Stability Analysis for Singularly Perturbed Differential Equations by the Upwind Difference Scheme

摘要

For solving singularly perturbed differential equations (SPDE), the upwind difference scheme (UDS) and the fitted difference method are chosen. The local refinements of grids are adopted in singular layers with the minimal meshspacing h_(min) = O(hε), where h is the maximal meshspacing, and ε(? 1) a very small parameter. The traditional condition number in 2-norm is given by Cond = O(h~(-2)ε~(-1)). For the infinitesimal ε(≤ 10~(-8)) in application, the huge Cond issues a dilemma regarding whether the numerical solutions by the UDS can be trusted. Although the UDS has been used for several decades, such a dilemma has not been clarified yet. The goal of this article is to clarify this dilemma. To this end, we solicit the effective condition number Cond_eff in Li et al. Numer Linear Algebra Appl 15 (2008), 575-690 Effective condition for Numerical Partial Different Equation, 2013, and develop a new actual condition number from the maximum principle. Both of them may offer much smaller bounds of the solution errors caused by perturbation, e.g., rounding, truncation, or discritization errors.We study the Dirichlet problems of SPDE by the UDS in a rectangle. When the Dirichlet boundary condition on the downwind side is homogeneous, we derive Cond_eff = O(1/h~(1/2)). When the entire Dirichlet boundary conditions are homogeneous, the extraordinary bound, Cond_eff = O(1), is achieved. Moreover, we derive the actual condition numbers as Cond_actual = O(1/h~(1/2)) and Cond_actual = O(1h) for the homogeneous and the nonhomogeneous SPDE, respectively. Note that these bounds do not depend on ε; this is distinct from the traditional Cond. Based on the analysis of this article, the existing dilemma caused by Cond has been removed, to grant a good stability of the UDS for SPDE.