Numerical analysis of an exponentially ill-conditioned boundary value problem with applications to metastable problems (Q2763937)

A differential problem is said to have a metastable behaviour when its solutions exhibit a exponentially slow time dependent motion towards the steady-state solution. Such a behaviour is often associated with exponentially ill-conditioned singularly perturbed problems. The viscous shock problem, the Fokker-Planck equation and phase separation models are among the problems classified as metastable. So far little is known about stability and convergence for the numerical schemes built up to generate approximations to the solutions of such problems. It is justifiable to expect the its associated numerical truncation error to be less than the order of the smallest eigenvalue related to the considered problem, but this turns out to be unbearable, as this eigenvalue is smaller than the machine precision. Neverthless, many conventional numerical schemes show a rather good performance when applied to these problems, even with moderate mesh sizes. The paper goal is to shed some light on this question and with this purpose he treats an exponentially ill-conditioned boundary layer resonance problem with three different algorithms, namely, the upwind scheme, the coupled scheme and the Il'in scheme. It is found that all of them are uniformly convergent on suitable meshes, in a sense the author makes precise. It is also deduced that their coefficient matrices inherit the extreme ill-conditioning associated with the continuous problem. NEWLINENEWLINENEWLINEAn important point is made: as long as sufficiently high precision arithmetic is employed, exponentially ill-conditioned singularly perturbed problems do not cause more hardships in numerical computations than other singular perturbation problems.