An Alternative Implementation of Variable Step-Size Multistep Formulas for Stiff ODEs

An alternative technique for the implementation of variable step-size multistep formulas is developed for the numerical solution of ordinary differential equations. Formulas based on this technique have the property that their leading coefficients are constant; thLs LS important for methods which solve stiff systems. In addition, both theoretmal and empirical results indicate that methods based on this techmque have stability properties similar to those of the corresponding variable coefficient implementations. As a particular example, we have nnplemented the backward differentiation formulas m this form, and the numerical results look very promising.

[1]  Kenneth Ronald Jackson.  Variable stepsize, variable order integrand approximation methods for the numerical solution of ordinary differential equations. , 1978 .

[2]  Kai-wen Tu.  Stability and convergence of general multistep and multivalue methods with variable step size , 1972 .

[3]  A. Hindmarsh,et al.  GEAR: ORDINARY DIFFERENTIAL EQUATION SYSTEM SOLVER. , 1971 .

[4]  R. Brayton,et al.  A new efficient algorithm for solving differential-algebraic systems using implicit backward differentiation formulas , 1972 .

[5]  C. W. Geart,et al.  THE EFFECT OF VARIABLE MESH SIZE ON THE STABILITY OF MULTISTEP METHODS , 1974 .

[6]  Kenneth R. Jackson,et al.  Comparative test results for two ODE solvers: EPISODE and GEAR , 1977 .

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

[8]  Lawrence F. Shampine,et al.  Local error and variable order Adams codes , 1975 .

[9]  W. H. Enright,et al.  Test Results on Initial Value Methods for Non-Stiff Ordinary Differential Equations , 1976 .

[10]  A. C. Hindmarsh,et al.  EPISODE: an effective package for the integration of systems of ordinary differential equations. [For stiff or non-stiff problems, in FORTRAN for CDC or IBM computers; TSTEP, core integrator routine; CONVRT, to change between single and double precision coding] , 1977 .

[11]  A. Sedgwick,et al.  An effective variable-order variable-step adams method. , 1973 .

[12]  W. H. Enright,et al.  Second Derivative Multistep Methods for Stiff Ordinary Differential Equations , 1974 .

[13]  Charles William Gear,et al.  The Stability of Automatic Programs for Numerical Problems , 1974 .

[14]  H. A. Watts,et al.  Solving Nonstiff Ordinary Differential Equations—The State of the Art , 1976 .

[15]  Kenneth R. Jackson,et al.  A comparison of two ode codes: gear and episode , 1977 .

[16]  T. E. Hull,et al.  Comparing Numerical Methods for the Solution of Stiff Systems of ODEs Arising in Chemistry , 1976 .

[17]  Alan C. Hindmarsh,et al.  A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations , 1975, TOMS.