Numerical Methods

  • Christian Kuehn
Part of the Applied Mathematical Sciences book series (AMS, volume 191)


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.


Boundary Value Problem Collocation Point Multiple Time Scale Slow Manifold Fast Subsystem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [AAF95]
    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.zbMATHMathSciNetGoogle Scholar
  2. [AAN11]
    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.zbMATHMathSciNetGoogle Scholar
  3. [AB86]
    U. Ascher and G. Bader. Stability of collocation at Gaussian points. SIAM J. Numer. Anal., 23(2): 412–422, 1986.zbMATHMathSciNetGoogle Scholar
  4. [Abd05]
    A. Abdulle. On a priori error analysis of fully discrete heterogeneous multiscale FEM. Multiscale Model. Simul., 4(2):447–459, 2005.zbMATHMathSciNetGoogle Scholar
  5. [Abd12]
    A. Abdulle. Explicit methods for stiff stochastic differential equations. Lecture Notes in Comput. Sci. Engineer., 82:1–22, 2012.MathSciNetGoogle Scholar
  6. [Abr12a]
    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.zbMATHMathSciNetGoogle Scholar
  7. [Abr13a]
    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.zbMATHMathSciNetGoogle Scholar
  8. [AC08]
    A. Abdulle and S. Cirilli. S-ROCK: Chebyshev methods for stiff stochastic differential equations. SIAM J. Sci. Comput., 30(2):997–1014, 2008.zbMATHMathSciNetGoogle Scholar
  9. [ACR79]
    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. [ACR81]
    U. Ascher, J. Christiansen, and R.D. Russell. Collocation software for boundary-value ODEs. ACM Trans. Math. Software, 7(2):209–222, 1981.zbMATHGoogle Scholar
  11. [AE03]
    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. [AEEVE12]
    A. Abdulle, W. E, B. Engquist, and E. Vanden-Eijnden. The heterogeneous multiscale method. Acta Numerica, 21:1–87, 2012.Google Scholar
  13. [AG13]
    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. [AHL10]
    A. Abdulle, Y. Hu, and T. Li. Chebyshev methods with discrete noise: the tau-ROCK methods. J. Comp. Math., 28(2):195–217, 2010.zbMATHMathSciNetGoogle Scholar
  15. [AJ89]
    U. Ascher and S. Jacobs. On collocation implementation for singularly perturbed two-point problems. SIAM J. Sci. Stat. Comput., 10(3):533–549, 1989.zbMATHMathSciNetGoogle Scholar
  16. [AK12]
    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.zbMATHMathSciNetGoogle Scholar
  17. [AKK74]
    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.zbMATHMathSciNetGoogle Scholar
  18. [AL08]
    A. Abdulle and T. Li. S-ROCK methods for stiff Itô SDEs. Comm. Math. Sci., 6(4):845–868, 2008.zbMATHMathSciNetGoogle Scholar
  19. [Ale77]
    R. Alexander. Diagonally implicit Runge–Kutta methods for stiff ODE’s. SIAM J. Numer. Anal., 14(6):1006–1021, 1977.zbMATHMathSciNetGoogle Scholar
  20. [ALS12]
    A. Abdulle, P. Lin, and A. Shapeev. Numerical methods for multilattices. Multiscale Model. Simul., 10(3):696–726, 2012.zbMATHMathSciNetGoogle Scholar
  21. [AM88]
    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.zbMATHMathSciNetGoogle Scholar
  22. [AMPS91]
    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.zbMATHMathSciNetGoogle Scholar
  23. [AMR87]
    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. [And87]
    J.L. Anderson. Equidistribution schemes, Poisson generators, and adaptive grids. Appl. Math. Comput., 24(3):211–227, 1987.zbMATHMathSciNetGoogle Scholar
  25. [AP06]
    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.zbMATHMathSciNetGoogle Scholar
  26. [AP12]
    A. Abdulle and G. Pavliotis. Numerical methods for stochastic partial differential equations with multiple scales. J. Comput. Phys., 231(6):2482–2497, 2012.zbMATHMathSciNetGoogle Scholar
  27. [AR81]
    U. Ascher and R.D. Russell. Reformulation of boundary value problems into standard form. SIAM Rev., 23(2):238–254, 1981.zbMATHMathSciNetGoogle Scholar
  28. [Asc85]
    U. Ascher. On some difference schemes for singular singularly-perturbed boundary value problems. Numer. Math., 46:1–30, 1985.zbMATHMathSciNetGoogle Scholar
  29. [AW83]
    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.zbMATHMathSciNetGoogle Scholar
  30. [AW84a]
    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.zbMATHMathSciNetGoogle Scholar
  31. [AW84b]
    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.zbMATHMathSciNetGoogle Scholar
  32. [BB79]
    K. Burrage and J.C. Butcher. Stability criteria for implicit Runge–Kutta methods. SIAM J. Numer. Anal., 16(1):46–57, 1979.zbMATHMathSciNetGoogle Scholar
  33. [BB80]
    K. Burrage and J.C. Butcher. Nonlinear stability of a general class of differential equation methods. BIT, 20(2):185–203, 1980.zbMATHMathSciNetGoogle Scholar
