Skip to main content

Numerical Methods

  • Chapter
  • First Online:
Multiple Time Scale Dynamics

Part of the book series: Applied Mathematical Sciences ((AMS,volume 191))

  • 6088 Accesses

Abstract

For the analysis of many nonlinear dynamical systems, numerical methods are indispensable. Fast–slow systems are no exception. In fact, multiscale differential equations provide a big challenge for efficient numerics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  1. S. Adjerid, M. Aiffa, and J.E. Flaherty. High-order finite element methods for singularly perturbed elliptic and parabolic problems. SIAM J. Appl. Math., 55(2):520–543, 1995.

    MATH  MathSciNet  Google Scholar 

  2. A. Arnold, N.B. Abdallah and C. Negulescu. WKB-based schemes for the oscillatory 1D Schrödinger equation in the semi-classical limit. SIAM J. Numer. Anal., 49(4):1436–1460, 2011.

    MATH  MathSciNet  Google Scholar 

  3. U. Ascher and G. Bader. Stability of collocation at Gaussian points. SIAM J. Numer. Anal., 23(2): 412–422, 1986.

    MATH  MathSciNet  Google Scholar 

  4. A. Abdulle. On a priori error analysis of fully discrete heterogeneous multiscale FEM. Multiscale Model. Simul., 4(2):447–459, 2005.

    MATH  MathSciNet  Google Scholar 

  5. A. Abdulle. Explicit methods for stiff stochastic differential equations. Lecture Notes in Comput. Sci. Engineer., 82:1–22, 2012.

    MathSciNet  Google Scholar 

  6. R.V. Abramov. A simple linear response closure approximation for slow dynamics of a multiscale system with linear coupling. Multiscale Model. Simul., 10(1):28–47, 2012.

    MATH  MathSciNet  Google Scholar 

  7. R.V. Abramov. A simple closure approximation for slow dynamics of a multiscale system: nonlinear and multiplicative coupling. Multiscale Model. Simul., 11(1):134–151, 2013.

    MATH  MathSciNet  Google Scholar 

  8. A. Abdulle and S. Cirilli. S-ROCK: Chebyshev methods for stiff stochastic differential equations. SIAM J. Sci. Comput., 30(2):997–1014, 2008.

    MATH  MathSciNet  Google Scholar 

  9. U. Ascher, J. Christiansen, and R.D. Russell. COLSYS-A collocation code for boundary-value problems. In Codes for Boundary-Value Problems in Ordinary Differential Equations, pages 164–185. Springer, 1979.

    Google Scholar 

  10. U. Ascher, J. Christiansen, and R.D. Russell. Collocation software for boundary-value ODEs. ACM Trans. Math. Software, 7(2):209–222, 1981.

    MATH  Google Scholar 

  11. A. Abdulle and W. E. Finite difference heterogeneous multi-scale method for homogenization problems. J. Comp. Phys., 191(1):18–39, 2003.

    Google Scholar 

  12. A. Abdulle, W. E, B. Engquist, and E. Vanden-Eijnden. The heterogeneous multiscale method. Acta Numerica, 21:1–87, 2012.

    Google Scholar 

  13. G.G. Avalos and N.B. Gallegos. Quasi-steady state model determination for systems with singular perturbations modelled by bond graphs. Math. Computer Mod. Dyn. Syst., pages 1–21, 2013. to appear.

    Google Scholar 

  14. A. Abdulle, Y. Hu, and T. Li. Chebyshev methods with discrete noise: the tau-ROCK methods. J. Comp. Math., 28(2):195–217, 2010.

    MATH  MathSciNet  Google Scholar 

  15. U. Ascher and S. Jacobs. On collocation implementation for singularly perturbed two-point problems. SIAM J. Sci. Stat. Comput., 10(3):533–549, 1989.

    MATH  MathSciNet  Google Scholar 

  16. D.F. Anderson and M. Koyama. Weak error analysis of approximate simulation methods for multi-scale stochastic chemical kinetic systems. Multiscale Model. Simul., 10(4):1493–1524, 2012.

    MATH  MathSciNet  Google Scholar 

  17. L.R. Abrahamsson, H.B. Keller, and H.-O. Kreiss. Difference approximations for singular perturbations of systems of ordinary differential equations. Numer. Math., 22:367–391, 1974.

    MATH  MathSciNet  Google Scholar 

  18. A. Abdulle and T. Li. S-ROCK methods for stiff Itô SDEs. Comm. Math. Sci., 6(4):845–868, 2008.

    MATH  MathSciNet  Google Scholar 

  19. R. Alexander. Diagonally implicit Runge–Kutta methods for stiff ODE’s. SIAM J. Numer. Anal., 14(6):1006–1021, 1977.

    MATH  MathSciNet  Google Scholar 

  20. A. Abdulle, P. Lin, and A. Shapeev. Numerical methods for multilattices. Multiscale Model. Simul., 10(3):696–726, 2012.

    MATH  MathSciNet  Google Scholar 

  21. U.M. Ascher and R.M.M. Mattheij. General framework, stability and error analysis for numerical stiff boundary value methods. Numer. Math., 54:355–372, 1988.

    MATH  MathSciNet  Google Scholar 

  22. U.M. Ascher, P.A. Markowich, P. Pietra, and C. Schmeiser. A phase plane analysis of transonic solutions for the hydrodynamic semiconductor model. Math. Mod. Meth. Appl. Sci., 1(3):347–376, 1991.

    MATH  MathSciNet  Google Scholar 

  23. U.M. Ascher, R.M.M. Mattheij, and R.D. Russell. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM, 1987.

    Google Scholar 

  24. J.L. Anderson. Equidistribution schemes, Poisson generators, and adaptive grids. Appl. Math. Comput., 24(3):211–227, 1987.

    MATH  MathSciNet  Google Scholar 

  25. N. Ben Abdallah and O. Pinaud. Multiscale simulation of transport in an open quantum system: Resonances and WKB interpolation. J. Comp. Phys., 213(1):288–310, 2006.

    MATH  MathSciNet  Google Scholar 

  26. A. Abdulle and G. Pavliotis. Numerical methods for stochastic partial differential equations with multiple scales. J. Comput. Phys., 231(6):2482–2497, 2012.

    MATH  MathSciNet  Google Scholar 

  27. U. Ascher and R.D. Russell. Reformulation of boundary value problems into standard form. SIAM Rev., 23(2):238–254, 1981.

    MATH  MathSciNet  Google Scholar 

  28. U. Ascher. On some difference schemes for singular singularly-perturbed boundary value problems. Numer. Math., 46:1–30, 1985.

    MATH  MathSciNet  Google Scholar 

  29. U. Ascher and R. Weiss. Collocation for singular perturbation problems I: first order systems with constant coefficients. SIAM J. Numer. Anal., 20(3):537–557, 1983.

    MATH  MathSciNet  Google Scholar 

  30. U. Ascher and R. Weiss. Collocation for singular perturbation problems II: linear first order systems without turning points. Mathematics of Computation, 43(167):157–187, 1984.

    MATH  MathSciNet  Google Scholar 

  31. U. Ascher and R. Weiss. Collocation for singular perturbation problems III: nonlinear problems without turning points. SIAM J. Sci. Stat. Comput., 5(4):811–829, 1984.

    MATH  MathSciNet  Google Scholar 

  32. K. Burrage and J.C. Butcher. Stability criteria for implicit Runge–Kutta methods. SIAM J. Numer. Anal., 16(1):46–57, 1979.

