Nonlinear Singular Perturbation Problems and One Sided Difference Schemes

Abstract

We consider the boundary value problem $\varepsilon y'' - a(y)y' - b(x,y) = F(x), - 1 \leqq x \leqq 1$, for $y( - 1)$ and $y(1)$ given and $b(x,0) \equiv 0$, $b_y (x,y) \geqq 0$. We construct a simple difference scheme approximating $\varepsilon u_{xx} - a(u)u_x - b(x,u) - F(x) = u_t $ for $t \geqq 0, - 1 \leqq x \leqq 1$ with the same boundary conditions and an arbitrary initial guess. As the numerical artificial time approaches infinity, we prove convergence to a unique steady state solution. As the mesh size $\Delta x$ and the parameter $\varepsilon $ approach zero in any order, we prove convergence to the solution of the corresponding O.D.E. We use one-sided differences to approximate the transport term. The method applies equally well to multidimensional analogues. We present numerical results verifying the theory.