Numerical solution of transport equation for applications in environmental hydraulics and hydrology

Abstract

The advective term in the one-dimensional transport equation, when numerically discretized, produces artificial diffusion. To minimize such artificial diffusion, which vanishes only for Courant number equal to unity, transport owing to advection has been modeled separately. The numerical solution of the advection equation for a Gaussian initial distribution is well established; however, large oscillations are observed when applied to an initial distribution with sleep gradients, such as trapezoidal distribution of a constituent or propagation of mass from a continuous input. In this study, the application of seven finite-difference schemes and one polynomial interpolation scheme is investigated to solve the transport equation for both Gaussian and non-Gaussian (trapezoidal) initial distributions. The results obtained from the numerical schemes are compared with the exact solutions. A constant advective velocity is assumed throughout the transport process. For a Gaussian distribution initial condition, all eight schemes give excellent results, except the Lax scheme which is diffusive. In application to the trapezoidal initial distribution, explicit finite-difference schemes prove to be superior to implicit finite-difference schemes because the latter produce large numerical oscillations near the steep gradients. The Warming-Kutler-Lomax (WKL) explicit scheme is found to be better among this group. The Hermite polynomial interpolation scheme yields the best result for a trapezoidal distribution among all eight schemes investigated. The second-order accurate schemes are sufficiently accurate for most practical problems, but the solution of unusual problems (concentration with steep gradient) requires the application of higher-order (e.g. third- and fourth-order) accurate schemes.