Computing spatial derivatives with BDF

[TheCanadian]

Using the method of lines, I am solving a system of equations of the form:
$$
\begin{aligned}
\frac {\partial E}{\partial t} &= u
\\
\frac {\partial u}{\partial t} &= u + \frac {\partial E}{\partial z} - \frac {\partial^2 E}{\partial z^2}
\end{aligned}
$$
Looking at the particular formulae involved, it seems straightforward. Although I am having a slight problem trying to actually implement this when I have spatial derivatives involved. For example, I could approximate the spatial derivatives by central differences (e.g. ##\frac {\partial^2 E}{\partial z^2} \approx \frac {E^{k+1}_i -2E^k_i +E^{k-1}_i}{(\Delta z)^2} ##), but then would need to find an approximation for the terms ##E^{k+1}_{i+1}##, which is a step beyond the current ##E^{k}_{i+1}## I am looking for. And the aforementioned central difference approximation is only of order 2. I have considered using the backward difference approximations instead of the central differences, but is such a routine suggested? If I wanted to stay consistent with the BDF order, it would become very cumbersome using higher order backward differences, no? Is there anything further I am missing which could help simplify the implementation, especially for higher order BDF?

 

[jvicens]

Thank you for your question. The method of lines is a commonly used approach for solving partial differential equations (PDEs) numerically. As you have correctly pointed out, the challenge in implementing this method lies in approximating the spatial derivatives accurately.

Firstly, I would like to clarify that using central differences for approximating the spatial derivatives is a valid approach. However, as you have mentioned, it is only second-order accurate and can lead to numerical instabilities for certain problems. Therefore, it is important to consider alternative methods for approximating the spatial derivatives.

One option is to use higher-order central difference approximations, such as the fourth-order accurate centered difference formula. This can be achieved by taking more points into account when calculating the derivative, for example:

$$
\frac {\partial^2 E}{\partial z^2} \approx \frac {E^{k+2}_i -4E^{k+1}_i +6E^k_i -4E^{k-1}_i +E^{k-2}_i}{(\Delta z)^4}
$$

This approach can improve the accuracy of the spatial derivatives, but it also increases the complexity of the implementation.

Another option is to use a spectral method, such as the Fourier method, to approximate the spatial derivatives. This approach is based on representing the solution as a sum of trigonometric functions and using the properties of these functions to calculate the derivatives. This method can achieve higher-order accuracy and is often used for problems with smooth solutions.

Regarding the use of backward difference approximations, it is not necessary to use them to stay consistent with the BDF order. As long as the time integration scheme is consistent with the spatial discretization scheme, the overall accuracy of the method should not be affected.

In conclusion, there is no one-size-fits-all solution for implementing the method of lines for PDEs. It is important to carefully consider the properties of the problem and choose an appropriate discretization scheme to ensure accurate and stable solutions. I hope this helps to clarify your doubts. Please feel free to reach out if you have any further questions.