A New Order Selection Strategy for Ordinary Differential Equation Solvers

Abstract

The normal order selection strategy used by a variable order, variable stepsize ordinary differential equation solver is based on selecting the formula that can use the largest stepsize while meeting the accuracy constraint. This strategy does not take into account the stiffness or nonstiffness of the differential system. By considering how the stiffness of the differential system influences order selection, a new selection strategy is proposed that is biased in the presence of stiffness toward the lower-order formulas. This new order selection strategy has been incorporated into a version of a widely used ordinary differential equation solver LSODE. The modified solver is shown to be greatly more efficient for problems that the original solver cannot solve efficiently, namely, problems where the Jacobian of the differential system has some eigenvalues of large modulus close to the imaginary axis. On other problems that the unmodified solver solves efficiently, the modified solver still performs with comparable efficiency.