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Runge-Kutta Methods for the Numerical Solution of Stiff Semilinear Systems

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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

  1. R. Alexander, The modified Newton method in the solution of stiff ordinary differential equations, Math. Comp., 57 (1991), pp. 673-701.

    Google Scholar 

  2. J. C. Butcher, The numerical analysis of ordinary differential equations, John Wiley, Chichester, 1987.

    Google Scholar 

  3. K. Burrage and J. C. Butcher, Stability criteria for implicit Runge-Kutta methods, SIAM J. Numer. Anal., 16 (1979), pp. 46-57.

    Google Scholar 

  4. K. Burrage, W. H. Hundsdorfer, and J. G. Verwer, A study of B-convergence of Runge-Kutta methods, Computing, 36 (1986), pp. 17-34.

    Google Scholar 

  5. K. Burrage and W. H. Hundsdorfer, The order of B-convergence of algebraically stable Runge-Kutta methods, BIT, 27 (1987), pp. 62-71.

    Google Scholar 

  6. 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.

    Google Scholar 

  7. M. Crouzeix, Sur la B-stabilité des méthodes de Runge-Kutta, Numer. Math., 32 (1979), pp. 75-82.

    Google Scholar 

  8. 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.

    Google Scholar 

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

  10. 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.

    Google Scholar 

  11. R. Frank, J. Schneid and C. W. Ueberhuber, The concept of B-convergence, SIAM J. Numer. Anal., 18 (1981), pp. 753-780.

    Google Scholar 

  12. 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.

    Google Scholar 

  13. E. Hairer and G. Wanner, Solving ordinary differential equations II, Springer-Verlag, Berlin, 1996.

    Google Scholar 

  14. 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.

    Google Scholar 

  15. J. F. B. M. Kraaijevanger, B-convergence of the implicit midpoint rule and the trapezoidal rule, BIT, 25 (1985), pp. 652-666.

    Google Scholar 

  16. J. F. B. M. Kraaijevanger and J. Schneid, On the unique solvability of the Runge-Kutta equations, Numer. Math., 59 (1991), pp. 129-157.

    Google Scholar 

  17. 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.

    Google Scholar 

  18. J. von Neumann, Eine Spektraltheorie f¨ur allgemeine Operatoren eines unit¨aren Raumes, Math. Nachrichten, 4 (1951), pp. 258-281.

    Google Scholar 

  19. 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.

    Google Scholar 

  20. 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.

    Google Scholar 

  21. B. A. Schmitt, Stability of implicit Runge-Kutta methods for nonlinear stiff differential equations, BIT, 28 (1988), pp. 884-897.

    Google Scholar 

  22. 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.

    Google Scholar 

  23. K. Strehmel and R. Weiner, B-convergence results for linearly implicit one step methods, BIT, 27, (1987), pp. 264-281.

    Google Scholar 

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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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