A parameter-uniform hybrid scheme designed for multi-point boundary value problems that are perturbed

Abstract

The numerical solution of a class of second-order singularly perturbed three-point boundary value problems (BVPs) in 1D is achieved using a uniformly convergent, stable, and efficient difference method on a piecewise-uniform mesh. The presence of a boundary layer(s) on one (or both) of the interval’s endpoints is caused by the presence of the tiny parameter in the highest order derivative. As the perturbation parameter approaches 0, traditional numerical techniques on the uniform mesh become insufficient, resulting in poor accuracy and large blows without the use of an excessive number of points. Specially customised techniques, such as fitted operator methods or methods linked to adapted or fitted meshes that solve essential characteristics such as boundary and/or inner layers, are necessary to overcome this drawback. We developed a fitted-mesh technique in this paper that works for all perturbation parameter values. The monotone hybrid technique, which includes midway upwinding in the outer area and centre differencing in the layer region on a fitted-mesh condensing in the border layer region, is the basis for our difference scheme. In a discrete \(L^\infty \) norm, uniform error estimates are constructed, and the technique is demonstrated to be parameter-uniform convergent of order two (up to a logarithmic factor). To show the effectiveness of the recommended technique and to corroborate the theoretical findings, a numerical example is presented. In practise, the convergence obtained matches the theoretical expectations.