  34. [BHR01]
    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.zbMATHMathSciNetGoogle Scholar
  35. [BJ78]
    T.A. Bickart and E.I. Jury. Arithmetic tests for A-stability, A[α]-stability, and stiff-stability. BIT, 18:9–21, 1978.zbMATHMathSciNetGoogle Scholar
  36. [BKVD12]
    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. [BL87]
    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.zbMATHMathSciNetGoogle Scholar
  38. [BO84]
    M. Berger and J. Oliger. Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys., 53(3):484–512, 1984.zbMATHMathSciNetGoogle Scholar
  39. [Bos96]
    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.zbMATHMathSciNetGoogle Scholar
  40. [Bré13b]
    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.zbMATHMathSciNetGoogle Scholar
  41. [BS73]
    C. De Boor and B. Swartz. Collocation at Gaussian points. SIAM J. Numer. Anal., 10:582–606, 1973.zbMATHMathSciNetGoogle Scholar
  42. [BT97b]
    L. Brugnano and D. Trigiante. A new mesh selection strategy for ODEs. Appl. Numer. Math., 24:1–21, 1997.zbMATHMathSciNetGoogle Scholar
  43. [But64]
    J.C. Butcher. Implicit Runge–Kutta processes. Math. Comput., 18(85):50–64, 1964.zbMATHMathSciNetGoogle Scholar
  44. [But75]
    J.C. Butcher. A stability property of implicit Runge–Kutta methods. BIT Numer. Math., 15(4): 358–361, 1975.zbMATHGoogle Scholar
  45. [But76]
    J.C. Butcher. On the implementation of implicit Runge–Kutta methods. BIT Numer. Math., 16(3): 237–240, 1976.zbMATHMathSciNetGoogle Scholar
  46. [Car54]
    G.F. Carrier. Boundary layer problems in applied mathematics. Comm. Pure Appl. Math., 7:11–17, 1954.zbMATHMathSciNetGoogle Scholar
  47. [Cas85]
    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.zbMATHMathSciNetGoogle Scholar
  48. [Cas88]
    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. [Cas03]
    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.zbMATHMathSciNetGoogle Scholar
  50. [CCM07]
    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.zbMATHMathSciNetGoogle Scholar
  51. [CDI09]
    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.zbMATHMathSciNetGoogle Scholar
  52. [CDI10a]
    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.zbMATHMathSciNetGoogle Scholar
  53. [CDI10b]
    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.zbMATHMathSciNetGoogle Scholar
  54. [CES05]
    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. [CGSR78]
    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.zbMATHMathSciNetGoogle Scholar
  56. [CH52]
    C.F. Curtiss and J. Hirschfelder. Integration of stiff equations. Proc. Natl. Acad. Sci. USA, 38(3): 235–243, 1952.zbMATHMathSciNetGoogle Scholar
  57. [Cha76]
    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.zbMATHMathSciNetGoogle Scholar
  58. [Che94]
    K. Chen. Error equidistribution and mesh adaptation. SIAM J. Sci. Comput., 15(4):798–818, 1994.zbMATHMathSciNetGoogle Scholar
  59. [CHL03]
    D. Cohen, E. Hairer, and C. Lubich. Modulated Fourier expansions of highly oscillatory differential equations. Found. Comput. Math., 3(4):327–345, 2003.zbMATHMathSciNetGoogle Scholar
  60. [CK94]
    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.zbMATHMathSciNetGoogle Scholar
  61. [CKS96]
    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.zbMATHMathSciNetGoogle Scholar
  62. [CM02]
    S.M. Cox and P.C. Matthews. Exponential time differencing for stiff systems. J. Comput. Phys., 176(2):430–455, 2002.zbMATHMathSciNetGoogle Scholar
  63. [CM05a]
    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.zbMATHMathSciNetGoogle Scholar
  64. [CMSS10]
    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.zbMATHMathSciNetGoogle Scholar
  65. [CMSS12]
    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.zbMATHMathSciNetGoogle Scholar
  66. [CMST06]
    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.zbMATHMathSciNetGoogle Scholar
  67. [CMW95]
    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.zbMATHMathSciNetGoogle Scholar
  68. [CS13]
    E.M. Constantinescu and A. Sandu. Extrapolated multirate methods for differential equations with multiple time scales. J. Sci. Comput., 56:28–44, 2013.zbMATHMathSciNetGoogle Scholar
  69. [CSS10]
    M.P. Calvo and J.M. Sanz-Serna. Heterogeneous multiscale methods for mechanical systems with vibrations. SIAM J. Sci. Comput., 32:2029–2046, 2010.zbMATHMathSciNetGoogle Scholar
  70. [Dah63]
    G. Dahlquist. A special stability problem for linear multistep methods. BIT, 3:27–43, 1963.zbMATHMathSciNetGoogle Scholar
  71. [DCD+07]
    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)., 2007.
