Higher Order Numerical Method on Layer Adapted Meshes for Singularly Per-turbed Parabolic Partial Differential Equation

Abstract

Here, we have considered the following class of singularly perturbed parabolic initial boundary value problem (IBVP): 8< : Lu(x) ut(x; t) 􀀀 "uxx(x; t) 􀀀 (a(x)u(x; t))x = f(x; t); G = (0; T]; = (0; 1); 􀀀 = GnG; u(x; 0) = s(x); x 2 ; u(0; t) = q1(t); u(1; t) = q2(t); t 2 [0; T]; (1) where 0 < " 1, a(x) is su ciently smooth and a(x) > 0. Under su cient smoothness and compatibility conditions on f and s, the IBVP (1) admits a unique solution which possesses a regular boundary layer of width O(") at x = 0. (refer [4]) Various types of numerical methods for stationary and non-stationary singularly perturbed convec- tion di usion problems on layer-adapted meshes have been developed by many researchers [2, 4]. Most of these methods are of rst order convergent. In order to get higher order accuracy, one should aim for a combination of higher order schemes known as hybrid schemes. In [3], a numerical scheme is developed which consists of classical backward-Euler method for the time discretization and a hybrid nite di erence scheme combining central di erence in layer region and midpoint scheme in outer region for the spatial discretization for time-dependent IBVP on Shishkin mesh resulting second order accuracy in space. But, this scheme has the demerit of a priori information about the location of the inner and outer region. Following the idea of [1], we devise a monotone hybrid scheme with variable weights for spatial dis- cretization having the proper combination of such higher order schemes. The weights are so chosen that the proposed scheme produces second order accuracy without the limitations described earlier. This scheme consists of classical backward-Euler method for time discretization and a monotone hybrid nite di erence scheme for the spatial discretization. The proposed scheme suggests a grad- ual switching from midpoint scheme to central di erence scheme as the mesh goes from coarse to ne while earlier this switching was sudden. The proposed scheme for IBVP (1) is analyzed on a layer adapted meshes in particular on Shishkin mesh, on Bakhvalov-Shishkin mesh and on adaptive grid (this adaptive grid has the advantage that it requires no information of width and lo- cation of boundary layer). It is shown to be uniformly convergent with respect to the perturbation parameter. Numerical results validate the theoretical ndings.