    MATH  MathSciNet  Google Scholar 

  33. K. Burrage and J.C. Butcher. Nonlinear stability of a general class of differential equation methods. BIT, 20(2):185–203, 1980.

    MATH  MathSciNet  Google Scholar 

  34. C.J. Budd, H. Huang, and R.D. Russell. Mesh selection for a nearly singular boundary value problem. J. Sci. Comput., 16(4):525–552, 2001.

    MATH  MathSciNet  Google Scholar 

  35. T.A. Bickart and E.I. Jury. Arithmetic tests for A-stability, A[α]-stability, and stiff-stability. BIT, 18:9–21, 1978.

    MATH  MathSciNet  Google Scholar 

  36. T. Bakri, Y.A. Kuznetsov, F. Verhulst, and E. Doedel. Multiple solutions of a generalized singular perturbed Bratu problem. Int. J. Bif. Chaos, 22(4), 2012.

    Google Scholar 

  37. D. Brown and J. Lorenz. A higher-order method for stiff boundary-value problems with turning points. SIAM J. Sci. Stat. Comp., 8:790–805, 1987.

    MATH  MathSciNet  Google Scholar 

  38. M. Berger and J. Oliger. Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys., 53(3):484–512, 1984.

    MATH  MathSciNet  Google Scholar 

  39. D.L. Bosley. An improved matching procedure for transient resonance layers in weakly nonlinear oscillatory systems. SIAM J. Appl. Math., 56(2):420–445, 1996.

    MATH  MathSciNet  Google Scholar 

  40. C.-E. Bréhier. Analysis of an HMM time-discretization scheme for a system of stochastic PDEs. SIAM J. Numer. Anal., 51(2):1185–1210, 2013.

    MATH  MathSciNet  Google Scholar 

  41. C. De Boor and B. Swartz. Collocation at Gaussian points. SIAM J. Numer. Anal., 10:582–606, 1973.

    MATH  MathSciNet  Google Scholar 

  42. L. Brugnano and D. Trigiante. A new mesh selection strategy for ODEs. Appl. Numer. Math., 24:1–21, 1997.

    MATH  MathSciNet  Google Scholar 

  43. J.C. Butcher. Implicit Runge–Kutta processes. Math. Comput., 18(85):50–64, 1964.

    MATH  MathSciNet  Google Scholar 

  44. J.C. Butcher. A stability property of implicit Runge–Kutta methods. BIT Numer. Math., 15(4): 358–361, 1975.

    MATH  Google Scholar 

  45. J.C. Butcher. On the implementation of implicit Runge–Kutta methods. BIT Numer. Math., 16(3): 237–240, 1976.

    MATH  MathSciNet  Google Scholar 

  46. G.F. Carrier. Boundary layer problems in applied mathematics. Comm. Pure Appl. Math., 7:11–17, 1954.

    MATH  MathSciNet  Google Scholar 

  47. J.R. Cash. Adaptive Runge–Kutta methods for nonlinear two-point boundary value problems with mild boundary layers. Comp. Maths. with Appls., 11(6):605–619, 1985.

    MATH  MathSciNet  Google Scholar 

  48. J.R. Cash. On the numerical integration of nonlinear two-point boundary value problems using iterated deferred corrections. Part 2: The development and analysis of highly stable deferred correction formulae. SIAM J. Numer. Anal., 25(4):862–882, 1988.

    Google Scholar 

  49. J.R. Cash. Efficient numerical methods for the solution of stiff initial-value problems and differential algebraic equations. Proc. R. Soc. Lond. A, 459:797–815, 2003.

    MATH  MathSciNet  Google Scholar 

  50. S. Capper, J. Cash, and F. Mazzia. On the development of effective algorithms for the numerical solution of singularly perturbed two-point boundary value problems. Int. J. Comput. Sci. Math., 1(1):42–57, 2007.

    MATH  MathSciNet  Google Scholar 

  51. M. Condon, A. Deano, and A. Iserles. On highly oscillatory problems arising in electronic engineering. ESIAM: Math. Model. Numer. Anal., 43(4):785–804, 2009.

    MATH  MathSciNet  Google Scholar 

  52. M. Condon, A. Deano, and A. Iserles. On second-order differential equations with highly oscillatory forcing terms. Proc. R. Soc. A, 466:1809–1828, 2010.

    MATH  MathSciNet  Google Scholar 

  53. M. Condon, A. Deano, and A. Iserles. On systems of differential equations with extrinsic oscillation. Discr. Cont. Dyn. Syst. A, 28(4):1345–1367, 2010.

    MATH  MathSciNet  Google Scholar 

  54. S. Chen, W. E, and C.W. Shu. The heterogeneous multiscale method based on the discontinuous Galerkin method for hyperbolic and parabolic problems. Multiscale Model. Simul., 3(4):871–894, 2005.

    Google Scholar 

  55. R.J. Clasen, D. Garfinkel, N.Z. Shapiro, and G.C. Roman. A method for solving certain stiff differential equations. SIAM J. Appl. Math., 34(4):732–742, 1978.

    MATH  MathSciNet  Google Scholar 

  56. C.F. Curtiss and J. Hirschfelder. Integration of stiff equations. Proc. Natl. Acad. Sci. USA, 38(3): 235–243, 1952.

    MATH  MathSciNet  Google Scholar 

  57. K.W. Chang. Singular perturbations of a boundary problem for a vector second order differential equation. SIAM J. Appl. Math., 30(1):42–54, 1976.

    MATH  MathSciNet  Google Scholar 

  58. K. Chen. Error equidistribution and mesh adaptation. SIAM J. Sci. Comput., 15(4):798–818, 1994.

    MATH  MathSciNet  Google Scholar 

  59. D. Cohen, E. Hairer, and C. Lubich. Modulated Fourier expansions of highly oscillatory differential equations. Found. Comput. Math., 3(4):327–345, 2003.

    MATH  MathSciNet  Google Scholar 

  60. A.R. Champneys and Yu.A. Kuznetsov. Numerical detection and continuation of codimension-two homoclinic bifurcations. Int. J. Bif. Chaos, 4(4):785–822, 1994.

    MATH  MathSciNet  Google Scholar 

  61. A.R. Champneys, Yu.A. Kuznetsov, and B. Sandstede. A numerical toolbox for homoclinic bifurcation analysis. Int. J. Bif. Chaos, 6(5):867–887, 1996.

    MATH  MathSciNet  Google Scholar 

  62. S.M. Cox and P.C. Matthews. Exponential time differencing for stiff systems. J. Comput. Phys., 176(2):430–455, 2002.

    MATH  MathSciNet  Google Scholar 

  63. J.R. Cash and F. Mazzia. A new mesh selection algorithm, based on conditioning, for two-point boundary value codes. J. Comp. Appl. Math., 184:362–381, 2005.

    MATH  MathSciNet  Google Scholar 

  64. P. Chartier, A. Murua, and J.M. Sanz-Serna. Higher-order averaging, formal series and numerical integration I: B-series. Found. Comput. Math., 10(6):695–727, 2010.

    MATH  MathSciNet  Google Scholar 

  65. P. Chartier, A. Murua, and J.M. Sanz-Serna. Higher-order averaging, formal series and numerical integration II: the quasi-periodic case. Found. Comput. Math., 12(4):471–508, 2012.

    MATH  MathSciNet  Google Scholar 

  66. J.R. Cash, F. Mazzia, N. Sumarti, and D. Trigiante. The role of conditioning in mesh selection algorithms for first order systems of linear two point boundary value problems. J. Comp. Appl. Math., 185:212–224, 2006.