  72. [DGK03]
    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.zbMATHMathSciNetGoogle Scholar
  73. [DGR00]
    A. Dutt, L. Greengard, and V. Rokhlin. Spectral deferred correction methods for ordinary differential equations. BIT Numer. Math., 40(2):241–266, 2000.zbMATHMathSciNetGoogle Scholar
  74. [DH83]
    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. [DK81]
    D.W. Decker and H.B. Keller. Path following near bifurcation. Comm. Pure Appl. Math., 34(2):149–175, 1981.zbMATHMathSciNetGoogle Scholar
  76. [DKK91a]
    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. [DKK91b]
    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. [DLO10a]
    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.zbMATHMathSciNetGoogle Scholar
  79. [DLO10b]
    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.zbMATHMathSciNetGoogle Scholar
  80. [Doe97]
    E.J. Doedel. Auto 97: Continuation and bifurcation software for ordinary differential equations., 1997.
  81. [Doe00]
    E.J. Doedel. Auto 2000: Continuation and bifurcation software for ordinary differential equations (with homcont)., 2000.
  82. [Doe07]
    E.J. Doedel. Lecture notes on numerical analysis of nonlinear equations., 2007.
  83. [DS09a]
    I. Dag and A. Sahin. Numerical solution of singularly perturbed problems. Int. J. Nonlin. Sci., 8(1): 32–39, 2009.zbMATHMathSciNetGoogle Scholar
  84. [DV84]
    K. Dekker and J.G. Verwer. Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations. North-Holland, 1984.Google Scholar
  85. [E03]
    W. E. Analysis of the heterogeneous multiscale method for ordinary differential equations. Comm. Math. Sci., 1(3):423–426, 2003.Google Scholar
  86. [E11]
    W. E. Principles of Multiscale Modeling. CUP, 2011.Google Scholar
  87. [EE03a]
    W. E and B. Engquist. The heterogeneous multiscale methods. Comm. Math.Sci., 1(1):87–132, 2003.Google Scholar
  88. [EE05]
    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. [EEH03]
    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. [EEL+07]
    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. [EFHI09]
    B. Engquist, A. Fokas, E. Hairer, and A. Iserles. Highly Oscillatory Problems. CUP, 2009.Google Scholar
  92. [EH09]
    Y.R. Efendiev and T.Y. Hou. Multiscale Finite Element Methods. Theory and Applications. Springer, 2009.Google Scholar
  93. [Ehl68]
    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.zbMATHMathSciNetGoogle Scholar
  94. [Ehl73]
    B.L. Ehle. A-stable methods and Padé approximations to the exponential. SIAM J. Math. Anal., 4(4):671–680, 1973.zbMATHMathSciNetGoogle Scholar
  95. [EHL75]
    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.zbMATHGoogle Scholar
  96. [EJL03]
    K. Eriksson, C. Johnson, and A. Logg. Explicit time-stepping for stiff ODEs. SIAM J. Sci. Comput., 25(4):1142–1157, 2003.zbMATHMathSciNetGoogle Scholar
  97. [EKAE06]
    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. [EL97]
    C. Engstler and C. Lubich. Multirate extrapolation methods for differential equations with different time scales. Computing, 58(2):173–185, 1997.zbMATHMathSciNetGoogle Scholar
  99. [ELVE05a]
    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. [ELVE05b]
    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. [ELVE07]
    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. [ELY06]
    W. E, J. Lu, and J.Z. Yang. Uniform accuracy of the quasicontinuum method. Phys. Rev. B., 74(21):214115, 2006.Google Scholar
  103. [EMZ05]
    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. [Enr74]
    W.H. Enright. Second derivative multistep methods for stiff ordinary differential equations. SIAM J. Numer. Anal., 11(2):321–331, 1974.zbMATHMathSciNetGoogle Scholar
  105. [ERVE09]
    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. [ET05]
    B. Engquist and Y.-H. Tsai. Heterogeneous multiscale methods for stiff ordinary differential equations. Math. Comput., 74(252):1707–1742, 2005.zbMATHMathSciNetGoogle Scholar
  107. [FHM+04]
    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.zbMATHMathSciNetGoogle Scholar
  108. [FHS96]
    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. [FJ91]
    T.F. Fairgrieve and A.D. Jepson. O.K. Floquet multipliers. SIAM J. Numer. Anal., 28(5):1446–1462, 1991.Google Scholar
  110. [FM80]
