zbMATH — the first resource for mathematics

Runge-Kutta methods for partial differential equations and fractional orders of convergence. (English) Zbl 0769.65068
This elegant paper studies the (possibly fractional) order behaviour of implicit Runge-Kutta methods when applied to linear partial differential equations \(u_ t(x,t) = Lu(x,t) + f(x,t)\), \({u(0) = u_ 0}\), where \(L\) is a differential operator with a pure point spectrum of nonpositive real part.
An \(A\)-stable Runge-Kutta method with stage order \(q\) and order of consistency \(p\) is shown to have global order given by \(\alpha_ r = \min\{p,q + 2 + \nu_ r\}\), where \(\nu_ r\) depends on the \(L_ r\) norm used to estimate the global error. In the case of the one- dimensional heat equation \(u_ t = u_{xx} + a(x)g(t)\) on \(\Omega = (0,1)\) with \(a(x)\) differentiable and nonvanishing on \(\partial\Omega\), \(\nu_ r = {1\over 2r}\).
A similar analysis for nonhomogeneous boundary conditions is given – in which case \(q+2\) is replaced by \(q+1\) or (in the case of even stage Gauss methods) \(q\). The results are numerically verified for a simple semi- discretized problem.

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65L05 Numerical methods for initial value problems involving ordinary differential equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
35K15 Initial value problems for second-order parabolic equations
Full Text: DOI
[1] Philip Brenner, Michel Crouzeix, and Vidar Thomée, Single-step methods for inhomogeneous linear differential equations in Banach space, RAIRO Anal. Numér. 16 (1982), no. 1, 5 – 26 (English, with French summary). · Zbl 0477.65040
[2] Philip Brenner, Vidar Thomée, and Lars B. Wahlbin, Besov spaces and applications to difference methods for initial value problems, Lecture Notes in Mathematics, Vol. 434, Springer-Verlag, Berlin-New York, 1975. · Zbl 0294.35002
[3] K. Burrage and W. H. Hundsdorfer, The order of \?-convergence of algebraically stable Runge-Kutta methods, BIT 27 (1987), no. 1, 62 – 71. · Zbl 0629.65075
[4] K. Burrage, W. H. Hundsdorfer, and J. G. Verwer, A study of \?-convergence of Runge-Kutta methods, Computing 36 (1986), no. 1-2, 17 – 34 (English, with German summary). · Zbl 0572.65053
[5] J. C. Butcher, The numerical analysis of ordinary differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1987. Runge\mhy Kutta and general linear methods. · Zbl 0616.65072
[6] M. Crouzeix, Sur l’approximation des équations différentielles opérationelles linéaires par des méthodes de Runge-Kutta, Thèse d’Etat, Université Paris VI, 1975.
[7] M. Crouzeix and P. A. Raviart, Méthodes de Runge-Kutta, Unpublished Lecture Notes, Université de Rennes, 1980.
[8] Reinhard Frank, Josef Schneid, and Christoph W. Ueberhuber, The concept of \?-convergence, SIAM J. Numer. Anal. 18 (1981), no. 5, 753 – 780. · Zbl 0467.65032
[9] E. Hairer, Ch. Lubich, and M. Roche, Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations, BIT 28 (1988), no. 3, 678 – 700. · Zbl 0657.65093
[10] E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations. I, Springer Series in Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1987. Nonstiff problems. · Zbl 0638.65058
[11] E. Hairer and G. Wanner, Solving ordinary differential equations. II, Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1991. Stiff and differential-algebraic problems. · Zbl 0729.65051
[12] Marie-Noëlle Le Roux, Semidiscretization in time for parabolic problems, Math. Comp. 33 (1979), no. 147, 919 – 931. · Zbl 0417.65049
[13] Marie-Noëlle Le Roux, Méthodes multipas pour des équations paraboliques non linéaires, Numer. Math. 35 (1980), no. 2, 143 – 162 (French, with English summary). · Zbl 0463.65067
[14] C. Lubich, On the convergence of multistep methods for nonlinear stiff differential equations, Numer. Math. 58 (1991), no. 8, 839 – 853. · Zbl 0729.65055
[15] A. Ostermann and M. Roche, Rosenbrock methods for partial differential equations and fractional orders of convergence, Submitted for publication. · Zbl 0780.65056
[16] A. Prothero and A. Robinson, On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Comp. 28 (1974), 145 – 162. · Zbl 0309.65034
[17] J. M. Sanz-Serna and J. G. Verwer, Stability and convergence at the PDE/stiff ODE interface, Appl. Numer. Math. 5 (1989), no. 1-2, 117 – 132. Recent theoretical results in numerical ordinary differential equations. · Zbl 0671.65078
[18] J. M. Sanz-Serna, J. G. Verwer, and W. H. Hundsdorfer, Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations, Numer. Math. 50 (1987), no. 4, 405 – 418. · Zbl 0589.65069
[19] S. Scholz, Order barriers for the B-convergence of SDIRK methods, Preprint, TU Dresden, 1987.
[20] H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. Hans Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978.
[21] J. G. Verwer, Convergence and order reduction of diagonally implicit Runge-Kutta schemes in the method of lines, Numerical analysis (Dundee, 1985) Pitman Res. Notes Math. Ser., vol. 140, Longman Sci. Tech., Harlow, 1986, pp. 220 – 237. · Zbl 0642.65066
[22] Robert M. Young, An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. · Zbl 0493.42001
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.