Application of the Multi-layer Method and Method of Continuing at the Best Parameter to the Solving of a Stiff Equation

Abstract

By using the example of one rigid boundary value problem for a second order differential equation on a sphere, we compare our two approaches for giving construction of approximate solutions. The first approach consists of using the method of prolongation of the solution by the best parameter. This approach allows us to substantially decrease the number of steps and increase stability of the calculation process with comparison to standard methods. The second approach is linked with the building of an approximated multi-layer solution to the problem and is based on the use of analytical recurrent rations. We build those recurrent ratios based on classic numerical methods, applied to an interval of variable length. In the result, an approximated solution in the form of a table of numbers is substituted with an approximated solution is the form of a function, which is easier to use for adaptation, building graphs and other purposes. Problems related to stiffness of material can be solved by using solution of singularly perturbed differential equations with applying standard methods for the numerical solution of ordinary differential equations, however it can lead to significant difficulties. The first difficulty is the loss of stability of the computational process, when small errors in individual steps lead to an uncontrolled increase in the error of computations as a whole. Another difficulty directly connected with the first one is the need to greatly reduce the integration step, which leads to a strong slowdown of the computational process. In this work, it is shown, that both of our approaches successfully cope with indicated difficulties.