Abstract
This paper studies the stability and convergence properties of general Runge-Kutta methods when they are applied to stiff semilinear systems y′(t) = J(t)y(t) + g(t, y(t)) with the stiffness contained in the variable coefficient linear part.
We consider two assumptions on the relative variation of the matrix J(t) and show that for each of them there is a family of implicit Runge-Kutta methods that is suitable for the numerical integration of the corresponding stiff semilinear systems, i.e. the methods of the family are stable, convergent and the stage equations possess a unique solution. The conditions on the coefficients of a method to belong to these families turn out to be essentially weaker than the usual algebraic stability condition which appears in connection with the B-stability and convergence for stiff nonlinear systems. Thus there are important RK methods which are not algebraically stable but, according to our theory, they are suitable for the numerical integration of semilinear problems.
This paper also extends previous results of Burrage, Hundsdorfer and Verwer on the optimal convergence of implicit Runge-Kutta methods for stiff semilinear systems with a constant coefficients linear part.
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References
R. Alexander, The modified Newton method in the solution of stiff ordinary differential equations, Math. Comp., 57 (1991), pp. 673-701.
J. C. Butcher, The numerical analysis of ordinary differential equations, John Wiley, Chichester, 1987.
K. Burrage and J. C. Butcher, Stability criteria for implicit Runge-Kutta methods, SIAM J. Numer. Anal., 16 (1979), pp. 46-57.
K. Burrage, W. H. Hundsdorfer, and J. G. Verwer, A study of B-convergence of Runge-Kutta methods, Computing, 36 (1986), pp. 17-34.
K. Burrage and W. H. Hundsdorfer, The order of B-convergence of algebraically stable Runge-Kutta methods, BIT, 27 (1987), pp. 62-71.
M. Calvo, S. Gonz´alez-Pinto, and J. I. Montijano, On the convergence of Runge-Kutta methods for stiff non linear differential equations, Numer. Math. 81 (1998), pp. 31-51.
M. Crouzeix, Sur la B-stabilité des méthodes de Runge-Kutta, Numer. Math., 32 (1979), pp. 75-82.
M. Crouzeix, W. H. Hundsdorfer, and M. N. Spijker, On the existence of solutions to the algebraic equations in implicit Runge-Kutta methods, BIT, 23 (1983), pp. 84-91.
K. Dekker and J. G. Verwer, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland, Amsterdam, 1984.
J. L. M. van Dorsselaer and M. N. Spijker, The error committed by stopping the Newton iteration in the numerical solution of stiff initial value problems, IMA J. Numer. Anal., 14 (1994), pp. 183-209.
R. Frank, J. Schneid and C. W. Ueberhuber, The concept of B-convergence, SIAM J. Numer. Anal., 18 (1981), pp. 753-780.
R. Frank, J. Schneid, and C. W. Ueberhuber, Order results for implicit Runge-Kutta methods applied to stiff systems, SIAM J. Numer. Anal., 22 (1985), pp. 515-534.
E. Hairer and G. Wanner, Solving ordinary differential equations II, Springer-Verlag, Berlin, 1996.
W. H. Hundsdorfer and M. N. Spijker, On the algebraic equations in implicit Runge-Kutta methods, SIAM J. Numer. Anal., 24 (1987), pp. 583-594.
J. F. B. M. Kraaijevanger, B-convergence of the implicit midpoint rule and the trapezoidal rule, BIT, 25 (1985), pp. 652-666.
J. F. B. M. Kraaijevanger and J. Schneid, On the unique solvability of the Runge-Kutta equations, Numer. Math., 59 (1991), pp. 129-157.
M. Z. Liu and J. F. B. M. Kraaijevanger, Solvability of the systems of equations arising in implicit Runge-Kutta methods, BIT, 28 (1988), pp. 825-838.
J. von Neumann, Eine Spektraltheorie f¨ur allgemeine Operatoren eines unit¨aren Raumes, Math. Nachrichten, 4 (1951), pp. 258-281.
O. Nevanlinna, Matrix valued versions of a result of von Neumann with an application to time discretization, J. Comput. Appl. Math, 12-13 (1985), pp. 475-489.
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), pp. 145-162.
B. A. Schmitt, Stability of implicit Runge-Kutta methods for nonlinear stiff differential equations, BIT, 28 (1988), pp. 884-897.
M. N. Spijker, On the error committed by stopping the Newton iteration in implicit Runge-Kutta methods, Ann. Numer. Math., 1 (1994), pp. 199-212.
K. Strehmel and R. Weiner, B-convergence results for linearly implicit one step methods, BIT, 27, (1987), pp. 264-281.
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Calvo, M., González-Pinto, S. & Montijano, J.I. Runge-Kutta Methods for the Numerical Solution of Stiff Semilinear Systems. BIT Numerical Mathematics 40, 611–639 (2000). https://doi.org/10.1023/A:1022332200092
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DOI: https://doi.org/10.1023/A:1022332200092