Need help solving 1st order pde numerically

[re444]

Hi everybody,

I need to solve a 1st order PDE for my thesis and I'm not a specialist in this field.
I've read some texts about this and know one method of solving a 1st order PDE is the method of characteristics. since my equation is nonlinear and a bit complicated, I'm going to solve it numerically.

Equation should be solved in x-z domain so I have to solve a system of five ODEs and is a boundary problem (Cauchy problem). According to the text found in internet the five are of the form:

main formula:

ODEs to be solved:

I omit the 5 boundary equations which are of the form for example for q, q(r,0)=g(r).

In the case of analytical solution at the end, u, the answer, will become: u=u(x,z). But when trying to solve numerically I'm a bit confused. How I should treat the parameter 'r' in ODEs? Software s like Matlab solves system of ODEs but this kind of ODEs with two parameters, 'r' and 's', seems strange to me!

Any suggestion and hint is really appreciated,
Reza,

 

[HallsofIvy]

These are all functions of r and s but you only have equations involving the derivative with respect to r? You will not be able to get a solution because you have no way to find the functions dependence on s.

 

[re444]

Then How should I treat with my problem? Should I test some other methods? How about "Vanishing Viscosity method" ?

 

[Chestermiller]

Hi everybody,

I need to solve a 1st order PDE for my thesis and I'm not a specialist in this field.
I've read some texts about this and know one method of solving a 1st order PDE is the method of characteristics. since my equation is nonlinear and a bit complicated, I'm going to solve it numerically.

Equation should be solved in x-z domain so I have to solve a system of five ODEs and is a boundary problem (Cauchy problem). According to the text found in internet the five are of the form:

main formula:

ODEs to be solved:

I omit the 5 boundary equations which are of the form for example for q, q(r,0)=g(r).

In the case of analytical solution at the end, u, the answer, will become: u=u(x,z). But when trying to solve numerically I'm a bit confused. How I should treat the parameter 'r' in ODEs? Software s like Matlab solves system of ODEs but this kind of ODEs with two parameters, 'r' and 's', seems strange to me!

Any suggestion and hint is really appreciated,
Reza,

What is the definition of r in this development? I assume that the equations you have listed were derived using the method of characteristics. Correct? Would it be possible to list the original PDEs and boundary conditions?

 

[re444]

What is the definition of r in this development? I assume that the equations you have listed were derived using the method of characteristics. Correct? Would it be possible to list the original PDEs and boundary conditions?

Yes sure, the original PDE is:

in which $\alpha_{i}$s are medium related parameters.

The formulation of the characteristic method I'm using is from here.

thanks,
Reza

 

[Chestermiller]

Thanks for sending me the write-up on the method of characteristics. It was a little complicated, and I didn't have the determination or time to go through the details. But I do understand the questions you are asking as well as the equations you are working with. The parameters r and s are independent variables, and x and y are parameterized in terms of r and s. You are integrating along lines of constant r in the s direction. You will be solving the set of differential equations over and over again, starting at location s = 0, but for different values of r. This is how you fill in the function space of all the dependent variables as a function of r and s. But, for this to work, you need to know the dependent variable values at the appropriate x and z values that correspond to the curve s = 0 at the different values of r along that contour. I hope this makes sense. The key is that you have to solve the ordinary differential equations over and over again for various values of r.

Chet