Numerical evaluation of systems of ODEs

[SirJac]

I'm looking to do numerical evaluation of a system of differential equations and would like to use the RK4 method however I'm having a problem as my differential equations are respect to different variables and I don't know how to adapt RK4 to allow for that. The general form of the equations of motion that I want to evaluate is given below, where $\sigma$ and $\omega$ are functions of both z and t and A,B,C,D and E are all real constants.

$$\frac{\delta \sigma}{\delta t} = iAz \sigma + iB \omega$$
$$\frac{\delta \omega}{\delta z} = -i(C+Dz) \omega + iE \sigma$$

If anyone can't point me in the direction of any resources that consider numerical evaluation of these types of ODE systems, I would greatly appreciate it.

 

[jvicens]

Hello,

Thank you for your post. I am a scientist with experience in numerical methods for solving differential equations. I understand your concern about adapting the RK4 method for systems of differential equations with respect to different variables.

The RK4 method is a popular choice for numerical evaluation of systems of differential equations due to its simplicity and accuracy. However, it is important to note that the method is typically used for systems of equations with only one independent variable. In your case, you have two independent variables (z and t) and this makes it more challenging to adapt the RK4 method.

One approach you could consider is to transform your system of equations into a single equation with one independent variable. This can be done by using a change of variables, such as introducing a new variable u = \sigma + i\omega. This would result in a single equation with respect to u, which can then be solved using the RK4 method.

Another approach is to use the RK4 method separately for each variable. In this case, you would need to solve the first equation with respect to t and then use the solution to solve the second equation with respect to z. This approach may be more computationally intensive, but it can still yield accurate results.

There are also other numerical methods that are specifically designed for systems of differential equations with multiple independent variables. Some examples include the Runge-Kutta-Nyström method and the Adams-Bashforth-Moulton method. These methods may be more complex, but they can handle systems of equations with multiple independent variables.

I suggest looking into these methods and seeing which one best suits your needs. There are also many resources available online that discuss numerical methods for solving systems of differential equations, including those with multiple independent variables. I recommend checking out some textbooks or online lecture notes for more in-depth explanations and examples.

I hope this helps and best of luck with your numerical evaluation.