A computational study of three numerical methods for some advection-diffusion problems

Abstract

Three numerical methods have been used to solve two problems described by advection-diffusion equations with specified initial and boundary conditions. The methods used are the third order upwind scheme [4], fourth order upwind scheme [4] and Non-Standard Finite Difference scheme (NSFD) [9]. We considered two test problems. The first test problem has steep boundary layers near x = 1 and this is challenging problem as many schemes are plagued by non-physical oscillation near steep boundaries [15]. Many methods suffer from computational noise when modelling the second test problem especially when the coefficient of diffusivity is very small for instance 0.01. We compute some errors, namely L 2 and L ∞ errors, dissipation and dispersion errors, total variation and the total mean square error for both problems and compare the computational time when the codes are run on a matlab platform. We then use the optimization technique devised by Appadu [1] to find the optimal value of the time step at a given value of the spatial step which minimizes the dispersion error and this is validated by some numerical experiments.

Optimized composite finite difference schemes for atmospheric flow modeling

“…Some applications of NSFD can be seen in [23,26] where they are used to discretize advection-diffusion equations with initial conditions described by Gaussian profiles or with steep boundary layer. Our aim is to use ideas from [25] to construct NSFD schemes for the partial differential equation given by Equation (1) which has three subequations which are as follows:…”

Numerical Solution of a 3-D Advection-Dispersion Model for Dissolved Oxygen Distribution in Facultative Ponds

“…The Forward Time Central Space Scheme in question is derivative approximation of time by using forward finite difference and space by using central finite difference. The derivative approximation by using finite difference FTCS is [7]:…”

“…[7] gives 3 numerical methods for solving 2 advection-diffusion problems with initial conditions and specific boundary conditions. [8] provides examples of numerical solutions of advectiondispersion equations 3-dimensional using finite difference methods FTCS (Forward Time Central Space) with different initial values and boundary conditions.…”

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

“…The main culprit is that when solute transport is advection‐dominated, the advection‐dispersion equation approximates to a hyperbolic‐type partial differential equation, and numerical approximation methods become prone to these problems. The upwind numerical scheme serves to correct some of these problems by damping responses to produce a more realistic solution in both heat transfer and fluid flow simulations …”

“…Numerous authors have investigated the upwind scheme for the numerical solution of the advection‐dispersion equation or, the more general form, the convection‐dispersion equation . For example, upstream weighting is often applied to finite element methods, where an element upstream of a node is weighted more heavily than the element located downstream, namely, the Petrov‐Galerkin finite element method .…”