Investigation of new methods for the integration of stiff ordinary differential systems

New methods for the integration of stiff ordinary differential systems are proposed. A new explicit exponential procedure which automatically partitions the variables at each step into stiff and nonstiff groups is considered. The stiff variables are subsequently integrated by a new explicit exponential method, whereas the nonstiff variables are integrated by a nonstiff explicit method. The new algorithm possesses excellent stability properties. It is very efficient for stiff problems where the integration step size is restricted mainly by stability rather than accuracy. Because of the automatic partitioning, the new algorithm is entirely suitable for both stiff and nonstiff problems.;A second explicit method is developed. The new first-order Euler/RK2 hybrid method requires only two derivative evaluations per step but has a stability region that is almost four times larger than that of RK2. The new hybrid method is most efficient on mildly and moderately stiff problems when low or intermediate accuracy is desired and has proven to be very successful with method-of-lines problems.;Two new implicit methods were also investigated. The first new implicit method is based on a variation of the general midpoint rule and is referred to as the backward midpoint rule. It is second-order accurate, L-stable and is more efficient than the classical midpoint rule for integrating highly stiff systems of ODEs when low or intermediate accuracy is desired.;Finally, a second new implicit method was investigated. The new Implicit Improved Euler is a second-order accurate method but has better stability properties (in terms of damping) than the trapezoidal rule. The new method is more efficient than the trapezoidal rule for integrating highly stiff systems of ODEs when operated at loose error tolerances and is generally always more efficient than the first-order backward Euler. For large stiff problems (e.g., method-of-lines problems), the new method is more efficient than the trapezoidal rule since it requires less work per step.;For the last three new methods, the use of global extrapolation is shown to improve the accuracy and efficiency of the methods and to provide global error estimates.