    MATH  MathSciNet  Google Scholar 

  67. J.R. Cash, G. Moore, and R.W. Wright. An automatic continuation strategy for the solution of singularly perturbed linear two-point boundary value problems. J. Comp. Phys., 122:266–279, 1995.

    MATH  MathSciNet  Google Scholar 

  68. E.M. Constantinescu and A. Sandu. Extrapolated multirate methods for differential equations with multiple time scales. J. Sci. Comput., 56:28–44, 2013.

    MATH  MathSciNet  Google Scholar 

  69. M.P. Calvo and J.M. Sanz-Serna. Heterogeneous multiscale methods for mechanical systems with vibrations. SIAM J. Sci. Comput., 32:2029–2046, 2010.

    MATH  MathSciNet  Google Scholar 

  70. G. Dahlquist. A special stability problem for linear multistep methods. BIT, 3:27–43, 1963.

    MATH  MathSciNet  Google Scholar 

  71. E.J. Doedel, A. Champneys, F. Dercole, T. Fairgrieve, Y. Kuznetsov, B. Oldeman, R. Paffenroth, B. Sandstede, X. Wang, and C. Zhang. Auto 2007p: Continuation and bifurcation software for ordinary differential equations (with homcont). http://cmvl.cs.concordia.ca/auto, 2007.

  72. A. Dhooge, W. Govaerts, and Yu.A. Kuznetsov. MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw., 29:141–164, 2003.

    MATH  MathSciNet  Google Scholar 

  73. A. Dutt, L. Greengard, and V. Rokhlin. Spectral deferred correction methods for ordinary differential equations. BIT Numer. Math., 40(2):241–266, 2000.

    MATH  MathSciNet  Google Scholar 

  74. E. Doedel and R.F. Heinemann. Numerical computation of periodic solution branches and oscillatory dynamics of the stirred tank reactor with A → B → C reactions. Chemical Engineering Science, 38(9):1493–1499, 1983.

    Google Scholar 

  75. D.W. Decker and H.B. Keller. Path following near bifurcation. Comm. Pure Appl. Math., 34(2):149–175, 1981.

    MATH  MathSciNet  Google Scholar 

  76. E. Doedel, H.B. Keller, and J.-P. Kernevez. Numerical analysis and control of bifurcation problems. I. Bifurcation in finite dimensions. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1(3):493–520, 1991.

    Google Scholar 

  77. E. Doedel, H.B. Keller, and J.-P. Kernevez. Numerical analysis and control of bifurcation problems. II. Bifurcation in infinite dimensions. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1(4):745–772, 1991.

    Google Scholar 

  78. M. Dobson, M. Luskin, and C. Ortner. Sharp stability estimates for the force-based quasicontinuum approximation of homogeneous tensile deformation. Multiscale Model. Simul., 8(3):782–802, 2010.

    MATH  MathSciNet  Google Scholar 

  79. M. Dobson, M. Luskin, and C. Ortner. Stability, instability, and error of the force-based quasicontinuum approximation. Arch. Rat. Mech. Anal., 197:179–202, 2010.

    MATH  MathSciNet  Google Scholar 

  80. E.J. Doedel. Auto 97: Continuation and bifurcation software for ordinary differential equations. http://indy.cs.concordia.ca/auto, 1997.

  81. E.J. Doedel. Auto 2000: Continuation and bifurcation software for ordinary differential equations (with homcont). http://cmvl.cs.concordia.ca/auto, 2000.

  82. E.J. Doedel. Lecture notes on numerical analysis of nonlinear equations. http://cmvl.cs.concordia.ca/publications/notes.ps.gz, 2007.

  83. I. Dag and A. Sahin. Numerical solution of singularly perturbed problems. Int. J. Nonlin. Sci., 8(1): 32–39, 2009.

    MATH  MathSciNet  Google Scholar 

  84. K. Dekker and J.G. Verwer. Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations. North-Holland, 1984.

    Google Scholar 

  85. W. E. Analysis of the heterogeneous multiscale method for ordinary differential equations. Comm. Math. Sci., 1(3):423–426, 2003.

    Google Scholar 

  86. W. E. Principles of Multiscale Modeling. CUP, 2011.

    Google Scholar 

  87. W. E and B. Engquist. The heterogeneous multiscale methods. Comm. Math.Sci., 1(1):87–132, 2003.

    Google Scholar 

  88. W. E and B. Engquist. The heterogeneous multi-scale method for homogenization problems. In Multiscale Methods in Science and Engineering, volume 44 of Lecture Notes Comput. Sci. Eng., pages 89–110. Springer, 2005.

    Google Scholar 

  89. W. E, B. Engquist, and Z. Huang. Heterogeneous multiscale method: a general methodology for multiscale modeling. Phys. Rev. B, 67(9):092101, 2003.

    Google Scholar 

  90. W. E, B. Engquist, X. Li, W. Ren, and E. Vanden-Eijnden. Heterogeneous multiscale methods: a review. Comm. Comp. Phys., 2(3):367–450, 2007.

    Google Scholar 

  91. B. Engquist, A. Fokas, E. Hairer, and A. Iserles. Highly Oscillatory Problems. CUP, 2009.

    Google Scholar 

  92. Y.R. Efendiev and T.Y. Hou. Multiscale Finite Element Methods. Theory and Applications. Springer, 2009.

    Google Scholar 

  93. B.L. Ehle. High order A-stable methods for the numerical solution of systems of DE’s. BIT Numer. Math., 8(4):276–278, 1968.

    MATH  MathSciNet  Google Scholar 

  94. B.L. Ehle. A-stable methods and Padé approximations to the exponential. SIAM J. Math. Anal., 4(4):671–680, 1973.

    MATH  MathSciNet  Google Scholar 

  95. W.H. Enright, T.E. Hull, and B. Lindberg. Comparing numerical methods for stiff systems of ODEs. BIT Numer. Math., 15(1):10–48, 1975.

    MATH  Google Scholar 

  96. K. Eriksson, C. Johnson, and A. Logg. Explicit time-stepping for stiff ODEs. SIAM J. Sci. Comput., 25(4):1142–1157, 2003.

    MATH  MathSciNet  Google Scholar 

  97. R. Erban, I.G. Kevrekisdis, D. Adalsteinsson, and T.C. Elston. Gene regulatory networks: a coarse-grained, equation-free approach to multiscale computation. J. Chem. Phys., 124(8): 084106, 2006.

    Google Scholar 

  98. C. Engstler and C. Lubich. Multirate extrapolation methods for differential equations with different time scales. Computing, 58(2):173–185, 1997.

    MATH  MathSciNet  Google Scholar 

  99. W. E, D. Liu, and E. Vanden-Eijnden. Analysis of multiscale methods for stochastic differential equations. Comm. Pure App. Math., 58:1544–1585, 2005.

    Google Scholar 

  100. W. E, D. Liu, and E. Vanden-Eijnden. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys., 123:194107, 2005.

    Google Scholar 

  101. W. E, D. Liu, and E. Vanden-Eijnden. Nested stochastic simulation algorithm for chemical kinetic systems with multiple time scales. J. Comp. Phys., 221(1):158–180, 2007.

    Google Scholar 

  102. W. E, J. Lu, and J.Z. Yang. Uniform accuracy of the quasicontinuum method. Phys. Rev. B., 74(21):214115, 2006.

    Google Scholar 

  103. W. E, P. Ming and P. Zhang. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc., 18(1):121–156, 2005.

    Google Scholar 

  104. W.H. Enright. Second derivative multistep methods for stiff ordinary differential equations. SIAM J. Numer. Anal., 11(2):321–331, 1974.

