A higher-order eulerian scheme for coupled advection-diffusion transport

A new accurate high‐order numerical method is presented for the coupled transport of a passive scalar (concentration) by advection and diffusion. Following the method of characteristics, the pure advection problem is first investigated. Interpolation of the concentration and its first derivative at the foot of the characteristic is carried out with a fifth‐degree polynomial. The latter is constructed by using as information the concentration and its first and second derivatives at computational points on current time level t in Eulerian co‐ordinates. The first derivative involved in the polynomial is transported by advection along the characteristic towards time level t + Δt in the same way as is the concentration itself. Second derivatives are obtained at the new time level t + Δt by solving a system of linear equations defined only by the concentrations and their derivatives at grid nodes, with the assumption that the third‐order derivatives are continuous. The approximation of the method is of sixth order. The results are extended to coupled transport by advection and diffusion. Diffusion of the concentration takes place in parallel with advection along the characteristic. The applicability and precision of the method are demonstrated for the case of a Gaussian initial distribution of concentrations as well as for the case of a steep advancing concentration front. The results of the simulations are compared with analytical solutions and some existing methods.