Melnik, R.V.N. and Gavrilyuk, I.P.
Time-dependent convection-diffusion models appear frequently in many branches of computational mechanics including, but not limited to, fluid dynamics and electro-magnetic theory. They also serve as building blocks for many complex applied problems requiring coupling
several different physical fields. The development of efficient discretisation methodologies for such models represents a challenging and important area of research in computational mechanics.
Earlier we proposed an efficient methodology for the full time-space discretisation convection-diffusion-absorption equations based on the Cayley transform technique. Recently a general framework based on the so-called strongly P-positive operator technique has been developed. Due to its ability to generate exponentially convergent algorithms easily amenable to efficient parallelisation strategies, this framework allows one to construct such algorithms for full time-space discretisations of evolutionary differential equations that have several important advantages over conventional algorithms (including those with polynomial convergence). In this paper we use this general framework to develop further our ideas in the context of time-dependent convection-diffusion problems. In particular, we construct exponentially convergent discretisations for these problems by using an efficient representation of the operator exponent and appropriate quadratures based on the Sinc approximations of corresponding integrals. We note that in this case the resulting quadratures, consisting of a sum of operator resolvents, can be effectively (at almost linear cost) approximated by the H-matrix technique. The situation where the diffusion coefficient is small due to the presence of boundary layers is also discussed. The algorithms presented in this paper admit parallel implementations in terms of both the computation of resolvents and the time stepping.
Key words: Cayley transform technique; geometric integration; Lie groups; full time-space discretization; operator exponents; Sinc approximations; H-matrix technique; parallel algorithms; resolvents; boundary layers; evolutionary differential equations; exponential convergence; convection-diffussion operators; time-stepping algorithms; dynamic adpatation; conservative numerical discretizations; Dunford-Cauchy integrals; non-selfadjoint operators.