    MATH  MathSciNet  Google Scholar 

  105. W. E., W. Ren, and E. Vanden-Eijnden. A general strategy for designing seamless multiscale methods. J. Comput. Phys., 228(15):5437–5433, 2009.

    Google Scholar 

  106. B. Engquist and Y.-H. Tsai. Heterogeneous multiscale methods for stiff ordinary differential equations. Math. Comput., 74(252):1707–1742, 2005.

    MATH  MathSciNet  Google Scholar 

  107. P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan, and G.I. Shishkin. Singularly perturbed convection-diffusion problems with boundary and weak interior layers. J. Comput. Appl. Math., 166:133–151, 2004.

    MATH  MathSciNet  Google Scholar 

  108. P.A. Farrell, P.W. Hemker, and G.I. Shishkin. Discrete approximations for singularly perturbed boundary value problems with parabolic layers. I. J. Comput. Math., 14:71–97, 1996.

    Google Scholar 

  109. T.F. Fairgrieve and A.D. Jepson. O.K. Floquet multipliers. SIAM J. Numer. Anal., 28(5):1446–1462, 1991.

    Google Scholar 

  110. J.E. Flaherty and W. Mathon. Collocation with polynomial and tension splines for singularly-perturbed boundary value problems. SIAM J. Sci. Stat. Comput., 1(2):260–289, 1980.

    MATH  MathSciNet  Google Scholar 

  111. P.A. Farrell, J.J. Miller, E. O’Riordan, and G.I. Shishkin. A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation. SIAM J. Numer. Anal., 33(3):1135–1149, 1996.

    MATH  MathSciNet  Google Scholar 

  112. J.E. Flaherty and R.E. O’Malley. The numerical solution of boundary value problems for stiff differential equations. Math. Comput., 31:66–93, 1977.

    MATH  MathSciNet  Google Scholar 

  113. J.E. Flaherty and R.E. O’Malley. Numerical methods for stiff systems of two-point boundary value problems. SIAM J. Sci. Stat. Comput., 5(4):865–886, 1984.

    MATH  MathSciNet  Google Scholar 

  114. G. Folland. Real Analysis - Modern Techniques and Their Applications. Wiley, 1999.

    Google Scholar 

  115. S. Franz and H.-G. Roos. The capriciousness of numerical methods for singular perturbations. SIAM Rev., 53(1):157–173, 2011.

    MATH  MathSciNet  Google Scholar 

  116. Walter Gautschi. Numerical Analysis. Birkhäuser Boston, 1997.

    Google Scholar 

  117. P.P.N. De Groen and P.W. Hemker. Error bounds for exponen- tially fitted Galerkin methods applied to stiff two-point boundary value problems. In P.W. Hemker and J.J.H. Miller, editors, Numerical Analysis of Singular Perturbation Problems, pages 217–249. Academic Press, 1979.

    Google Scholar 

  118. J. Guckenheimer, K. Hoffman, and W. Weckesser. Numerical computation of canards. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10(12):2669–2687, 2000.

    MATH  MathSciNet  Google Scholar 

  119. M. Garbey and H.G. Kaper. Heterogeneous domain decomposition for singularly perturbed elliptic boundary value problems. SIAM J. Numer. Anal., 34(4):1513–1544, 1997.

    MATH  MathSciNet  Google Scholar 

  120. C.W. Gear and I.G. Kevrekidis. Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum. SIAM J. Sci. Comput., 24(4):1091–1106, 2003.

    MATH  MathSciNet  Google Scholar 

  121. J. Guckenheimer and C. Kuehn. Computing slow manifolds of saddle-type. SIAM J. Appl. Dyn. Syst., 8(3):854–879, 2009.

    MATH  MathSciNet  Google Scholar 

  122. J. Guckenheimer and C. Kuehn. Homoclinic orbits of the FitzHugh–Nagumo equation: Bifurcations in the full system. SIAM J. Appl. Dyn. Syst., 9:138–153, 2010.

    MATH  MathSciNet  Google Scholar 

  123. D. Givon, I.G. Kevrekidis, and R. Kupferman. Strong convergence of projective integration schemes for singularly perturbed stochastic differential systems. Comm. Math. Sci., 4(4):707–729, 2006.

    MATH  MathSciNet  Google Scholar 

  124. C.W. Gear, I.G. Kevrekidis, and C. Theodoropoulos. Coarse-integration/bifurcation analysis via microscopic simulators: micro-Galerkin methods. Comput. Chem. Eng., 26(7):941–963, 2002.

    Google Scholar 

  125. J. Guckenheimer and D. LaMar. Periodic orbit continuation in multiple time scale systems. In Understanding Complex Systems: Numerical continuation methods for dynamical systems, pages 253–267. Springer, 2007.

    Google Scholar 

  126. C.W. Gear, J. Li, and I.G. Kevrekidis. The gap-tooth method in particle simulations. Phys. Lett. A, 316(3):190–195, 2003.

    MATH  MathSciNet  Google Scholar 

  127. W.F. Govaerts. Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM, Philadelphia, PA, 1987.

    Google Scholar 

  128. P.P.N. De Groen. A finite element method with a large mesh-width for a stiff two-point boundary value problem. J. Comput. Appl. Math., 7(1):3–15, 1981.

    MATH  MathSciNet  Google Scholar 

  129. R. Gobbi and R. Spigler. Comparing Shannon to autocorre- lation-based wavelets for solving singularly perturbed elliptic BV problems. BIT Numer. Math., 52:21–43, 2012.

    MATH  MathSciNet  Google Scholar 

  130. Y. Hu, A. Abdulle, and T. Li. Boosted hybrid method for solving chemical reaction systems with multiple scales in time and population size. Comm. Comp. Phys., 12:981–1005, 2012.

    MathSciNet  Google Scholar 

  131. P. Henrici. Discrete Variable Methods in Ordinary Differential Equations. Wiley, 1962.

    Google Scholar 

  132. E. Hairer and C. Lubich. Extrapolation at stiff differential equations. Numer. Math., 52(4):377–400, 1988.

    MATH  MathSciNet  Google Scholar 

  133. E. Hairer and C. Lubich. Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal., 38(2):414–441, 2000.

    MATH  MathSciNet  Google Scholar 

  134. E. Hairer, C. Lubich, and M. Roche. Error of Runge–Kutta methods for stiff problems studied via differential algebraic equations. BIT, 28(3):678–700, 1988.

    MATH  MathSciNet  Google Scholar 

  135. S.P. Hastings and J.B. McLeod. Classical Methods in Ordinary Differential Equations: With Applications to Boundary Value Problems. AMS, 2012.

    Google Scholar 

  136. A.F. Hegarty, J.J. Miller, E. O’Riordan, and G.I. Shishkin. Special meshes for finite difference approximations to an advection–diffusion equation with parabolic layers. J. Comput. Phys., 117: 47–54, 1995.

    MATH  MathSciNet  Google Scholar 

  137. A. Haselbacher, F.M. Najjar, L. Massa, and R.D. Moser. Slow-time acceleration for modeling multiple-time-scale problems. J. Comput. Phys., 229(2):325–342, 2010.

    MATH  MathSciNet  Google Scholar 

  138. W. Huang, Y. Ren, and R.D. Russell. Moving mesh partial differential equations (MMPDES) based on the equidistribution principle. SIAM J. Numer. Anal., 31(3):709–730, 1994.

