A Time Adaptive Scheme for the Solution of the Advection Equation in the Presence of a Transient Flow Velocity

A Fourier analysis conducted on both the spatial and the temporal discretizations of the governing partial differential equation shows that the Courant number as well as the time marching scheme have significant influences on the numerical behaviour of a Modified Least Squares (MLS) method for the solution of the advection equation. The variations of the amplification factor and the relative phase velocity with the Courant number and the dimensionless wave number indicate that when Courant number is equal to unity, the MLS method with the specified time-weighting and upwind function gives accurate results. This conclusion is confirmed by the numerical computation of the problem of the onedimensional advective transport with constant flow velocity, carried out with different Courant numbers. Based on this observation, a time-adaptive scheme is developed to examine the problem where the advective transport has a time-dependent velocity. The time step is selected adaptively using the Courant number criterion Cr = 1. The time-adaptive scheme is applied to analyze the advective transport problem in which the flow velocity is governed by a pressure transient, resulting from the consideration of the compressibilities of the pore fluid and the porous skeleton, as well as the transient hydraulic boundary conditions. keyword: Modified Least Squares method, Fourier analysis, advection equation, Courant number, timeadaptive scheme.