Automatic Solution of Differential Equations Based on the User of Linear Multistep Methods

Variable-stepsize variable-formula methods (VSVFMs) are often used in the numerical integration of systems of ordinary differential equations. In this way, roughly speaking, one attempts to minimize the number of steps, Le., to select the largest possible stepsize according to a prescribed error tolerance. Very often, however, the selectmn of the stepsize depends not so much on the accuracy requirements but rather on the absolute stabdlty properties of the formulas included in the particular VSVFM. Therefore, at least for problems where the absolute stability requirements dominate the accuracy requirements, it Is unportant to use only formulas with the best possible absolute stability characteristics in the VSVFM. Moreover, it is important to find an algorithm which predicts the largest possible stepslzes so that the next steps will be successful (the local truncation error estimator is enurely unable to do this when the absolute stability requirements are dominant). An attempt to use formulas with large absolute stability regions and to apply a strategy which normally will ensure stable computations is discussed.

[1]  R. L. Crane,et al.  A Predictor-Corrector Algorithm with an Increased Range of Absolute Stability , 1965, JACM.

[2]  R. W. Klopfenstein,et al.  Numerical stability of a one-evaluation predictor-corrector algorithm for numerical solution of ordinary differential equations , 1968 .

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

[4]  C. W. Gear,et al.  Algorithm 407: DIFSUB for solution of ordinary differential equations [D2] , 1971, Commun. ACM.

[5]  F. T. Krogh.  Variable order integrators for the numerical solution of ordinary differential equations , 1971 .

[6]  Fred T. Krogh,et al.  On Testing a Subroutine for the Numerical Integration of Ordinary Differential Equations , 1973, JACM.

[7]  D. J. Rodabaugh,et al.  Adams-type methods with increased ranges of stability , 1978 .

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

[9]  L. Shampine,et al.  Computer solution of ordinary differential equations : the initial value problem , 1975 .

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

[11]  D. J. Rodabaugh,et al.  Corrector methods with increased ranges of stability , 1977 .

[12]  H. Stetter.  Analysis of Discretization Methods for Ordinary Differential Equations , 1973 .

[13]  L. F. Shampine.  Solving ordinary differential equations for simulation , 1978 .

[14]  Lawrence F. Shampine.  Limiting precision in differential equation solvers. II Sources of trouble and starting a code , 1978 .

[15]  Zahari Zlatev,et al.  Stability properties of variable stepsize variable formula methods , 1978 .

[16]  Kenneth Schoen.  Fifth and Sixth Order PECE Algorithms with Improved Stability Properties , 1971 .

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

[18]  Charles William Gear,et al.  Stability and convergence of variable order multistep methods , 1974 .

[19]  Fred T. Krogh.  Predictor-Corrector Methods of High Order With Improved Stability Characteristics , 1966, JACM.

[20]  J. Lambert.  Computational Methods in Ordinary Differential Equations , 1973 .

[21]  Lawrence F. Shampine,et al.  Stiffness and Non-Stiff Differential Equation Solvers , 1975 .

[22]  Peter Piotrowski.  Stability, consistency and convergence of variable K-step methods for numerical integration of large systems of ordinary differential equations , 1969 .

[23]  T. E. Hull,et al.  Comparing Numerical Methods for Ordinary Differential Equations , 1972 .