Finite difference numerical solution of Troesch’s problem on a piecewise uniform Shishkin mesh

Abstract

Finding numerical solutions to the Troesch’s problem is known to be challenging, especially when the sensitivity parameter $\lambda$ is large. In this manuscript, we propose a numerical method for solving the Troesch’s problem which combines efficiency and accuracy, even for large sensitivity parameter. Our method can be summarized as a finite difference method formulated on a Shishkin mesh, which is piecewise uniform with smaller step size in the boundary layer in the neighborhood of the point $x=1$. In fact, we treat the Troesch’s problem as a singularly perturbed problem with a special transition point on the mesh. The transition point for the Shishkin mesh is first defined and computed. The application of the finite difference method on the piecewise uniform mesh leads to a nonlinear matrix system, which is solved numerically using the generalized Newton’s method. While, existing numerical solvers suffer significantly when sensitivity parameter $\lambda$ becomes large and fail to provide accurate numerical solutions for $\lambda >20$ (Chang, Appl Math Comput 216(11):3303–3306, 2010; Khuri and Sayfy, Math Comput Model 54(9):1907–1918, 2011; Raja, Inf Sci 279:860–873, 2014; Temimi, Appl Math Comput 219(2):521–529, 2012), our method provides accurate numerical solution for large values of this sensitivity parameter, up to $\lambda =100$. Moreover, the implementation of the method is straight forward and the computation cost is low. Numerical experiments show that the method provides accurate solutions for different values of the sensitivity parameter $\lambda$.