On some developments and evaluation of an Eulerian-Lagrangian method for the transport equation

Abstract

The modelling of typical engineering problems in industry, such as water-jet cooling of hot-rolled steel strip products, directly involves the solution of a transport (advection-diffusion) equation for the cooling characteristics of the strip. The non-linear nature of the heat conduction involved aggravates the difficulty of the problem. Traditional Finite Difference techniques for the solution of this advection dominated transport equation incur severe Courant number stability restrictions as well as instabilities in the presence of temperature discontinuities. Eulerian-Lagrangian Methods (ELM's) solve the transport equation in Lagrangian form `along' backward characteristics effectively decoupling the advection and diffusion terms but retaining the convenience of fixed computational grids. Typical interpolation methods used to obtain the values at the feet of characteristic lines lead to spurious oscillations, numerical diffusion, peak clipping and phase errors. Through the use of `peak tracking', by the forward-tracking of Eulerian nodal points, this paper attempts to alleviate these errors. A comparison of 1-D benchmark tests from the Convection-Diffusion Forum as well as appropriate error measures, are shown to produce appreciable improvements over the standard methods for a range of time steps, very large Peclet numbers and Courant numbers in excess of one.