AN ADAPTIVE THETA-METHOD FOR THE SOLUTION OF STIFF AND NONSTIFF DIFFERENTIAL-EQUATIONS

This paper describes a new adaptive method that has been developed to give improved efficiency for solving differential equations where the degree of stiffness varies during the course of the integration or is not known beforehand. The method is a modification of the theta method, in which the new adaptive strategy is to automatically select the value of theta and to switch between functional iteration and Newton iteration for the solution of the nonlinear equations arising at each integration step. The criteria for selecting theta and for switching are established by optimising the permissible step size. The performance of the adaptive methods is demonstrated on a range of test problems including one arising from the method of lines solution of a convection-dominated partial differential equation. In some cases the new approach halves the amount of computational work.