Graded-mesh difference schemes for singularly perturbed two-point boundary value problems

Abstract:

This paper is concerned with the numerical approximation by compact finite-difference schemes of differential operators of the form ${L_\varepsilon }u = \varepsilon {u^{(m)}} + \Sigma _{\nu = 0}^{m - 1}{a_\nu }{u^{(\nu )}}$ without turning points. The stability of ${L_\varepsilon }$ combined with various auxiliary conditions is discussed, and a representation result for solutions of problems involving it is proven. This representation decomposes the solution into a smooth outer component plus a decaying exponential layer term along the lines of the Method of Multiple Scales. The stability of compact difference analogues of ${L_\varepsilon }$ is studied, and a stability result is proven which generalizes earlier work. This result encompasses, for example, discretizations of second-order problems that fail to possess a maximum principle. It allows for standard polynomial-based differences in outer regions (away from boundary layers) with uniform meshes, even though such schemes admit oscillatory solutions. A family of finite-difference schemes based on an exponentially graded mesh and local polynomial basis functions is discussed. These schemes can be constructed to have arbitrarily high uniform order of convergence. To achieve a scheme of order $O({h^K})$, roughly K times as many points are distributed inside the layer as outside. The high order is achieved by using extra local evaluations of the coefficient functions and source term of the problem. A rigorous discretization error analysis of these schemes, using the established stability and representation results, is given. Numerical results exhibiting the performance of these schemes are presented and generalizations of the results in the paper are discussed.