1992.

Numerical methods for ODEs (General)

• K. E. Brenan, S. L. Campbell, and L. R. Petzold. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. SIAM, Philadelphia, PA, 1996. (reprint of 1989 edition).
• R. Bulirsch and J. Stoer. Numerical treatment of ordinary differential equations by extrapolation methods. Numer. Math., 8:1-13, 1966.
• G. D. Byrne and A. C. Hindmarsh. Stiff ODE solvers: A review of current and coming attractions. J. Comput. Phys., 70:1-62, 1987.
• Enright, W.H. (2000). Continuous numerical methods for ODEs with defect control. J. Comp. Appl. Math. 125, 159–170.
• Enright, W.H. (1989). A new error–control for initial value solvers. Appl. Math. Comput. 31, 588–599
• C. W. Gear. Numerical Initial Value Problems in Ordinary Differential Equations. Prentice Hall, Englewood Cliffs, NJ, 1971.
• C. W. Gear. Numerical solution of ordinary differential equations: Is there anything left to do? SIAM Review, 23:10-24, 1981
• P. Henrici. Discrete Variable Methods in Ordinary Differential Equations. John Wiley & Sons, New York, 1962.
• N. J. Higham. Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia, PA, 1996.
• D.J. Higham, (1989). Robust defect control with Runge–Kutta schemes. SIAM J. Numer. Anal. 26, 1175–1183.
• D.J. Higham, (1991). Runge–Kutta defect control using Hermite–Birkho? interpolation. SIAM J. Sci. Stat. Comput. 12, 991–999
• A. Iserles. A First Course in the Numerical Analysis of Differential Equations. Cambridge University Press, New York, 1996.
• H. Karcher, A Gronwall argument for error estimates of ODE integration schemes, http://www.math.uni-bonn.de/people/karcher /ODEerrorViaGronwall.pdf
• W. E. Milne: Numerical Solutions of Differential Equations, 2nd ed. Dover, 1970
• J. Muscat:Ordinary Differential Equations, http://staff.um.edu.mt/jmus1/diffeq1.pdf
• L. F. Shampine. What everyone solving differential equations numerically should know. In I. Gladwell and D. K. Sayers, editors, Computational Techniques for ODEs, pages 1-17. Academic, New York, 1980.
• L. F. Shampine. Numerical Solution of Ordinary Differential Equations. Chapman and Hall, New York, 1994.
• L. F. Shampine and C. W. Gear. A user's view of solving stiff ordinary differential equations. SIAM Review, 21:1-17, 1979.
• L. F. Shampine, H. A. Watts, and S. M. Davenport. Solving nonstiff ordinary differential equations--the state of the art. SIAM Review, 18:376-411, 1976
• L.F. Shampine, Solving ODEs and DDEs with residual control, http://faculty.smu.edu/lshampin/residuals.pdf

Multistep methods

• G. Dahlquist and Å. Björck. Numerical Methods. Prentice Hall, Englewood Cliffs, NJ, 1974.
• R. D. Skeel. Equivalent forms of multistep methods. Math. Comp., 33:1229-1250, 1979.

Runge-Kutta methods

• J. C. Butcher. The Numerical Analysis of Ordinary Differential Equations. John Wiley & Sons, New York, 1987
• E. Hairer, S. Norsett, and G. Wanner. Solving Ordinary Differential Equations. Springer-Verlag, New York, 1987.

