Parameter-uniform hybrid numerical scheme for time-dependent convection-dominated initial-boundary-value problems (Q836945)

In the region \((0,1)\times(0,T)\) the authors consider the singularly perturbed parabolic convection-diffusion equation \[ u_t-\varepsilon u_{xx}+a(x)u_x+b(x)u=f(x,t) \] under initial-boundary conditions \(u(x,0)=u_0(x)\), \(x\in (0,1)\), \(u(0,t)=u(1,t)\), \(t\in [0,T]\). Here \(0<\varepsilon\ll 1\) is a small parameter and smooth functions \(a\), \(b\) satisfy conditions \(a(x)\geq\alpha>0\), \(b(x)\geq\beta>0\) on \([0,1]\). The unique solution to the above problem exhibits a regular boundary layer of width \(O(\varepsilon)\) at \(x=1\). Firstly, the authors approximate the problem with respect to time by the backward-Euler method. The obtained semidiscrete problem is then spatial discretized on a piecewise-uniform Shishkin mesh by a hybrid finite difference scheme (a suitable combination of a classical central difference scheme and a midpoint upwind one). The \(\varepsilon\)-uniform convergence in the supremum norm of the fully discrete scheme is proved. The method is almost second-order accurate in spatial variable. Two test problems are analyzed numerically.