Applying the Best Parameterization Method and Its Modifications for Numerical Solving of Some Classes of Singularly Perturbed Problems

Abstract

The chapter deals with the numerical solution of initial and boundary value problems for systems of nonlinear singularly perturbed ordinary differential equations. Similar problems arise in many application areas, such as structural mechanics, nuclear power and mechanical engineering. The main difficulty in the numerical integration of singularly perturbed equations is the presence of sections of fast change in the integral curves, replaced by sections of a slow one (for example, boundary and interior layers, contrast structures). Explicit schemes are ineffective for this class of problems. At the same time, implicit schemes are more efficient, but they are significantly inferior to explicit schemes in performance. We propose a new approach for solving singularly perturbed initial and boundary value problems based on the method of solution continuation with respect to the best argument and its modifications. The solution continuation method allows to eliminate or smooth out the sections of fast change in the integral curves, making it possible to increase the efficiency of applying explicit schemes to singularly perturbed problems. The developed approach is approved on the test initial value problem with several limiting singular points and the applied boundary value problem of supersonic flow. We compare numerical results with the exact solutions of considered problems and give conditions for choosing a solution continuation argument.