Methods for Hamiltonian Systems

• J. H. E. Cartwright and O. Piro The Dynamics of Runge--Kutta Methods, Int. J. Bifurcation and Chaos, 2, 427-449, 1992
• Aiken, R. C., editor [1985] Stiff Computation (Oxford University Press).
• Aronson, D. G., Chory, M. A., Hall, G. R. & McGehee, R. P. [1983] ``Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study,'' Commun. Math. Phys. 83, 303.
• Arrowsmith, D. K., Cartwright, J. H. E., Lansbury, A. N. & Place, C. M. [1993] ``The Bogdanov map: Bifurcations, mode locking, and chaos in a dissipative system,'' Int. J. Bifurcation and Chaos 3, 803--42.
• Auerbach, S. P. & Friedmann, A. [1991] ``Long-term behaviour of numerically computed orbits: Small and intermediate timestep analysis of one-dimensional systems,'' J. Comput. Phys. 93, 189.
• Beyn, W.-J. [1987a] ``On invariant closed curves for one-step methods,'' Numer. Math. 51, 103.
• Beyn, W.-J. [1987b] ``On the numerical approximation of phase portraits near stationary points,'' SIAM J. Num. Anal. 24, 1095.
• Butcher, J. C. [1987] The Numerical Analysis of Ordinary Differential Equations: Runge--Kutta and General Linear Methods (Wiley).
• Candy, J. & Rozmus, W. [1991] ``A symplectic integration algorithm for separable Hamiltonian systems,'' J. Comput. Phys. 92, 230.
• Channell, P. J. & Scovel, C. [1990] ``Symplectic integration of Hamiltonian systems,'' Nonlinearity 3, 231.
• Chua, L. O. & Lin, P. M. [1975] Computer-Aided Analysis of Electronic Circuits: Algorithms and Computational Techniques (Prentice-Hall).
• Devaney, R. L. [1989] An Introduction to Chaotic Dynamical Systems (Addison--Wesley) second edition.
• Earn, D. J. D. & Tremaine, S. [1992] ``Exact numerical studies of Hamiltonian Maps: Iterating without roundoff error,'' Physica D 56, 1.
• Feng, K. [1986] ``Difference schemes for Hamiltonian formalism and symplectic geometry,'' J. Comput. Math. 4, 279.
• Feng, K. & Qin, M.--z. [1991] ``Hamiltonian algorithms for Hamiltonian systems and a comparative numerical study,'' Comput. Phys. Commun. 65, 173.
• Forest, E. & Ruth, R. D. [1990] ``Fourth order symplectic integration,'' Physica D 43, 105.
• Gardini, L., Lupini, R., Mammana, C. & Messia, M. G. [1987] ``Bifurcations and transition to chaos in the three-dimensional Lotka--Volterra map,'' SIAM J. Appl. Math. 47, 455.
• E. Hairer, C. Lubich, G. Wanner. Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Comput. Mathematics, Vol. 31, Springer-Verlag 2002.
• Hall, G. & Watt, J. M. [1976] Modern Numerical Methods for Ordinary Differential Equations (Oxford University Press).
• Hammel, S. M., Yorke, J. A. & Gregobi, C. [1988] ``Numerical orbits of chaotic processes represent true orbits,'' Bull. Am. Math. Soc. 19, 465.
• Iserles, A. [1990] ``Stability and dynamics of numerical methods for nonlinear ordinary differential equations,'' IMA J. Num. Anal. 10, 1.
• Itoh, T. & Abe, K. [1988] ``Hamiltonian-conserving discrete canonical equations based on variational difference quotients,'' J. Comput. Phys. 76, 85.
