×

Modern convergence theory for stiff initial-value problems. (English) Zbl 0782.65087

This is a brief review on theoretical results about convergence of discretization schemes when applied to stiff ordinary differential equations. The authors consider problems satisfying a one-sided Lipschitz-condition and also singular perturbation problems. They mention that the results on \(B\)-convergence extend to problems of the form \(y' = J(t)y + g(t,y)\) where \(J(t)\) is diagonalizable with a well-conditioned, smoothly varying eigensystem and \(g\) is a Lipschitz-bounded nonlinearity.
Reviewer: E.Hairer (Genève)

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34E13 Multiple scale methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Auzinger, W., On the error structure of the implicit Euler scheme applied to stiff systems of differential equations, Computing, 43, 115-131, (1989) · Zbl 0685.65062
[2] Auzinger, W., On error structures and extrapolation for stiff systems, with application in the method of lines, Computing, 44, 331-356, (1990) · Zbl 0716.65061
[3] Auzinger, W.; Frank, R., Asymptotic expansions of the global discretization error for stiff problems, SIAM J. sci. statist. comput., 10, 950-963, (1989) · Zbl 0677.65077
[4] Auzinger, W.; Frank, R.; Kirlinger, G., Asymptotic error expansions for stiff equations: applications, Computing, 43, 223-253, (1989) · Zbl 0688.65057
[5] Auzinger, W.; Frank, R.; Kirlinger, G., A note on convergence concepts for stiff problems, Computing, 44, 197-208, (1990) · Zbl 0719.65058
[6] Auzinger, W.; Frank, R.; Kirlinger, G., An extension of B-convergence for runge—kutta methods, Appl. numer. math., 9, 2, 91-109, (1992) · Zbl 0747.65056
[7] Auzinger, W.; Frank, R.; Macsek, F., Asymptotic error expansions for stiff equations: the implicit Euler scheme, SIAM J. numer. anal., 27, 67-104, (1990) · Zbl 0688.65056
[8] Burrage, K.; Butcher, J.C., Stability criteria for implicit runge—kutta methods, SIAM J. numer. anal., 16, 46-57, (1979) · Zbl 0396.65043
[9] Butcher, J.C., A stability property of implicit runge—kutta methods, Bit, 15, 358-361, (1975) · Zbl 0333.65031
[10] Crouzeix, M., Sur la B-stabilité des méthodes de runge—kutta, Numer. math., 32, 75-82, (1979) · Zbl 0431.65052
[11] Dahlquist, G.; Dahlquist, G., Stability and error bounds in the numerical integration of ordinary differential equations, Ph.D. thesis, Trans. roy. inst. technol. Stockholm, 130, (1959), also · Zbl 0085.33401
[12] Dahlquist, G., A special stability problem for linear multistep methods, Bit, 3, 27-43, (1963) · Zbl 0123.11703
[13] Dahlquist, G., Error analysis for a class of methods for stiff nonlinear initial value problems, (), 60-72, Lecture Notes in Math.
[14] Dahlquist, G., G-stability is equivalent to A-stability, Bit, 18, 384-401, (1978) · Zbl 0413.65057
[15] Dekker, K.; Verwer, J.G., Stability of runge—kutta methods for stiff nonlinear differential equations, 2, (1984), North-Holland Amsterdam, CWI Monographs · Zbl 0571.65057
[16] Frank, R.; Schneid, J.; Ueberhuber, C.W., Einseitige lipschitzbedingungen für gewöhnliche differentialgleichunge, ()
[17] Frank, R.; Schneid, J.; Ueberhuber, C.W., The concept of B-convergence, SIAM J. numer. anal., 18, 753-780, (1981) · Zbl 0467.65032
[18] Frank, R.; Schneid, J.; Ueberhuber, C.W., Stability properties of implicit runge—kutta methods, SIAM J. numer. anal., 22, 497-515, (1985) · Zbl 0577.65055
[19] Frank, R.; Schneid, J.; Ueberhuber, C.W., Order results for implicit runge—kutta methods applied to stiff systems, SIAM J. numer. anal., 22, 515-534, (1985) · Zbl 0577.65056
[20] Frank, R.; Schneid, J.; Ueberhuber, C.W., B-convergence: A survey, Appl. numer. math., 5, 1-2, 51-61, (1989) · Zbl 0674.65056
[21] Hairer, E.; Lubich, Ch., Extrapolation at stiff differential equations, Numer. math., 52, 377-400, (1988) · Zbl 0643.65034
[22] Hairer, E.; Lubich, Ch.; Roche, M., Error of runge—kutta methods for stiff problems studied via differential algebraic equations, Bit, 28, 678-700, (1988) · Zbl 0657.65093
[23] Kraaijevanger, J.F.B.M., Contractivity of runge—kutta methods, () · Zbl 0763.65059
[24] Le Roux, M.-N., Méthodes multipas pour des équations paraboliques non linéaires, Numer. math., 35, 143-162, (1980) · Zbl 0463.65067
[25] Leveque, R.J.; Trefethen, L.N., On the resolvent condition in the kreiss matrix theorem, Bit, 24, 584-591, (1984) · Zbl 0559.15018
[26] Lubich, Ch., On the convergence of multistep methods for nonlinear stiff differential equations, Numer. math., 58, 839-853, (1991) · Zbl 0729.65055
[27] Lubich, Ch.; Nevanlinna, O., On resolvent conditions and stability estimates, () · Zbl 0731.65043
[28] O’Malley, R.E., Introduction to singular perturbations, (1974), Academic Press New York · Zbl 0287.34062
[29] Prothero, A.; Robinson, A., On the stability and accuracy of one step methods for solving stiff systems of ordinary differential equations, Math. comp., 28, 145-162, (1974) · Zbl 0309.65034
[30] Reddy, S.C.; Trefethen, L.N., Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues, Comput. methods appl. mech. engrg., 80, 147-164, (1990) · Zbl 0735.65070
[31] Sanz-Serna, J.M.; Verwer, J.G., Stability and convergence at the PDE/stiff ODE interface, Appl. numer. math., 5, 1-2, 117-132, (1989) · Zbl 0671.65078
[32] Spijker, M.N., Contractivity in the numerical solution of initial value problems, Numer. math., 42, 271-290, (1983) · Zbl 0504.65030
[33] Stetter, H.J., Asymptotic expansions for the error of discretization algorithms for nonlinear functional equations, Numer. math., 7, 18-31, (1965) · Zbl 0148.39003
[34] van Veldhuizen, R., Asymptotic expansions of the global error for the implicit midpoint rule (stiff case), Computing, 33, 185-192, (1984) · Zbl 0547.65055
[35] Widlund, O.B., A note on unconditionally stable linear multistep mehtods, Bit, 7, 65-70, (1967) · Zbl 0178.18502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.