Solving STIFF systems by Taylor series

摘要

The analysis of long Taylor series for the solutions of stiff problems gives new insight into their true character and leads to new solution algorithms. Some ODE's are difficult to solve because of singularities close to the real axis. This is not true for stiff problems. There is a linear ODE hidden within the stiff system of ODE's. When the hidden linear ODE is of order one, there is an exponential component in the solution. Most stiff problems are of this type. This exponential forces the integration stepsize to be very small relative to the domain of the stiff problems. This exponential component is readily made apparent by analysis of the Taylor series for the solutions.We analyze in detail a stiff problem from chemistry, which is very difficult to solve. The portions of the equations giving rise to the stiffness in this problem are identified. In this problem, there are more than one hidden linear ODE. Artificial stiff problems with up to stiff-order four have been created by combining linear ODE's with a non-linear problem with known solutions. The Taylor series algorithm for solving stiff problems is an integral part of the ATOMFT system for the automatic solutions of ODE's.