Introduction to Numerical Methods for Time Dependent Differential Equations

CHAPTER 9

APPROXIMATIONS OF 1-PERIODIC SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS

In this chapter we introduce basic finite difference operators that approximate space derivatives and spectral derivatives that are based on Fourier theory. Then the method of lines, a very useful approach that consists of approximating the space derivatives but keeping the time continuous, is explained. Under this approach the partial differential equations become systems of ordinary differential equations, and therefore we can use the results of the first part of the book to analyze the model problems.

9.1 Approximations of space derivatives

Let M be a positive integer and h = (2M + 1)⁻¹ a grid size. The corresponding grid points are denoted by x_j = jh, j = 0, ±1, ±2, …. A grid function f_j = f(x_j) is simply a function defined on the grid. It is called 1-periodic if

and therefore any 1-periodic grid function is well defined by its 2M + 1 values f₀, f₁, …, f_2M.

Figure 9.1 Grid.

The most common difference operators approximating d/dx are D₊, D₋, D₀, which are defined by

(9.1)

and which are called forward, backward, and centered difference operators, respectively. Clearly, if f_j is 1-periodic, then ...