Numerical approximations to a singularly perturbed convection-diffusion problem with a discontinuous initial condition

A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified which matches the discontinuity in the initial condition and also satisfies the homogenous parabolic differential equation associated with the problem. The difference between this analytical function and the solution of the parabolic problem is approximated numerically, using an upwind finite difference operator combined with an appropriate layer-adapted mesh. The numerical method is shown to be parameter-uniform. Numerical results are presented to illustrate the theoretical error bounds established in the paper. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature

An a-posteriori adaptive mesh technique for singularly perturbed convection-diffusion problems with a moving interior layer

We study numerical approximations for a class of singularly perturbed problems of convection-diffusion type with a moving interior layer. In a domain (a segment) with a moving interface between two subdomains, we consider an initial boundary value problem for a singularly perturbed parabolic convection-diffusion equation. Convection fluxes on the subdomains are directed towards the interface. The solution of this problem has a moving transition layer in the neighbourhood of the interface. Unlike problems with a stationary layer, the solution exhibits singular behaviour also with respect to the time variable. Well-known upwind finite difference schemes for such problems do not~converge $eps$-uniformly in the uniform norm, even under the condition $N^{-1}+N_0^{-1}approx eps$, where $eps$ is the perturbation parameter and $N$ and $N_0$ denote the number of mesh points with respect to $x$ and $t$. In the case of rectangular meshes which are ({it a~priori,} or {it a~posteriori,}) locally refined in the transition layer, there are no schemes that convergence uniformly in $eps$ even under the {it very restrictive,} condition $N^{-2}+N_0^{-2} approx eps$. However, the condition for convergence can be {it essentially weakened} if we take the geometry of the layer into account, i.e., if we introduce a new coordinate system which captures the interface. For the problem in such a coordinate system, one can use either an {it a~priori,}, or an {it a~posteriori,} adaptive mesh technique. Here we construct a scheme on {it a~posteriori,} adaptive meshes (based on the gradient of the solution), whose solution converges `almost $eps$-uniformly', viz., under the condition $N^{-1}=o(eps^{ u})$, where $u>0$ is an arbitrary number from the half-open interval $(0,1]$

Shock-layer bounds for a singularly perturbed equation

The size of the shock-layer governed by a conservation law is studied. The conservation law is a parabolic reaction-convection-diffusion equation with a small parameter multiplying the diffusion term and convex flux. Rigorous upper and lower bounding functions for the solution of the conservation law are established based on maximum-principle arguments. The bounding functions demonstrate that the size of the shock-layer is proportional to the parameter multiplying the diffusion term