    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.zbMATHMathSciNetGoogle Scholar
  111. [FMOS96]
    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.zbMATHMathSciNetGoogle Scholar
  112. [FO77]
    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.zbMATHMathSciNetGoogle Scholar
  113. [FO84]
    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.zbMATHMathSciNetGoogle Scholar
  114. [Fol99]
    G. Folland. Real Analysis - Modern Techniques and Their Applications. Wiley, 1999.Google Scholar
  115. [FR11]
    S. Franz and H.-G. Roos. The capriciousness of numerical methods for singular perturbations. SIAM Rev., 53(1):157–173, 2011.zbMATHMathSciNetGoogle Scholar
  116. [Gau97]
    Walter Gautschi. Numerical Analysis. Birkhäuser Boston, 1997.Google Scholar
  117. [GH79]
    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. [GHW00]
    J. Guckenheimer, K. Hoffman, and W. Weckesser. Numerical computation of canards. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10(12):2669–2687, 2000.zbMATHMathSciNetGoogle Scholar
  119. [GK97]
    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.zbMATHMathSciNetGoogle Scholar
  120. [GK03]
    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.zbMATHMathSciNetGoogle Scholar
  121. [GK09a]
    J. Guckenheimer and C. Kuehn. Computing slow manifolds of saddle-type. SIAM J. Appl. Dyn. Syst., 8(3):854–879, 2009.zbMATHMathSciNetGoogle Scholar
  122. [GK10b]
    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.zbMATHMathSciNetGoogle Scholar
  123. [GKK06]
    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.zbMATHMathSciNetGoogle Scholar
  124. [GKT02]
    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. [GL07]
    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. [GLK03]
    C.W. Gear, J. Li, and I.G. Kevrekidis. The gap-tooth method in particle simulations. Phys. Lett. A, 316(3):190–195, 2003.zbMATHMathSciNetGoogle Scholar
  127. [Gov87b]
    W.F. Govaerts. Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM, Philadelphia, PA, 1987.Google Scholar
  128. [Gro81]
    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.zbMATHMathSciNetGoogle Scholar
  129. [GS12a]
    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.zbMATHMathSciNetGoogle Scholar
  130. [HAL12]
    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.MathSciNetGoogle Scholar
  131. [Hen62]
    P. Henrici. Discrete Variable Methods in Ordinary Differential Equations. Wiley, 1962.Google Scholar
  132. [HL88]
    E. Hairer and C. Lubich. Extrapolation at stiff differential equations. Numer. Math., 52(4):377–400, 1988.zbMATHMathSciNetGoogle Scholar
  133. [HL00b]
    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.zbMATHMathSciNetGoogle Scholar
  134. [HLR88]
    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.zbMATHMathSciNetGoogle Scholar
  135. [HM12]
    S.P. Hastings and J.B. McLeod. Classical Methods in Ordinary Differential Equations: With Applications to Boundary Value Problems. AMS, 2012.Google Scholar
  136. [HMOS95]
    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.zbMATHMathSciNetGoogle Scholar
  137. [HNMM10]
    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.zbMATHMathSciNetGoogle Scholar
  138. [HRR94]
    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.zbMATHMathSciNetGoogle Scholar
  139. [HSS97]
    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. [HSS00]
    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. [HV03]
    W. Hundsdorfer and J.G. Verwer. Numerical Solution of Time-dependent Advection-Diffusion-Reaction Equations. Springer, 2003.Google Scholar
  142. [HW91a]
    E. Hairer and G. Wanner. Solving Ordinary Differential Equations I. Springer, 1991.Google Scholar
  143. [HW91b]
    E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Springer, 1991.Google Scholar
  144. [Il’69]
    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. [Ise77]
    A. Iserles. Functional fitting - new family of schemes for integration of stiff ODE. Math. Comput., 31:112–123, 1977.zbMATHMathSciNetGoogle Scholar
  146. [Ise81]
    A. Iserles. Quadrature methods for stiff ordinary differential systems. Math. Comput., 36:171–182, 1981.zbMATHMathSciNetGoogle Scholar
  147. [Ise84]
    A. Iserles. Composite methods for numerical solution of stiff systems of ODEs. SIAM J. Num. Anal., 21:340–351, 1984.zbMATHMathSciNetGoogle Scholar
