Analysis and implementation of TR-BDF2

Abstract Bank et al. (1985) developed a one-step method, TR-BDF2, for the simulation of circuits and semiconductor devices based on the trapezoidal rule and the backward differentiation formula of order 2 that provides some of the important advantages of BDF2 without the disadvantages of a memory. Its success and popularity in the context justify its study and further development for general-purpose codes. Here the method is shown to be strongly S-stable. It is shown to be optimal in a class of practical one-step methods. An efficient, globally C 1 interpolation scheme is developed. The truncation error estimate of Bank et al. (1985) is not effective when the problem is very stiff. Coming to an understanding of this leads to a way of correcting the estimate and to a more effective implementation. These developments improve greatly the effectiveness of the method for very stiff problems.

[1]  G. D. Byrne,et al.  VODE: a variable-coefficient ODE solver , 1989 .

[2]  A. C. Hindmarsh,et al.  Applications of EPISODE: an experimental package for the integration of systems of ordinary differential equations. [In FORTRAN for CDC 7600] , 1975 .

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

[4]  Andre Vladimirescu,et al.  The Spice Book , 1994 .

[5]  L. Shampine.  Implementation of Implicit Formulas for the Solution of ODEs , 1980 .

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

[7]  T. E. Hull,et al.  Comparing numerical methods for stiff systems of O.D.E:s , 1975 .

[8]  M. E. Hosea.  A new recurrence for computing Runge-Kutta truncation error coefficients , 1995 .

[9]  P. G. Thomsen,et al.  Local error control inSDIRK-methods , 1986 .

[10]  Randolph E. Bank,et al.  Transient simulation of silicon devices and circuits , 1985, IEEE Transactions on Electron Devices.

[11]  Niels Houbak,et al.  Displacement or Residual Test in the Application of Implicit Methods for Stiff Problems , 1985 .

[12]  A. Prothero,et al.  On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations , 1974 .

[13]  Lawrence F. Shampine,et al.  The MATLAB ODE Suite , 1997, SIAM J. Sci. Comput..

[14]  J. Carroll.  A composite integration scheme for the numerical solution of systems of ordinary differential equations , 1989 .

[15]  J. Carroll.  A composite integration scheme for the numerical solution of systems of parabolic PDEs in one space dimension , 1993 .

[16]  M. Berzins,et al.  An adaptive theta method for the solution of stiff and nonstiff differential equations , 1992 .

[17]  R. E. Scraton,et al.  Some L-stable methods for stiff differential equations , 1981 .

[18]  R. Alexander.  Diagonally implicit runge-kutta methods for stiff odes , 1977 .

[19]  T. Steihaug,et al.  An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations , 1979 .