    MATH  MathSciNet  Google Scholar 

  139. P.W. Hemker, G.I Shishkin, and L.P. Shishkina. The use of defect correction for the solution of parabolic singular perturbation problems. Z. Angew. Math. Mech., 77(1):59–74, 1997.

    Google Scholar 

  140. P.W. Hemker, G.I Shishkin, and L.P. Shishkina. ε-uniform schemes with high-order time-accuracy for parabolic singular perturbation problems. IMA J. Numer. Anal., 20(1):99–121, 2000.

    Google Scholar 

  141. W. Hundsdorfer and J.G. Verwer. Numerical Solution of Time-dependent Advection-Diffusion-Reaction Equations. Springer, 2003.

    Google Scholar 

  142. E. Hairer and G. Wanner. Solving Ordinary Differential Equations I. Springer, 1991.

    Google Scholar 

  143. E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Springer, 1991.

    Google Scholar 

  144. A.M. Il’in. Differencing scheme for a differential equation with a small parameter affecting the highest derivative. Math. Notes Acad. Sci. USSR, 6(2):596–602, 1969.

    Google Scholar 

  145. A. Iserles. Functional fitting - new family of schemes for integration of stiff ODE. Math. Comput., 31:112–123, 1977.

    MATH  MathSciNet  Google Scholar 

  146. A. Iserles. Quadrature methods for stiff ordinary differential systems. Math. Comput., 36:171–182, 1981.

    MATH  MathSciNet  Google Scholar 

  147. A. Iserles. Composite methods for numerical solution of stiff systems of ODEs. SIAM J. Num. Anal., 21:340–351, 1984.

    MATH  MathSciNet  Google Scholar 

  148. A. Iserles. A First Course in the Numerical Analysis of Differential Equations. CUP, 1996.

    Google Scholar 

  149. A. Iserles. On the global error of discretization methods for highly-oscillatory ordinary differential equations. BIT, 42:561–599, 2002.

    MATH  MathSciNet  Google Scholar 

  150. T. Jahnke. Long-time-step integrators for almost-adiabatic quantum dynamics. SIAM J. Sci. Comput., 25:2145–2164, 2004.

    MATH  MathSciNet  Google Scholar 

  151. J. Jansson, C. Johnson, and A. Logg. Computational modeling of dynamical systems. Math. Mod. Meth. Appl. Sci., 15(3):471, 2005.

    Google Scholar 

  152. T. Jahnke and C. Lubich. Numerical integrators for quantum dynamics close to the adiabatic limit. Numerische Mathematik, 94:289–314, 2003.

    MATH  MathSciNet  Google Scholar 

  153. Z. Jia and B. Leimkuhler. A parallel multiple time-scale reversible integrator for dynamics simulation. Future Gen. Comp. Syst., 19:415–424, 2003.

    Google Scholar 

  154. Z. Jia and B. Leimkuhler. Geometric integrators for multiple timescale simulation. J. Phys. A, 439:5379–5403, 2006.

    MathSciNet  Google Scholar 

  155. D.A. Knoll, L. Chacon, L.G. Margolin, and V.A. Mousseau. On balanced approximations for time integration of multiple time scale systems. J. Comput. Phys., 185(2):583–611, 2003.

    MATH  Google Scholar 

  156. H. Keller. Accurate difference methods for nonlinear two-point boundary value problems. SIAM J. Numer. Anal., 11(2): 305–320, 1974.

    MATH  MathSciNet  Google Scholar 

  157. H. Keller. The bordering algorithm and path following near singular points of higher nullity. SIAM J. Sci. Comput., 4(4): 573–582, 1983.

    MATH  Google Scholar 

  158. I.G. Kevrekidis, C.W. Gear, J.M. Hyman, P.G. Kevrekidis, O. Runborg, and C. Theodoropoulos. Equation-free, coarse-grained multiscale computation: enabling mocroscopic simulators to perform system-level analysis. Comm. Math. Sci., 1(4):715–762, 2003.

    MATH  MathSciNet  Google Scholar 

  159. I.G. Kevrekidis, C.W. Gear, and G. Hummer. Equation-free: the computer-aided analysis of complex multiscale systems. AIChE Journal, 50(7):1346–1355, 2004.

    Google Scholar 

  160. R. Kirby. On the convergence of high resolution methods with multiple time scales for hyperbolic conservation laws. Math. Comp., 72(243):1239–1250, 2003.

    MATH  MathSciNet  Google Scholar 

  161. B. Kreiss and H.-O. Kreiss. Numerical methods for singular perturbation problems. SIAM J. Numer. Anal., 18(2):262–276, 1981.

    MATH  MathSciNet  Google Scholar 

  162. H.-O. Kreiss, N.K. Nichols, and D.L. Brown. Numerical methods for stiff two-point boundary value problems. SIAM J. Numer. Anal., 18(2):325–386, 1986.

    MathSciNet  Google Scholar 

  163. B. Krauskopf, H.M. Osinga, and J. Galán-Vique, editors. Numerical Continuation Methods for Dynamical Systems: Path following and boundary value problems. Springer, 2007.

    Google Scholar 

  164. M.K. Kadalbajoo and K.C. Patidar. A survey of numerical techniques for solving singularly perturbed ordinary differential equations. Appl. Math. Comp., 130(2):457–510, 2002.

    MATH  MathSciNet  Google Scholar 

  165. P.E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations. Springer, 2010.

    Google Scholar 

  166. P. Kim, X. Piao, and S.D. Kim. An error-corrected Euler method for solving stiff problems based on Chebyshev collocation. SIAM J. Numer. Anal., 49(6):2211–2230, 2011.

    MATH  MathSciNet  Google Scholar 

  167. B. Krauskopf and T. Riess. A Lin’s method approach to finding and continuing heteroclinic connections involving periodic orbits. Nonlinearity, 21(8):1655–1690, 2008.

    MATH  MathSciNet  Google Scholar 

  168. D. Kushnir and V. Rokhlin. A highly accurate solver for stiff ordinary differential equations. SIAM J. Sci. Comput., 34(3): A1296–A1315, 2012.

    MATH  MathSciNet  Google Scholar 

  169. H.-O. Kreiss. Central difference schemes and stiff boundary value problems. BIT, 24:560–567, 1984.

    MATH  MathSciNet  Google Scholar 

  170. N. Kopteva and M. Stynes. A robust adaptive method for a quasi-linear one-dimensional convection-diffusion problem. SIAM J. Numer. Anal., 39(4):1446–1467, 2001.

    MATH  MathSciNet  Google Scholar 

  171. I.G. Kevrekidis and G. Samaey. Equation-free multiscale computation: algorithms and applications. Ann. Rev. Phys. Chem., 60:321–344, 2009.

    Google Scholar 

  172. A.K. Kassam and L.N. Trefethen. Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput., 26(4):1214–1233, 2005.

    MATH  MathSciNet  Google Scholar 

  173. Yu.A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer, New York, NY, 3rd edition, 2004.

    MATH  Google Scholar 

  174. T. Li, A. Abdulle, and W. E. Effectiveness of implicit methods for stiff stochastic differential equations. Comm. Comp. Phys., 3(2):295–307, 2008.

    Google Scholar 

  175. X.-B. Lin. Using Melnikov’s method to solve Shilnikov’s problems. Proc. Roy. Soc. Edinburgh, 116: 295–325, 1990.

    MATH  Google Scholar 

  176. P. Lin. A numerical method for quasilinear singular perturbation problems with turning points. Computing, 46(2):155–164, 1991.