  148. [Ise96]
    A. Iserles. A First Course in the Numerical Analysis of Differential Equations. CUP, 1996.Google Scholar
  149. [Ise02]
    A. Iserles. On the global error of discretization methods for highly-oscillatory ordinary differential equations. BIT, 42:561–599, 2002.zbMATHMathSciNetGoogle Scholar
  150. [Jah04]
    T. Jahnke. Long-time-step integrators for almost-adiabatic quantum dynamics. SIAM J. Sci. Comput., 25:2145–2164, 2004.zbMATHMathSciNetGoogle Scholar
  151. [JJL05]
    J. Jansson, C. Johnson, and A. Logg. Computational modeling of dynamical systems. Math. Mod. Meth. Appl. Sci., 15(3):471, 2005.Google Scholar
  152. [JL03a]
    T. Jahnke and C. Lubich. Numerical integrators for quantum dynamics close to the adiabatic limit. Numerische Mathematik, 94:289–314, 2003.zbMATHMathSciNetGoogle Scholar
  153. [JL03b]
    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. [JL06]
    Z. Jia and B. Leimkuhler. Geometric integrators for multiple timescale simulation. J. Phys. A, 439:5379–5403, 2006.MathSciNetGoogle Scholar
  155. [KCMM03]
    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.zbMATHGoogle Scholar
  156. [Kel74]
    H. Keller. Accurate difference methods for nonlinear two-point boundary value problems. SIAM J. Numer. Anal., 11(2): 305–320, 1974.zbMATHMathSciNetGoogle Scholar
  157. [Kel83]
    H. Keller. The bordering algorithm and path following near singular points of higher nullity. SIAM J. Sci. Comput., 4(4): 573–582, 1983.zbMATHGoogle Scholar
  158. [KGH+03]
    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.zbMATHMathSciNetGoogle Scholar
  159. [KGH04]
    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. [Kir03]
    R. Kirby. On the convergence of high resolution methods with multiple time scales for hyperbolic conservation laws. Math. Comp., 72(243):1239–1250, 2003.zbMATHMathSciNetGoogle Scholar
  161. [KK81]
    B. Kreiss and H.-O. Kreiss. Numerical methods for singular perturbation problems. SIAM J. Numer. Anal., 18(2):262–276, 1981.zbMATHMathSciNetGoogle Scholar
  162. [KNB86]
    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.MathSciNetGoogle Scholar
  163. [KOGV07]
    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. [KP02]
    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.zbMATHMathSciNetGoogle Scholar
  165. [KP10]
    P.E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations. Springer, 2010.Google Scholar
  166. [KPK11]
    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.zbMATHMathSciNetGoogle Scholar
  167. [KR08]
    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.zbMATHMathSciNetGoogle Scholar
  168. [KR12]
    D. Kushnir and V. Rokhlin. A highly accurate solver for stiff ordinary differential equations. SIAM J. Sci. Comput., 34(3): A1296–A1315, 2012.zbMATHMathSciNetGoogle Scholar
  169. [Kre84]
    H.-O. Kreiss. Central difference schemes and stiff boundary value problems. BIT, 24:560–567, 1984.zbMATHMathSciNetGoogle Scholar
  170. [KS01a]
    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.zbMATHMathSciNetGoogle Scholar
  171. [KS09]
    I.G. Kevrekidis and G. Samaey. Equation-free multiscale computation: algorithms and applications. Ann. Rev. Phys. Chem., 60:321–344, 2009.Google Scholar
  172. [KT05]
    A.K. Kassam and L.N. Trefethen. Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput., 26(4):1214–1233, 2005.zbMATHMathSciNetGoogle Scholar
  173. [Kuz04]
    Yu.A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer, New York, NY, 3rd edition, 2004.zbMATHGoogle Scholar
  174. [LAE08]
    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. [Lin90b]
    X.-B. Lin. Using Melnikov’s method to solve Shilnikov’s problems. Proc. Roy. Soc. Edinburgh, 116: 295–325, 1990.zbMATHGoogle Scholar
  176. [Lin91]
    P. Lin. A numerical method for quasilinear singular perturbation problems with turning points. Computing, 46(2):155–164, 1991.zbMATHMathSciNetGoogle Scholar
  177. [Lin03]
    P. Lin. Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comput., 72(242):657–675, 2003.zbMATHGoogle Scholar
  178. [LJL05]
    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.zbMATHMathSciNetGoogle Scholar
  179. [LLS13]
    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.zbMATHGoogle Scholar
  180. [LNS95]
    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.zbMATHMathSciNetGoogle Scholar
