Help with the numerical method of characteristics

[kobjob]

I have a system of first order pdes of the form
dm/dt+f(m,n,a)dm/dx = g1(m,n,a),
dn/dt+f(m,n,a)dn/dx = g2(m,n,a),
da/dt+f(m,n,a)da/dx = g3(m,n,a).
(those d's are partials)

I want to solve them with the numerical MOC's so I put dx/dt = f(m,n,a), dm/dt = g1(m,n,a) etc and solve this system of 4 odes with the explicit euler method. I don't think I'm doing it properly though because my solution is way too different from the one I got using the upwinding method. I'm also not sure how to treat the moving bounday... in fact I'm not really sure about any of it so if someone could help me out/point me towards an appropriate book or website i would be really grateful.

Thanks,
kate

 

[jvicens]

Dear Kate,

Thank you for sharing your question on the forum. Solving systems of first order partial differential equations (PDEs) using the method of characteristics (MOCs) can be a challenging task, and it is important to carefully consider the approach you are taking. Here are some suggestions and resources that may help you in solving your system of PDEs.

1. Check your boundary conditions: The boundary conditions are crucial in solving any PDE, and it is important to make sure that they are properly specified for your problem. In the case of a moving boundary, you may need to use a different approach, such as the method of lines, to discretize your domain and solve the equations.

2. Understand the MOCs: Before applying the MOCs, it is important to have a good understanding of what they are and how they work. The MOCs are a method for solving first order PDEs, and they involve finding curves (characteristics) along which the solution is constant. These curves are determined by the equations themselves, and can be used to reduce the PDEs into a system of ordinary differential equations (ODEs). You can find more information on the MOCs in textbooks or online resources such as this one: https://math.berkeley.edu/~evans/OCam-Notes/OCam-Notes.html.

3. Consider using a different discretization method: While the explicit Euler method is a commonly used method for solving ODEs, it may not be the most appropriate choice for solving your system of PDEs. You may want to consider using a more accurate and stable method, such as the implicit Euler method or a Runge-Kutta method. These methods can be found in most numerical analysis textbooks.

4. Consult a numerical analysis textbook: There are many excellent textbooks that cover numerical methods for solving PDEs, such as "Numerical Methods for Partial Differential Equations" by George F. Pinder and "Numerical Methods for Partial Differential Equations" by William F. Ames. These textbooks provide a thorough introduction to numerical methods for PDEs and can serve as a helpful reference for your problem.

I hope these suggestions are helpful in guiding you towards an appropriate solution for your system of PDEs. It is also a good idea to consult with your colleagues or a supervisor for additional guidance and support. Good luck with your research!