    MATH  MathSciNet  Google Scholar 

  177. P. Lin. Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comput., 72(242):657–675, 2003.

    MATH  Google Scholar 

  178. K. Lorenz, T. Jahnke, and C. Lubich. Adiabatic integrators for highly oscillatory second-order linear differential equations with time-varying eigendecomposition. BIT, 45:91–115, 2005.

    MATH  MathSciNet  Google Scholar 

  179. F. Legoll, T. Lelièvre, and G. Samaey. A micro-macro parareal algorithm: application to singularly perturbed ordinary differential equations. SIAM J. Sci. Comput., 35(4):A1951–A1986, 2013.

    MATH  Google Scholar 

  180. Ch. Lubich, K. Nipp, and D. Stoffer. Runge–Kutta solutions of stiff differential equations near stationary points. SIAM J. Numer. Anal., 32(4):1296–1307, 1995.

    MATH  MathSciNet  Google Scholar 

  181. M. Luskin and C. Ortner. An analysis of node-based cluster summation rules in the quasicontinuum method. SIAM J. Numer. Anal., 47(4):3070–3086, 2009.

    MATH  MathSciNet  Google Scholar 

  182. M. Luskin and C. Ortner. Atomistic-to-continuum coupling. Acta Numerica, 22:397–508, 2013.

    MATH  MathSciNet  Google Scholar 

  183. A.M. Lentini and V. Pereyra. An adaptive finite difference solver for nonlinear two-point boundary value problems with mild boundary layers. SIAM J. Numer. Anal., 14:91–111, 1977.

    MATH  MathSciNet  Google Scholar 

  184. B. Leimkuhler and S. Reich. A reversible averaging integrator for multiple time-scale dynamics. J. Comput. Phys., 171:95–114, 2001.

    MATH  MathSciNet  Google Scholar 

  185. B. Leimkuhler and S. Reich. Simulating Hamiltonian Dynamics. CUP, 2004.

    Google Scholar 

  186. T. Linß, H.-G. Roos, and R. Vulanovic. Uniform pointwise convergence on Shishkin-type meshes for quasi-linear convection-diffusion problems. SIAM J. Numer. Anal., 38(3):897–912, 2000.

    MATH  MathSciNet  Google Scholar 

  187. T. Linß and M. Stynes. Asymptotic analysis and Shishkin-type decomposition for an elliptic convection–diffusion problem. J. Math. Anal. Appl., 261(2):604–632, 2001.

    MATH  MathSciNet  Google Scholar 

  188. T. Linß and M. Stynes. The SDFEM on Shishkin meshes for linear convection–diffusion problems. Numer. Math., 87(3):457–484, 2001.

    MATH  MathSciNet  Google Scholar 

  189. Ch. Lubich. Convolution quadrature and discretized operational calculus. II. Numer. Math., 52(4): 413–415, 1988.

    Google Scholar 

  190. Ch. Lubich. On the convergence of multistep methods for nonlinear stiff differential equations. Numer. Math., 58(1):839–853, 1990.

    MathSciNet  Google Scholar 

  191. Ch. Lubich. Integration of stiff mechanical systems by Runge–Kutta methods. Z. Angew. Math. Phys., 44(6):1022–1053, 1993.

    MATH  MathSciNet  Google Scholar 

  192. C. Lubich. From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis. EMS, 2008.

    Google Scholar 

  193. K. Lust. Improved numerical Floquet multipliers. Int. J. Bif. Chaos, 11:2389–2410, 2001.

    MATH  MathSciNet  Google Scholar 

  194. W. Liniger and R.A. Willoughby. Efficient integration methods for stiff systems of ordinary differential equations. SIAM J. Numer. Anal., 7(1):47–66, 1970.

    MATH  MathSciNet  Google Scholar 

  195. P. De Maesschalck and M. Desroches. Numerical continuation techniques for planar slow–fast systems. SIAM J. Appl. Dyn. Syst., 12(3):1159–1180, 2013.

    MATH  MathSciNet  Google Scholar 

  196. J.M. Melenk. hp-Finite Element Methods for Singular Perturbations, volume 1796 of Lecture Notes in Mathematics. Springer, 2003.

    Google Scholar 

  197. J. MacLean and G.A. Gottwald. On the convergence of the projective integration method for stiff ordinary differential equations. arXiv:1301:6851v1, pages 1–22, 2013.

    Google Scholar 

  198. W.L. Miranker. Numerical Methods for Stiff Equations and Singular Perturbation Problems. Kluwer, 1981.

    Google Scholar 

  199. S. MacLachlan and N. Madden. Robust solution of singularly perturbed problems using multigrid methods. SIAM J. Sci. Comput., 35(5):A2225–A2254, 2013.

    MATH  MathSciNet  Google Scholar 

  200. A.G. Makeev, D. Maroudas, and I.G. Kevrekidis. “Coarse” stability and bifurcation analysis using stochastic simulators: kinetic Monte Carlo examples. J. Chem. Phys., 116(23):10083–10091, 2002.

    Google Scholar 

  201. J. Mohapatra and S. Natesan. Parameter-uniform numerical methods for singularly perturbed mixed boundary value problems using grid equidistribution. J. Appl. Math. Comput., 37:247–265, 2011.

    MATH  MathSciNet  Google Scholar 

  202. K. Mukherjee and S. Natesan. Optimal error estimate of upwind scheme on Shishkin-type meshes for singularly perturbed parabolic problems with discontinuous convection coefficients. BIT Numer. Math., 51:289–315, 2011.

    MATH  MathSciNet  Google Scholar 

  203. P. Morin, R.H. Nochetto, and K.G. Siebert. Convergence of adaptive finite element methods. SIAM Rev., 44(4):631–658, 2002.

    MATH  MathSciNet  Google Scholar 

  204. J.J. Miller, E. O’Riordan, and G.I. Shishkin. Fitted Numerical Methods for Singular Perturbation Problems. World Scientific, 1996.

    Google Scholar 

  205. S. Matthews, E. O’Riordan, and G.I. Shishkin. A numerical method for a system of singularly perturbed reaction–diffusion equations. J. Comput. Appl. Math., 145:151–166, 2002.

    MATH  MathSciNet  Google Scholar 

  206. P.K. Moore and L.R. Petzold. A stepsize control strategy for stiff systems of ordinary differential equations. Appl. Numer. Math., 15(4):449–463, 1994.

    MATH  MathSciNet  Google Scholar 

  207. T. Mei, J. Roychowdhury, T.S. Coffey, S.A. Hutchinson, and D.M. Day. Robust, stable time-domain methods for solving MPDEs of fast/slow systems. IEEE Trans. Computer-Aided Desg. Integr. Circ. Syst., 24(2):226–239, 2005.

    Google Scholar 

  208. J.B. McLeod and S. Sadhu. Existence of solutions and asymptotic analysis of a class of singularly perturbed ODEs with boundary conditions. Adv. Differential Equat., 18(9):825–848, 2013.

    MATH  MathSciNet  Google Scholar 

  209. C. Marschler, J. Sieber, R. Berkemer, A. Kawamoto, and J. Starke. Implicit methods for equation-free analysis: convergence results and analysis of emergent waves in microscopic traffic models. arXiv:1301.6640v1, pages 1–30, 2013.

    Google Scholar 

  210. F. Mazzia and D. Trigiante. A hybrid mesh selection strategy based on conditioning for boundary value ODE problems. Numerical Algorithms, 36:169–187, 2004.