• Jackson, E. A. [1989] Perspectives of Nonlinear Dynamics, vol. 1 (Cambridge University Press).
• Kloeden, P. E. & Lorenz, J. [1986] ``Stable attracting sets in dynamical systems and their one-step discretizations,'' SIAM J. Num. Anal. 23, 986.
• Lasagni, F. M. [1988] ``Canonical Runge--Kutta methods,'' ZAMP 39, 952.
• MacKay, R. S. [1990] ``Some aspects of the dynamics and numerics of Hamiltonian systems,'' in Dynamics of Numerics and Numerics of Dynamics IMA.
• Maclachlan, R. I. & Atela, P. [1992] ``The accuracy of symplectic integrators,'' Nonlinearity 5, 541.
• Marsden, J. E., O'Reilly, O. M., Wicklin, F. W. & Zombro, B. W. [1991] ``Symmetry, stability, geometric phases and mechanical integrators,'' Preprint.
• Menyuk, C. R. [1984] ``Some properties of the discrete Hamiltonian method,'' Physica D 11, 109.
• Miller, R. H. [1991] ``A horror story about integration methods,'' J. Comput. Phys. 93, 469.
• Parker, T. S. & Chua, L. O. [1989] Practical Numerical Algorithms for Chaotic Systems (Springer).
• Peitgen, H.-O. & Richter, P. H. [1986] The Beauty of Fractals, chapter 8 `A Discrete Volterra--Lotka System', ,p. 125 (Springer).
• Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. A. [1988] Numerical Recipes in C (Cambridge University Press).
• Prüfer, M. [1985] ``Turbulence in multistep methods for initial value problems,'' SIAM J. Appl. Math. 45, 32.
• Santillan Iturres, A., Domenech, G., El Hasi, C., Vucetich, H. & Piro, O. [1992] Preprint.
• Sanz-Serna, J. M. [1988] ``Runge--Kutta schemes for Hamiltonian systems,'' BIT 28, 877.
• Sanz-Serna, J. M. & Abia, L. [1991] ``Order conditions for canonical Runge--Kutta schemes,'' SIAM J. Num. Anal. 28, 1081.
• Sanz-Serna, J. M. & Vadillo, F. [1987] ``Studies in numerical nonlinear instability iii: Augmented Hamiltonian systems,'' SIAM J. Appl. Math. 47, 92.
• Sauer, T. & Yorke, J. A. [1991] ``Rigorous verification of trajectories for the computer simulation of dynamical systems,'' Nonlinearity 4, 961.
• Stetter, H. J. [1973] Analysis of Discretization Methods for Ordinary Differential Equations (Springer).
• Stewart, I. [1992] ``Numerical methods: Warning—handle with care!,'' Nature 355, 16.
• Thompson, J. M. T. & Stewart, H. B. [1986] Nonlinear Dynamics and Chaos (Wiley).
• Tomita, K. [1986] ``Periodically forced nonlinear oscillators,'' in Holden, A. V., editor, Chaos (Manchester University Press).
• Ushiki, S. [1982] ``Central difference scheme and chaos,'' Physica D 4, 407.
• Yamaguti, M. & Ushiki, S. [1981] ``Chaos in numerical analysis of ordinary differential equations,'' Physica D 3, 618.
• Yee, H. C., Sweby, P. K. & Griffiths, D. F. [1991] ``Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. i. The dynamics of time discretization and its implications for algorithmic development in computational fluid dynamics,'' J. Comput. Phys. 47, 249.
• Yoshida, H. [1990] ``Construction of higher order symplectic integrators,'' Phys. Lett. A 150, 262.
• Zhong, G. & Marsden, J. [1988] ``Lie--Poisson Hamilton--Jacobi theory and Lie--Poisson integrators,'' Phys. Lett. A 133, 134.

