Uniform Approximation Property of Implicit Methods for a Stiff Family of Differential Equations
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DepartmentMathematics and Statistics
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Stiff systems are characterized by the presence of multiple time scales where the fast scales are stable. The presence of a scaling factor, ?, in the system creates slow and fast components which lead to the distinct time scales. Conventional stability analysis shows that numerical solutions using explicit Taylor methods need a step size that is smaller than the fast time scale to get a stable solution. With implicit Taylor methods, the step size can be larger in comparison without affecting the stability of the solution. Most of the analysis done in regard to stiff systems tends to fix the step size and look at the stability of the numerical solution as the number of steps goes to infinity. This thesis presents a new form of analysis for numerical methods for stiff systems. We examine the numerical solutions over a finite time interval as the step size goes to zero over an entire range of the scaling factors ??1. We show that uniform convergence is a better indicator than unconditional stability for the effectiveness of a numerical solution to a stiff system. We investigate the uniform convergence of the numerical solutions to the true solution for a family of systems. We begin with an analysis of a family of scalar linear real equations to show that explicit Euler and trapezoidal methods are not uniformly convergent. Using Dini's theorem, we can show that the numerical solution using an implicit Taylor's method converges uniformly to the true solution as the step size decreases. In addition, we were able to show that when using the implicit Euler method, the uniform convergence is of the order O(h) where h is the step size. We then extend the results to higher dimensions to show that for a family of linear non-stiff systems, the explicit and implicit Euler converge uniformly. In addition, we show that the implicit Euler method converges uniformly for a family of stiff two-dimensional linear systems.