Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments (Q526714)

The authors develop a numerical method for the following class of time-dependent singularly perturbed convection-diffusion-reaction problems with delay \[ \frac{\partial u}{\partial t}-\epsilon\frac{\partial^{2_{\mathcal{U}}}}{\partial x^{2}}+a(x)\frac{\partial u}{\partial x}+c(x)u(x-\delta,\;t)+b(x)u(x,\;t)+d(x)u(x+\eta,\;t)=f(x,\;t), \] subject to the interval boundary conditions \(u(x,\;t) =\phi(x,\;t)\), \(\forall(x,\;t)\in\Omega_{1}=\)\{\((x,\;t):-\delta\leq x\leq 0\) and \(0\leq t\leq T\)\}, \(u(x,\;t) =\psi(x,\;t)\), \(\forall(x,\;t) \in\Omega_{2}=\)\{\((x,\;t):1\leq x\leq 1+\eta\) and \(0\leq t\leq T\)\}, and the initial condition \(u(x,\;0)=u(x)\), \(\forall x\in\overline{D}\). The proposed numerical method is based on the backward Euler finite difference scheme for time discretization and the exact three-point finite difference scheme on a uniform grid for space discretization. The authors provide error and stability estimates of the method. Several numerical examples are considered.