  181. [LO09]
    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.zbMATHMathSciNetGoogle Scholar
  182. [LO13b]
    M. Luskin and C. Ortner. Atomistic-to-continuum coupling. Acta Numerica, 22:397–508, 2013.zbMATHMathSciNetGoogle Scholar
  183. [LP77]
    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.zbMATHMathSciNetGoogle Scholar
  184. [LR01]
    B. Leimkuhler and S. Reich. A reversible averaging integrator for multiple time-scale dynamics. J. Comput. Phys., 171:95–114, 2001.zbMATHMathSciNetGoogle Scholar
  185. [LR04a]
    B. Leimkuhler and S. Reich. Simulating Hamiltonian Dynamics. CUP, 2004.Google Scholar
  186. [LRV00]
    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.zbMATHMathSciNetGoogle Scholar
  187. [LS01a]
    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.zbMATHMathSciNetGoogle Scholar
  188. [LS01b]
    T. Linß and M. Stynes. The SDFEM on Shishkin meshes for linear convection–diffusion problems. Numer. Math., 87(3):457–484, 2001.zbMATHMathSciNetGoogle Scholar
  189. [Lub88]
    Ch. Lubich. Convolution quadrature and discretized operational calculus. II. Numer. Math., 52(4): 413–415, 1988.Google Scholar
  190. [Lub90]
    Ch. Lubich. On the convergence of multistep methods for nonlinear stiff differential equations. Numer. Math., 58(1):839–853, 1990.MathSciNetGoogle Scholar
  191. [Lub93]
    Ch. Lubich. Integration of stiff mechanical systems by Runge–Kutta methods. Z. Angew. Math. Phys., 44(6):1022–1053, 1993.zbMATHMathSciNetGoogle Scholar
  192. [Lub08]
    C. Lubich. From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis. EMS, 2008.Google Scholar
  193. [Lus01]
    K. Lust. Improved numerical Floquet multipliers. Int. J. Bif. Chaos, 11:2389–2410, 2001.zbMATHMathSciNetGoogle Scholar
  194. [LW70]
    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.zbMATHMathSciNetGoogle Scholar
  195. [MD13]
    P. De Maesschalck and M. Desroches. Numerical continuation techniques for planar slow–fast systems. SIAM J. Appl. Dyn. Syst., 12(3):1159–1180, 2013.zbMATHMathSciNetGoogle Scholar
  196. [Mel03]
    J.M. Melenk. hp-Finite Element Methods for Singular Perturbations, volume 1796 of Lecture Notes in Mathematics. Springer, 2003.Google Scholar
  197. [MG13a]
    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. [Mir81]
    W.L. Miranker. Numerical Methods for Stiff Equations and Singular Perturbation Problems. Kluwer, 1981.Google Scholar
  199. [MM13a]
    S. MacLachlan and N. Madden. Robust solution of singularly perturbed problems using multigrid methods. SIAM J. Sci. Comput., 35(5):A2225–A2254, 2013.zbMATHMathSciNetGoogle Scholar
  200. [MMK02]
    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. [MN11a]
    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.zbMATHMathSciNetGoogle Scholar
  202. [MN11b]
    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.zbMATHMathSciNetGoogle Scholar
  203. [MNS02]
    P. Morin, R.H. Nochetto, and K.G. Siebert. Convergence of adaptive finite element methods. SIAM Rev., 44(4):631–658, 2002.zbMATHMathSciNetGoogle Scholar
  204. [MOS96]
    J.J. Miller, E. O’Riordan, and G.I. Shishkin. Fitted Numerical Methods for Singular Perturbation Problems. World Scientific, 1996.Google Scholar
  205. [MOS02]
    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.zbMATHMathSciNetGoogle Scholar
  206. [MP94]
    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.zbMATHMathSciNetGoogle Scholar
  207. [MRC+05]
    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. [MS13]
    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.zbMATHMathSciNetGoogle Scholar
  209. [MSB+13]
    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. [MT04]
    F. Mazzia and D. Trigiante. A hybrid mesh selection strategy based on conditioning for boundary value ODE problems. Numerical Algorithms, 36:169–187, 2004.zbMATHMathSciNetGoogle Scholar
  211. [MXO13]
    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.zbMATHMathSciNetGoogle Scholar
  212. [MY09]
    P. Ming and J.Z. Yang. Analysis of a one-dimensional nonlocal quasi-continuum method. Multiscale Model. Simul., 7(4):1838–1875, 2009.zbMATHMathSciNetGoogle Scholar
  213. [Neg08]
    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.zbMATHMathSciNetGoogle Scholar
  214. [Nip91]
