Abstract
A class of half-explicit methods for index 2 differential-algebraic systems in Hessenberg form is proposed, which takes advantage of the partitioned structure of such problems. For this family of methods, which we call partitioned half-explicit Runge-Kutta methods, a better choice in the parameters of the method than for previously available half-explicit Runge-Kutta methods can be made. In particular we construct a family of 6-stage methods of order 5, and determine its parameters (based on the coefficients of the successful explicit Runge-Kutta method DOPRI5) in order to optimize the local error coefficients. Numerical experiments demonstrate the efficiency of this method for the solution of constrained multi-body systems.
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References
Arnold, M.: Half-explicit Runge-Kutta Methods with explicit stages for differential-algebraic systems of index 2. Submitted to BIT (1995).
Ascher, U., Petzold, L. R.: Projected implicit Runge-Kutta methods for differential-algebraic equations. SIAM J. Numer. Anal.4, 1097–1120 (1991).
Brasey, V.: Half-explicit methods for semi-explicit differential-algebraic equations of index 2. Ph. D. Thesis, Université de Genève, Genève 1994.
Brasey, V., Hairer, E.: Half-explicit Runge-Kutta methods for differential-algebraic systems of index 2. SIAM J. Numer. Anal.30, 538–552 (1993).
Brasey, V.: A half-explicit Runge-Kutta method of order 5 for solving constrained mechanical systems. Computing48, 191–201 (1992).
Brenan, K. E., Campbell, S. L., Petzold, L. R.: Numerical solution of initial-value problems in differential-algebraic equations. New York: North-Holland, 1989.
Dormand, J. R., Prince, P. J.: A family of embedded Runge-Kutta formulae. J. Comp. Appl. Math.6, 19–26 (1980).
Hairer, E., Lubich, Ch., Roche, M.: The numerical solution of differential algebraic systems by Runge-Kutta Methods. Lecture Notes in Mathematics1409. Berlin Heidelberg New York Tokyo: Springer, 1989.
Hairer, E, Nørsett, S. P., Wanner, G.: Solving ordinary differential equations I. Nonstiff Problems, 2nd ed. Berlin Heidelberg New York Tokyo: Springer, 1993.
Hairer, E., Wanner, G.: Solving ordinary differential equations II. Stiff and differential-algebraic problems, 2nd ed. Berlin Heidelberg New York Tokyo: Springer, 1996.
Jay, L.: Convergence of a class of Runge-Kutta methods for differential-algebraic systems of index 2. BIT33, 137–150 (1993).
Murua, A.: Partitioned Runge-Kutta methods for semi-explicit differential-algebraic systems of index 2. Technical Report, EHU-KZAA-IKT-196 (1996).
Murua, A.: Formal series and numerical integrators. Preprint.
Potra, F. A., Rheinboldt, W. C.: On the numerical solution of Euler-Lagrange equations. J. Mech. Struct. Mach.19, 1–18 (1991).
Simeon, B., Grupp, F., Führer, C., Rentrop, P.: A nonlinear truck model and its treatment as a multi-body system. J. Comput. Appl. Math.50, 523–532 (1994).
Simeon, B.: On the numerical solution of a wheel suspension benchmark problem, Comp. Appl. Math.66, 443–456 (1995).
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Murua, A. Partitioned half-explicit Runge-Kutta methods for differential-algebraic systems of index 2. Computing 59, 43–61 (1997). https://doi.org/10.1007/BF02684403
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DOI: https://doi.org/10.1007/BF02684403