3 - Finite Difference Method

Summary

Introduction

All models that are utilized in atmospheric science are formulated based on fundamental physical principles, such as conservation of mass, conservation of momentum, and conservation of energy. These basic physical principles embodied in the atmospheric models are expressed mathematically as a coupled system of nonlinear partial differential equations. As these mathematical equations are complex and cannot be solved analytically, one has to resort to numerical methods to solve them. Numerical methods approximate a continuous model (system of partial differential equations) to a discrete model (system of difference equations). The process of approximating a continuous model to a discrete model by employing a numerical method is called “numerical discretization.”

Method of Finite Difference

Consider a function f that depends on a single dependent variable x, say f = f (x). Assume that x spans an interval L. Let the interval be partitioned by N+1 equally spaced grid points (including the two end points at the limits of the interval). The grid length is then defined as Δx=L=N and the grid points are located at x_j = jΔx, where j = 0,1,2,...,N are integers. Let the value of f at x j be represented by fj. Mathematically, the derivative of a function f(x) is defined as

which is shown graphically in Figure 3.1. As in the finite difference method, the value of Δx is finite and does not go to zero, the derivative and the finite difference (the RHS of Equation 3.1) are not exactly identical. Hence, the latter is also called “finite difference approximation” and an error is associated with replacing the derivative of a function with its “finite difference approximation.” The value of the derivative of a function f(x) at a grid point x_j can be obtained in three different ways (refer Figure 3.2):

Each one of the aforementioned finite difference approximations can be obtained from the Taylor series expansion of f(x+Δx) about the value of x, which is given by

Forward difference scheme

Using the Taylor series (Equation (3.5)) with f(x+ Δx) as f_j+1 and f(x) as fj, the following equation is obtained: