Question

Explain why the local discretization error of the finite difference method is proportional to the square of the step size. Also explain why the global discretization error is proportional to the step size itself.

Short answer

Question: Explain the relationship between the local and global discretization errors in the finite difference method and the step size. Answer: In the finite difference method, the local discretization error, which occurs at each step of the process, is proportional to the square of the step size. The global discretization error, which represents the accumulated error over all the steps, is proportional to the step size itself. As the step size decreases, both the local and global discretization errors decrease.

Step-by-step solution

Understand Local and Global Discretization Errors

To begin, let's understand what local and global discretization errors are in the context of the finite difference method. The finite difference method is an approximate numerical technique used for solving differential equations. However, since it is an approximation, there will be errors associated with this method. Local discretization error refers to the error that occurs at each step of the finite difference method, while global discretization error represents the accumulated error over all the steps.

Local Discretization Error and Step Size

To derive the relationship between the local discretization error and the square of the step size, we will consider the Taylor series expansion. The Taylor series is an infinite series of terms that represent the function as a sum of its derivatives in powers of the step size. For a given function f(x), we can represent its Taylor series expansion as: \[f(x + h) = f(x) + hf'(x) + \frac{h^2}{2!}f''(x) + \frac{h^3}{3!}f'''(x) + \cdots\] where \(h = \Delta{x}\) is the step size, and \(f'(x), f''(x),\) etc. represent the first, second, and higher-order derivatives of the function. For the finite difference method, we mainly consider the first few terms for the approximation of a derivative. Now, let's take the forward difference approximation: \[f'(x)\approx\frac{f(x+h)-f(x)}{h}\] Rearranging the Taylor series expansion to obtain the error term, we have: \[\frac{f(x + h) - f(x)}{h} = f'(x)+\frac{h}{2!}f''(x) + \cdots = f'(x) + e_L\] where \(e_L\) is the local discretization error. It can be seen that the local discretization error is proportional to the square of the step size (\(h^2\)). The local discretization error decreases as the square of the step size decreases.

Global Discretization Error and Step Size

Now, let's focus on the global discretization error. When applying the finite difference method over an interval, we divide the interval into equal partitions with step size h. The total error will be the sum of the local discretization errors at each step. Let's denote the global discretization error as \(e_G\). Since there are N steps in total and the local discretization error at each step is proportional to \(h^2\), we have: \[e_G \propto N \times h^2 = \frac{(b - a)}{h} \times h^2 = h \times (b - a)\] Here, (b - a) represents the width of the interval, \((x_0,x_N)\). It can be seen that the global discretization error is proportional to the step size h itself. As the step size decreases, the global discretization error also decreases. In summary, the local discretization error of the finite difference method is proportional to the square of the step size, while the global discretization error is proportional to the step size itself.