    K. Nipp. Numerical integration of stiff ODE’s of singular perturbation type. Zeitschr. Appl. Math. Phys., 42:54–79, 1991.MathSciNetGoogle Scholar
  215. [Nip02]
    K. Nipp. Numerical integration of differential algebraic systems and invariant manifolds. BIT, 42(2):408–439, 2002.zbMATHMathSciNetGoogle Scholar
  216. [NS95]
    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.zbMATHMathSciNetGoogle Scholar
  217. [NS96]
    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.zbMATHMathSciNetGoogle Scholar
  218. [NS03]
    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.zbMATHMathSciNetGoogle Scholar
  219. [OCK03]
    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.zbMATHMathSciNetGoogle Scholar
  220. [OQ11]
    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.zbMATHMathSciNetGoogle Scholar
  221. [OS91]
    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.zbMATHMathSciNetGoogle Scholar
  222. [PJY97]
    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. [PR74]
    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.MathSciNetGoogle Scholar
  224. [PTVF07]
    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. [RC78]
    R.D. Russell and J. Christiansen. Adaptive mesh selection strategies for solving boundary value problems. SIAM J. Numer. Anal., 15(1):59–80, 1978.zbMATHMathSciNetGoogle Scholar
  226. [Rei99b]
    S. Reich. Preservation of adiabatic invariants under symplectic discretization. Appl. Numer. Math., 29:45–56, 1999.zbMATHMathSciNetGoogle Scholar
  227. [Rin84]
    C.A. Ringhofer. On collocation schemes for quasilinear singularly perturbed boundary value problems. SIAM J. Numer. Anal., 21:864–882, 1984.zbMATHMathSciNetGoogle Scholar
  228. [RL99]
    H.-G. Roos and T. Linß. Sufficient conditions for uniform convergence on layer-adapted grids. Computing, 63(1):27–45, 1999.zbMATHMathSciNetGoogle Scholar
  229. [RM80]
    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.zbMATHMathSciNetGoogle Scholar
  230. [Rob86]
    S.M. Roberts. An approach to singular perturbation problems insoluble by asymptotic methods. J. Optimization Theory and Applications, 48(2):325–339, 1986.zbMATHGoogle Scholar
  231. [Rob09]
    A.J. Roberts. Model dynamics across multiple length and time scales on a spatial multigrid. Multiscale Model. Simul., 7(4):1525–1548, 2009.zbMATHMathSciNetGoogle Scholar
  232. [Roo94]
    H.-G. Roos. Ten ways to generate the Il’in and related schemes. J. Comput. Appl. Math., 53(1):43–59, 1994.zbMATHMathSciNetGoogle Scholar
  233. [Roo98]
    H.-G. Roos. Layer-adapted grids for singular perturbation problems. Z. Angew. Math. Mech., 78(5): 291–309, 1998.zbMATHMathSciNetGoogle Scholar
  234. [RR92]
    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.zbMATHMathSciNetGoogle Scholar
  235. [RST96]
    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. [RTK02]
    O. Runborg, C. Theodoropoulos, and I.G. Kevrekidis. Effective bifurcation analysis: a time-stepper-based approach. Nonlinearity, 15(2):491–511, 2002.zbMATHMathSciNetGoogle Scholar
  237. [Rus77]
    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.zbMATHMathSciNetGoogle Scholar
  238. [Rus79]
    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. [SB04]
    B.F. Smith and P.E. Børstad. Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. CUP, 2004.Google Scholar
  240. [SG79]
    L.F. Shampine and C.W. Gear. A user’s view of solving stiff ordinary differential equations. SIAM Rev., 21(1):1–17, 1979.zbMATHMathSciNetGoogle Scholar
  241. [SGK03]
    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. [SH87]
    R. Seydel and V. Hlavaceka. Role of continuation in engineering analysis. Chem. Eng. Sci., 42(6): 1281–1295, 1987.Google Scholar
  243. [Shi91]
    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.zbMATHMathSciNetGoogle Scholar
  244. [Shi97]
    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.zbMATHMathSciNetGoogle Scholar
  245. [Shi05]
    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.zbMATHMathSciNetGoogle Scholar
  246. [Shi12]
    A. Shilnikov. Complete dynamical analysis of a neuron model. Nonlinear Dyn., 68:305–328, 2012.zbMATHMathSciNetGoogle Scholar
  247. [SK07]
    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.zbMATHMathSciNetGoogle Scholar
  248. [Ske82]
