Strongly A-stable first stage explicit collocation methods with stepsize control for stiff and differential-algebraic equations

A variable stepsize implementation of a recently introduced one-parameter family of high order strongly A-stable Runge-Kutta collocation methods with the first internal stage of explicit type is presented. The so-called SAFERK(@a,s) methods with free parameter @a and s stages are well-suited for the integration of stiff and differential-algebraic systems, and they are computationally equivalent to the (s-1)-stage Radau IIA method, since they all have a similar amount of implicitness. For the same number of implicit stages, both SAFERK(@a,s) and Radau IIA(s-1) methods possess algebraic order 2s-3, whereas the stage order is one unit higher for SAFERK methods. Although there are no L-stable schemes in this method family, the free parameter @a can be selected in order to minimize the error coefficients or to maximize the numerical dissipation. Besides a general discussion of the method class, it is shown here how the 4-stage methods can be endowed with an embedded third order formula, and an implementation based on the perfected RADAU5 code with an adaptive stepsize controller proves to be competitive for a wide selection of test problems including electric circuit analysis, constrained mechanical systems, and time-dependent partial differential equations treated by the method of lines.

[1]  Lawrence F. Shampine,et al.  Error estimators for stiff differential equations , 1984 .

[2]  Owe Axelsson,et al.  A class ofA-stable methods , 1969 .

[3]  S. González-Pinto,et al.  Global error estimates for a uniparametric family of stiffly accurate Runge-Kutta collocation methods on singularly perturbed problems , 2011 .

[4]  Bernd Simeon,et al.  Order reduction of stiff solvers at elastic multibody systems , 1998 .

[5]  J. Butcher.  Numerical methods for ordinary differential equations , 2003 .

[6]  E. Hairer,et al.  Stiff differential equations solved by Radau methods , 1999 .

[7]  D. Hernández-Abreu,et al.  An efficient family of strongly A-stable Runge-Kutta collocation methods for stiff systems and DAEs. Part II: Convergence results , 2012 .

[8]  Laurent O. Jay,et al.  Convergence of a class of runge-kutta methods for differential-algebraic systems of index 2 , 1993 .

[9]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[10]  Gustaf Söderlind,et al.  On the construction of error estimators for implicit Runge-Kutta methods , 1997 .

[11]  D. Hernández-Abreu,et al.  An efficient family of strongly A-stable Runge-Kutta collocation methods for stiff systems and DAEs. Part I: Stability and order results , 2010, J. Comput. Appl. Math..

[12]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[13]  L. Shampine,et al.  Numerical Solution of Ordinary Differential Equations. , 1995 .

[14]  J. Dormand,et al.  A family of embedded Runge-Kutta formulae , 1980 .

[15]  J. Butcher.  Implicit Runge-Kutta processes , 1964 .

[16]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[17]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[18]  Desmond J. Higham,et al.  Global error versus tolerance for explicit Runge-Kutta methods , 1991 .