Numerical methods for PDEs

• W. F. Ames. Numerical Methods for Partial Differential Equations. Academic, New York, 3d edition, 1992.
• N. Bakaev, A. Ostermann, Long-term stability of variable stepsize approximations of semigroups, Math. Comp. 71 (2002) 1545–1567.
• J. Becker, A second order backward difference method with variable steps for a parabolic problem, BIT 38 (1998) 644–662.A. Brandt, A., Multi-level adaptive technique (MLAT) for fast numerical solutions to boundary value problems, in Proc. 3rd Int. Conf. on Numerical Methods in Fluid Mechanics (Cabannes, H. and Temam, R., eds.), Lecture Notes in Physics 18, Springer-Verlag, 1973, pp. 82–89
• M. Calvo, R.D. Grigorieff, Time discretisation of parabolic problems with the variable 3-step BDF, BIT 42 (2002) 689–701.
• E. Emmrich, Stability and error of the variable two-step BDF for semilinear parabolic problems, Extended version of Preprint No. 703, TU Berlin, Berlin, 2001.B. Fornberg. A Practical Guide to Pseudospectral Methods. Cambridge University Press, New York, 1996.
• C.W. Gear, K.W. Tu, The effect of variable mesh size on the stability of multistep methods, SIAM J. Numer. Anal. 11 (1974) 1025–1043. K. Godunov and V. S. Ryabenki: Difference Schemes: an introduction to the underlying theory, North Holland, 1987
• J. C. Goswami, R. E. Miller, R. D. Nevels, Wavelet Methods for Solving Integral and Differential Equations, in Wiley Encyclopedia of Electrical and Electronics Engineering, Wiley 1999.
• C. González, A. Ostermann, C. Palencia, M. Thalhammer, Backward Euler discretization of fully nonlinear parabolic problems, Math. Comp. 71 (2002) 125–145.
• R.D. Grigorieff, Stability of multistep-methods on variable grids, Numer. Math. 42 (1983) 359–377.
• B. Gustafsson, H.-O. Kreiss, and J. Oliger. Time Dependent Problems and Difference Methods. John Wiley & Sons, New York, 1995. E. Hairer, S.P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, second revised ed., Springer, Berlin, 1993.
• D. Henry, Geometric Theory of Semilinear Parabolic Equations, in: Lecture Notes in Math., vol. 840, Springer, Berlin, 1981.
• A.T. Hill, E. Süli, Upper semicontinuity of attractors for linear multistep methods approximating sectorial evolution equations, Math. Comp. 64 (1995) 1097–1122.C. Johnson. Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, New York, 1987.
• W. Kahan and Ren-Cang Li, Unconventional Schemes for a Class of Ordinary Differential Equations -- With Applications to the Korteweg-deVries Equation, J. Computational Physics, Vol. 134, 1997, pp. 316-331.
• R. J. LeVeque. Numerical Methods for Conservation Laws. Birkhäuser, Boston, 2d edition, 1992.
• A. Majda and A. Bertozzi, Vorticity and Incompressible Flow. Cambridge University Press, New York, 2002.
• K. W. Morton and D. F. Mayers. Numerical Solution of Partial Differential Equations. Cambridge University Press, New York, 1994. A. Ostermann, M. Thalhammer, Convergence of Runge–Kutta methods for nonlinear parabolic problems, Appl. Numer. Math. (2002) 367–380.
• C. Palencia, A stability result for sectorial operators in Banach spaces, SIAM J. Numer. Anal. 30 (1993) 1373–1384.
• C. Palencia, On the stability of variable stepsize rational approximations of holomorphic semigroups, Math. Comp. 62 (1994) 93–103. A. Ostermann M. Thalhammer, G. Kirlinger, Stability of linear multistep methods and applications to nonlinear parabolic problems, Applied Numerical Mathematics 48 (2004) 389–407 Elsevier
• R. Richtmyer and K. W. Morton. Difference Methods for Initial-Value Problems. John Wiley & Sons, New York, 2d edition, 1967.
• J. Strikwerda: Finite Difference Schemes and Partial Differential Equations, Wadsworth and Brooks/Cole, 1989.

Numerical methods for boundary value problems

• W. L. Briggs. A Multigrid Tutorial. SIAM, Philadelphia, PA, 1987.
• W. L. Briggs and V. E. Henson. The DFT: An Owner's Manual for the Discrete Fourier Transform. SIAM, Philadelphia, PA, 1995.
• J. W. Demmel: Applied Numerical Linear Algebra, SIAM, 1997
• H. Keller: Numerical Methods for Two-Point Boundary Value Problems, SIAM, 1976; Dover [reprint], 1992.
• V. Pereyra. Finite difference solution of boundary value problems in ordinary differential equations. In G. H. Golub, editor, Studies in Numerical Analysis, pages 243-269. Math. Assoc. Amer., Washington, DC, 1984.
• M. Pickering. An Introduction to Fast Fourier Transform Methods for Partial Differential Equations, with Applications. John Wiley & Sons, New York, 1986.
• G. Strang and G. Fix. An Analysis of the Finite Element Method. Prentice Hall, Englewood Cliffs, NJ, 1973.
• L. N. Trefethen and D. Bau. Numerical Linear Algebra. SIAM, Philadelphia, PA, 1997.

Solitons

• Lamb, G.L., Elements of soliton theory, Wiley, New York (1980)
• Whitham, G.B., Linear and nonlinear waves, Wiley, New York (1974).