Augmented upwind numerical schemes for the groundwater transport advection‐dispersion equation with local operators

Abstract

Summary When solute transport is advection‐dominated, the advection‐dispersion equation approximates to a hyperbolic‐type partial differential equation, and finite difference and finite element numerical approximation methods become prone to artificial oscillations. The upwind scheme serves to correct these responses to produce a more realistic solution. The upwind scheme is reviewed and then applied to the advection‐dispersion equation with local operators for the first‐order upwinding numerical approximation scheme. The traditional explicit and implicit schemes, as well as the Crank‐Nicolson scheme, are developed and analyzed for numerical stability to form a comparison base. Two new numerical approximation schemes are then proposed, namely, upwind–Crank‐Nicolson scheme, where only for the advection term is applied, and weighted upwind‐downwind scheme. These newly developed schemes are analyzed for numerical stability and compared to the traditional schemes. It was found that an upwind–Crank‐Nicolson scheme is appropriate if the Crank‐Nicolson scheme is only applied to the advection term of the advection‐dispersion equation. Furthermore, the proposed explicit weighted upwind‐downwind finite difference numerical scheme is an improvement on the traditional explicit first‐order upwind scheme, whereas the implicit weighted first‐order upwind‐downwind finite difference numerical scheme is stable under all assumptions when the appropriate weighting factor (θ) is assigned.

Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems

“…The development and analysis of numerical solutions for the fractional advectiondispersion equation has remained relevant, to continually pursue improved numerical solution methods [32,34,31,17,2,3,7,13,18]. Upwind finite difference numerical methods have been applied to fractional partial differential equations [38,36,37], and recently [1] presented augmented upwind schemes for the advectiondispersion equation with local operators, where an upwind Crank-Nicolson and weighted upwind-downwind finite difference schemes were developed. The numerical schemes were found to be an improvement on the traditional upwind approach, and are thus selected for application to the fractional advection-dispersion equation.…”

“…The first-order upwind scheme is applied to the one-dimensional, nonreactive fractional advection-dispersion equation for numerical approximation. Similar to the classical advection-dispersion equation, the upwind scheme for the fractional advection-dispersion equation finite difference approximation influences the advection term, where backward or forward differences are considered depending on the direction of the transporting velocity [1]. Applying the Caputo definition to the fractional advection-dispersion equation,…”

“…First-order upwind advection Crank-Nicolson scheme. The upwind Crank-Nicolson scheme for the Caputo fractional advection-dispersion equation is considered here [1]. The time component remains the same as with the implicit and explicit schemes, but the space components change to,…”

“…Considering both the upwind and downwind direction for the advection term, a ratio of upwind to downwind is controlled by θ, where 0 ≤ θ ≤ 1. The advection component can thus be represented as follows for the explicit upwind-downwind weighted scheme [1]:…”

“…As with the explicit formulations, both the upwind and downwind direction for the advection term is consdered for the implicit upwind-downwind weighted approximation. The advection component is approximated as follows using the Caputo fractional derivative [1]:…”