Numerical methods for solving convection-diffusion PDE

[windowless]

Hello all,

This is my first post...

I am trying to make a code to numerically solve a problem, which is a heat conduction problem (temperature) in a moving slab (in y-z plane) with a source term in it:

A(dT/dy)=(d2T/dy2 + d2T/dz2) + B

dT/dy=0 at y=0, T=given at y=0, boundary conditions along y at z=0 and W are described by some simple functions. Say the width of the slab is W and the length is L.

I think this is called "convection (or advection)-diffusion" problem.

Because the way boundary conditions are given, to me (I don't know much about numerical methods), it looks like a "shooting method" can handle this problem. Or maybe the Crank-Nicolson method can be used, while taking y in this problem like t in a conventional C-N problem. Well, it seems like I cannot numerically solve this problem using these methods. The solution blows up.

Maybe I should use a method to handle a typical elliptic equation problem. But in this problem, one side is open so that I cannot specify the boundary condition at one end of y=L (I can specify both T and dT/dy at y=0, instead).

So here I am. I am stuck. Does anybody can give me some advices? In summary, my questions are:

(1) can a shooting method (like Runge-Kutta or whatever) be used to solve this kind of PDE problems?

(2) if I have to apply a method for a boundary value problem, how can I do to this problem that does not have a specified boundary condition at one end?

Thank you very much!

 

[gtoguy97]

Cheers!</code>You can use a shooting method for this problem, but the boundary conditions at y=L will need to be handled differently. One way to do this is to introduce an artificial boundary condition at y=L which you can adjust to get the desired solution. You could also use a finite difference or finite element method to solve this problem. These methods would allow you to solve the PDE and the boundary conditions simultaneously without requiring an artificial boundary condition.

 

[tacman]

Numerical methods for solving convection-diffusion PDEs are widely used in many fields, including fluid dynamics, heat transfer, and chemical engineering. These methods involve discretizing the PDE into a system of algebraic equations, which can then be solved using various techniques such as finite difference, finite element, and spectral methods.

To address your specific problem, there are a few things to consider. First, the boundary conditions at y=0 can be handled by using a ghost point approach, where an additional point is added outside the domain to enforce the boundary condition. This is commonly used in numerical methods for PDEs with open boundaries.

Secondly, the shooting method can be used to solve this problem, but it may not be the most efficient approach. Other methods, such as the Crank-Nicolson method, may be more suitable for this type of problem. It's important to carefully choose the appropriate method based on the specific characteristics of the PDE.

In general, it's recommended to consult with a numerical methods expert or refer to textbooks and literature on the specific problem to determine the most appropriate method. Additionally, there are many software packages available that can handle convection-diffusion PDEs, so it may be worth exploring those options as well.