    MATH  MathSciNet  Google Scholar 

  211. J.M. Melenk, C. Xenophontos, and L. Oberbroeckling. Robust exponential convergence of hp FEM for singularly perturbed reaction–diffusion systems with multiple scales. IMA J. Numer. Anal., 33(2):609–628, 2013.

    MATH  MathSciNet  Google Scholar 

  212. P. Ming and J.Z. Yang. Analysis of a one-dimensional nonlocal quasi-continuum method. Multiscale Model. Simul., 7(4):1838–1875, 2009.

    MATH  MathSciNet  Google Scholar 

  213. C. Negulescu. Numerical analysis of a multiscale finite element scheme for the resolution of the stationary Schrödinger equation. Numer. Math., 108(4):625–652, 2008.

    MATH  MathSciNet  Google Scholar 

  214. K. Nipp. Numerical integration of stiff ODE’s of singular perturbation type. Zeitschr. Appl. Math. Phys., 42:54–79, 1991.

    MathSciNet  Google Scholar 

  215. K. Nipp. Numerical integration of differential algebraic systems and invariant manifolds. BIT, 42(2):408–439, 2002.

    MATH  MathSciNet  Google Scholar 

  216. K. Nipp and D. Stoffer. Invariant manifolds and global error estimates of numerical integration schemes applied to stiff systems of singular perturbation type - Part I: RK-methods. Numer. Math., 70:245–257, 1995.

    MATH  MathSciNet  Google Scholar 

  217. K. Nipp and D. Stoffer. Invariant manifolds and global error estimates of numerical integration schemes applied to stiff systems of singular perturbation type - Part II: Linear multistep methods. Numer. Math., 74:305–323, 1996.

    MATH  MathSciNet  Google Scholar 

  218. M.C. Natividad and M. Stynes. Richardson extrapolation for a convection-diffusion problem using a Shishkin mesh. Appl. Numer. Math., 45(2):315–329, 2003.

    MATH  MathSciNet  Google Scholar 

  219. B.E. Oldeman, A.R. Champneys, and B. Krauskopf. Homoclinic branch switching: a numerical implementation of Lin’s method. Int. J. Bif. Chaos, 13(10):2977–2999, 2003.

    MATH  MathSciNet  Google Scholar 

  220. E. O’Riordan and J. Quinn. Parameter-uniform numerical methods for some linear and nonlinear singularly perturbed convection diffusion boundary turning point problems. BIT Numer. Math., 51: 317–337, 2011.

    MATH  MathSciNet  Google Scholar 

  221. E. O’Riordan and M. Stynes. A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions. Math. Comput., 57(195):47–62, 1991.

    MATH  MathSciNet  Google Scholar 

  222. L.R. Petzhold, L.O. Jay, and J. Yen. Numerical solution of highly oscillatory ordinary differential equations. Acta Numerica, 6:437–483, 1997.

    Google Scholar 

  223. A. Prothero and A. Robinson. On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Maths. Comput., 28:145–162, 1974.

    MathSciNet  Google Scholar 

  224. W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical Recipes 3rd Edition: The Art of Scientific Computing. CUP, 2007.

    Google Scholar 

  225. R.D. Russell and J. Christiansen. Adaptive mesh selection strategies for solving boundary value problems. SIAM J. Numer. Anal., 15(1):59–80, 1978.

    MATH  MathSciNet  Google Scholar 

  226. S. Reich. Preservation of adiabatic invariants under symplectic discretization. Appl. Numer. Math., 29:45–56, 1999.

    MATH  MathSciNet  Google Scholar 

  227. C.A. Ringhofer. On collocation schemes for quasilinear singularly perturbed boundary value problems. SIAM J. Numer. Anal., 21:864–882, 1984.

    MATH  MathSciNet  Google Scholar 

  228. H.-G. Roos and T. Linß. Sufficient conditions for uniform convergence on layer-adapted grids. Computing, 63(1):27–45, 1999.

    MATH  MathSciNet  Google Scholar 

  229. J. Rinzel and R.N. Miller. Numerical calculation of stable and unstable periodic solutions to the Hodgkin–Huxley equations. Math. Biosci., 49(1):27–59, 1980.

    MATH  MathSciNet  Google Scholar 

  230. S.M. Roberts. An approach to singular perturbation problems insoluble by asymptotic methods. J. Optimization Theory and Applications, 48(2):325–339, 1986.

    MATH  Google Scholar 

  231. A.J. Roberts. Model dynamics across multiple length and time scales on a spatial multigrid. Multiscale Model. Simul., 7(4):1525–1548, 2009.

    MATH  MathSciNet  Google Scholar 

  232. H.-G. Roos. Ten ways to generate the Il’in and related schemes. J. Comput. Appl. Math., 53(1):43–59, 1994.

    MATH  MathSciNet  Google Scholar 

  233. H.-G. Roos. Layer-adapted grids for singular perturbation problems. Z. Angew. Math. Mech., 78(5): 291–309, 1998.

    MATH  MathSciNet  Google Scholar 

  234. Y. Ren and R.D. Russell. Moving mesh techniques based upon equidistribution, and their stability. SIAM J. Sci. Stat. Comput., 13(6):1265–1286, 1992.

    MATH  MathSciNet  Google Scholar 

  235. H.-G. Roos, M. Stynes, and L. Tobiska. Numerical Methods for Singularly perturbed Differential Equations: Convection-Diffusion and Flow Problems. Springer, 1996.

    Google Scholar 

  236. O. Runborg, C. Theodoropoulos, and I.G. Kevrekidis. Effective bifurcation analysis: a time-stepper-based approach. Nonlinearity, 15(2):491–511, 2002.

    MATH  MathSciNet  Google Scholar 

  237. R.D. Russell. A comparison of collocation and finite differences for two-point boundary value problems. SIAM J. Numer. Anal., 14(1):19–39, 1977.

    MATH  MathSciNet  Google Scholar 

  238. R.D. Russell. Mesh selection methods. In Codes for Boundary-Value Problems in Ordinary Differential Equations, volume 74 of Lecture Notes in Computer Science, pages 228–242. Springer, 1979.

    Google Scholar 

  239. B.F. Smith and P.E. Børstad. Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. CUP, 2004.

    Google Scholar 

  240. L.F. Shampine and C.W. Gear. A user’s view of solving stiff ordinary differential equations. SIAM Rev., 21(1):1–17, 1979.

    MATH  MathSciNet  Google Scholar 

  241. C.I. Siettos, M.D. Graham, and I.G. Kevrekidis. Coarse Brownian dynamics for nematic liquid crystals: bifurcation, projective integration, and control via stochastic simulation. J. Chem. Phys., 118(22):10149–10156, 2003.

    Google Scholar 

  242. R. Seydel and V. Hlavaceka. Role of continuation in engineering analysis. Chem. Eng. Sci., 42(6): 1281–1295, 1987.

    Google Scholar 

  243. G.I. Shishkin. Grid approximation of singularly perturbed boundary value problem for quasi-linear parabolic equations in the case of complete degeneracy in spatial variables. Russ. J. Numer. Anal. Math. Mod., 6(3):243–262, 1991.

    MATH  MathSciNet  Google Scholar 

  244. G.I. Shishkin. On finite difference fitted schemes for singularly perturbed boundary value problems with a parabolic boundary layer. J. Math. Anal. Appl., 208(1):181–204, 1997.

    MATH  MathSciNet  Google Scholar 

  245. G.I. Shishkin. Robust novel high-order accurate numerical methods for singularly perturbed convection–diffusion problems 1. Math. Mod. Anal., 10(4):393–412, 2005.