    R.D. Skeel. A theoretical framework for proving accuracy results for deferred corrections. SIAM J. Numer. Anal., 19(1):171–196, 1982.zbMATHMathSciNetGoogle Scholar
  249. [SKR04]
    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. [SKR06]
    G. Samaey, I.G. Kevrekidis, and D. Roose. Patch dynamics with buffers for homogenization problems. J. Comput. Phys., 213(1):264–287, 2006.zbMATHMathSciNetGoogle Scholar
  251. [SKR07]
    G. Samaey, I.G. Kevrekidis, and D. Roose. Patch dynamics: macroscopic simulation of multiscale systems. PAMM, 7(1):1025803–1025804, 2007.Google Scholar
  252. [SM03]
    E. Süli and D. Mayers. An Introduction to Numerical Analysis. CUP, 2003.Google Scholar
  253. [SO97b]
    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.zbMATHMathSciNetGoogle Scholar
  254. [Spo00b]
    B. Sportisse. An analysis of operator splitting techniques in the stiff case. J. Comput. Phys., 161(1):140–168, 2000.zbMATHMathSciNetGoogle Scholar
  255. [SR97a]
    L.F. Shampine and M.W. Reichelt. The MatLab ODE suite. SIAM J. Sci. Comput., 18(1):1–22, 1997.zbMATHMathSciNetGoogle Scholar
  256. [SR97b]
    M. Stynes and H.-G. Roos. The midpoint upwind scheme. Appl. Numer. Math., 23(3):361–374, 1997.zbMATHMathSciNetGoogle Scholar
  257. [SRK05]
    G. Samaey, D. Roose, and I.G. Kevrekidis. The gap-tooth scheme for homogenization problems. Multiscale Model. Simul., 4(1):278–306, 2005.zbMATHMathSciNetGoogle Scholar
  258. [SS95]
    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. [SS08a]
    J.M. Sanz-Serna. Mollified impulse methods for highly oscillatory differential equations. SIAM J. Numer. Anal., 46(2):1040–1059, 2008.zbMATHMathSciNetGoogle Scholar
  260. [SSV06]
    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.zbMATHMathSciNetGoogle Scholar
  261. [ST03]
    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.zbMATHMathSciNetGoogle Scholar
  262. [STE05]
    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. [Sty05]
    M. Stynes. Steady-state convection-diffusion problems. Acta Numerica, 14:445–508, 2005.zbMATHMathSciNetGoogle Scholar
  264. [SW79]
    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.zbMATHMathSciNetGoogle Scholar
  265. [SWC89]
    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.zbMATHMathSciNetGoogle Scholar
  266. [SZB96]
    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. [TQK00]
    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.zbMATHGoogle Scholar
  268. [VAR07]
    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.zbMATHMathSciNetGoogle Scholar
  269. [VE03]
    E. Vanden-Eijnden. Numerical techniques for multiscale dynamical systems with stochastic effects. Comm. Math. Sci., 1:385–391, 2003.zbMATHMathSciNetGoogle Scholar
  270. [Ver76]
    J.G. Verwer. S-stability properties for generalized Runge–Kutta methods. Numer. Math., 27(4): 359–370, 1976.MathSciNetGoogle Scholar
  271. [Ver94]
    R. Verfürth. A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math., 50:67–83, 1994.zbMATHMathSciNetGoogle Scholar
  272. [Ver96]
    R. Verfürth. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, 1996.Google Scholar
  273. [Ver09]
    J.G. Verwer. Runge–Kutta methods and viscous wave equations. Numer. Math., 112(3):485–507, 2009.zbMATHMathSciNetGoogle Scholar
  274. [VMMM09]
    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.zbMATHMathSciNetGoogle Scholar
  275. [vV78]
    M. van Veldhuizen. Higher order methods for a singularly perturbed problem. Numer. Math., 30(3):267–279, 1978.zbMATHMathSciNetGoogle Scholar
  276. [vV83]
    M. van Veldhuizen. On D-stability and B-stability. Numer. Math., 42(3):349–357, 1983.zbMATHMathSciNetGoogle Scholar
  277. [WCM94]
    R. Wright, J. Cash, and G. Moore. Mesh selection for stiff two-point boundary value problems. Numer. Algorithms, 7:205–224, 1994.zbMATHMathSciNetGoogle Scholar
  278. [Wei84]
    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. [Wid67]
    O.B. Widlund. A note on unconditionally stable linear multistep methods. BIT Numer. Math., 7(1): 65–70, 1967.zbMATHMathSciNetGoogle Scholar
  280. [WS72]
    H.A. Watts and L. Shampine. A-stable block implicit one-step methods. BIT Numer. Math., 12(2): 252–266, 1972.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Christian Kuehn
    • 1
  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria

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