    MATH  MathSciNet  Google Scholar 

  246. A. Shilnikov. Complete dynamical analysis of a neuron model. Nonlinear Dyn., 68:305–328, 2012.

    MATH  MathSciNet  Google Scholar 

  247. J. Sieber and B. Krauskopf. Control-based continuation of periodic orbits with a time-delayed difference scheme. Int. J. Bif. Chaos, 17(8):2579–2593, 2007.

    MATH  MathSciNet  Google Scholar 

  248. R.D. Skeel. A theoretical framework for proving accuracy results for deferred corrections. SIAM J. Numer. Anal., 19(1):171–196, 1982.

    MATH  MathSciNet  Google Scholar 

  249. G. Samaey, I.G. Kevrekidis, and D. Roose. Damping factors for the gap-tooth scheme. In Multiscale Modelling and Simulation, volume 39 of Lecture Notes Comput. Sci.Eng., pages 93–102. Springer, 2004.

    Google Scholar 

  250. G. Samaey, I.G. Kevrekidis, and D. Roose. Patch dynamics with buffers for homogenization problems. J. Comput. Phys., 213(1):264–287, 2006.

    MATH  MathSciNet  Google Scholar 

  251. G. Samaey, I.G. Kevrekidis, and D. Roose. Patch dynamics: macroscopic simulation of multiscale systems. PAMM, 7(1):1025803–1025804, 2007.

    Google Scholar 

  252. E. Süli and D. Mayers. An Introduction to Numerical Analysis. CUP, 2003.

    Google Scholar 

  253. M. Stynes and E. O’Riordan. A uniformly convergent Galerkin method on a Shishkin mesh for a convection–diffusion problem. J. Math. Anal. Appl., 214(1):36–54, 1997.

    MATH  MathSciNet  Google Scholar 

  254. B. Sportisse. An analysis of operator splitting techniques in the stiff case. J. Comput. Phys., 161(1):140–168, 2000.

    MATH  MathSciNet  Google Scholar 

  255. L.F. Shampine and M.W. Reichelt. The MatLab ODE suite. SIAM J. Sci. Comput., 18(1):1–22, 1997.

    MATH  MathSciNet  Google Scholar 

  256. M. Stynes and H.-G. Roos. The midpoint upwind scheme. Appl. Numer. Math., 23(3):361–374, 1997.

    MATH  MathSciNet  Google Scholar 

  257. G. Samaey, D. Roose, and I.G. Kevrekidis. The gap-tooth scheme for homogenization problems. Multiscale Model. Simul., 4(1):278–306, 2005.

    MATH  MathSciNet  Google Scholar 

  258. G. Sun and M. Stynes. Finite-element methods for singularly perturbed high-order elliptic two-point boundary value problems. I: reaction–diffusion-type problems. IMA J. Numer. Anal., 15:117–139, 1995.

    Google Scholar 

  259. J.M. Sanz-Serna. Mollified impulse methods for highly oscillatory differential equations. SIAM J. Numer. Anal., 46(2):1040–1059, 2008.

    MATH  MathSciNet  Google Scholar 

  260. L.F. Shampine, B.P. Sommeijer, and J.G. Verwer. IRKC: an IMEX solver for stiff diffusion-reaction PDEs. J. Comput. Appl. Math., 196(2):485–497, 2006.

    MATH  MathSciNet  Google Scholar 

  261. M. Stynes and L. Tobiska. The SDFEM for a convection-diffusion problem with a boundary layer: optimal error analysis and enhancement of accuracy. SIAM J. Numer. Anal., 41(5):1620–1642, 2003.

    MATH  MathSciNet  Google Scholar 

  262. R. Sharp, Y.-H. Tsai, and B. Engquist. Multiple time scale numerical methods for the inverted pendulum problem. In Multiscale Methods in Science and Engineering, pages 241–261. Springer, 2005.

    Google Scholar 

  263. M. Stynes. Steady-state convection-diffusion problems. Acta Numerica, 14:445–508, 2005.

    MATH  MathSciNet  Google Scholar 

  264. T. Steihaug and A. Wolfbrandt. An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations. Math. Comput., 33:521–534, 1979.

    MATH  MathSciNet  Google Scholar 

  265. K. Strehmel, R. Weiner, and H. Claus. Stability analysis of linearly implicit one-step interpolation methods for stiff retarded differential equations. SIAM J. Numer. Anal., 26(5):1158–1174, 1989.

    MATH  MathSciNet  Google Scholar 

  266. S.J. Stuart, R. Zhou, and B.J. Berne. Molecular dynamics with multiple time scales: the selection of efficient reference system propagators. J. Chem. Phys., 105:1426–1436, 1996.

    Google Scholar 

  267. C. Theodoropoulos, Y.H. Qian, and I.G. Kevrekidis. “Coarse” stability and bifurcation analysis using time-steppers: a reaction–diffusion example. Proc. Natl. Acad. Sci., 97(18):9840–9843, 2000.

    MATH  Google Scholar 

  268. J. Vigo-Aguiar and H. Ramos. A family of A-stable Runge–Kutta collocation methods of higher order for initial-value problems. IMA J. Numer. Anal., 27(4):798–817, 2007.

    MATH  MathSciNet  Google Scholar 

  269. E. Vanden-Eijnden. Numerical techniques for multiscale dynamical systems with stochastic effects. Comm. Math. Sci., 1:385–391, 2003.

    MATH  MathSciNet  Google Scholar 

  270. J.G. Verwer. S-stability properties for generalized Runge–Kutta methods. Numer. Math., 27(4): 359–370, 1976.

    MathSciNet  Google Scholar 

  271. R. Verfürth. A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math., 50:67–83, 1994.

    MATH  MathSciNet  Google Scholar 

  272. R. Verfürth. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, 1996.

    Google Scholar 

  273. J.G. Verwer. Runge–Kutta methods and viscous wave equations. Numer. Math., 112(3):485–507, 2009.

    MATH  MathSciNet  Google Scholar 

  274. N. Vaissmoradi, A. Malek, and S.H. Momeni-Masuleh. Error analysis and applications of the Fourier–Galerkin Runge–Kutta schemes for high-order stiff PDEs. J. Comput. Appl. Math., 231(1):124–133, 2009.

    MATH  MathSciNet  Google Scholar 

  275. M. van Veldhuizen. Higher order methods for a singularly perturbed problem. Numer. Math., 30(3):267–279, 1978.

    MATH  MathSciNet  Google Scholar 

  276. M. van Veldhuizen. On D-stability and B-stability. Numer. Math., 42(3):349–357, 1983.

    MATH  MathSciNet  Google Scholar 

  277. R. Wright, J. Cash, and G. Moore. Mesh selection for stiff two-point boundary value problems. Numer. Algorithms, 7:205–224, 1994.

    MATH  MathSciNet  Google Scholar 

  278. R. Weiss. An analysis of the box and trapezoidal schemes for linear singularly perturbed boundary value problems. Math. Comp., 42:537–557, 1984.

    Google Scholar 

  279. O.B. Widlund. A note on unconditionally stable linear multistep methods. BIT Numer. Math., 7(1): 65–70, 1967.

    MATH  MathSciNet  Google Scholar 

  280. H.A. Watts and L. Shampine. A-stable block implicit one-step methods. BIT Numer. Math., 12(2): 252–266, 1972.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Kuehn, C. (2015). Numerical Methods. In: Multiple Time Scale Dynamics. Applied Mathematical Sciences, vol 191. Springer, Cham. https://doi.org/10.1007/978-3-319-12316-5_10

Download citation

Publish with us

Policies and ethics