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I need to solve the following hyperbolic equation in x and phi co-ordinates

$$\frac{\partial \left ( -s/f \right )}{\partial \varphi }+\frac{\partial \left ( 1/f \right )}{\partial x}=0$$

$$\varphi >0\rightarrow f=\left ( 1+\frac{\left ( 1-s \right )\left ( 1+0.4\varphi e^{-x} \right )}{10s} \right )^{-1} $$

$$\varphi <0\rightarrow f=\left ( 1+\frac{1-s}{10s} \right )^{-1}$$

$$\varphi =-0.1x\rightarrow s=0.1,f=0.526$$ $$x=0,\varphi >0\rightarrow s=1,f=1$$

in domain $$0<x<1$$ $$-0.1x<\varphi <1 $$

I am searching for the hyperbolic package that can solve this equation like clawpack, but I don't know how to define my equation there thanks

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  • $\begingroup$ yes David , you are right, in the problem I want to solve at initial condition f=0 and 1/f tends to infinity but lets here assume at t=0 s=0.1 and f=0.5 $\endgroup$ – Sara Borazjani Oct 22 '14 at 6:26
  • $\begingroup$ David I edit the equation a little bit, this is exactly the one that I need to solve as you can see its in x and phi coordinate instead of x and time, and I need to deal with different f when phi is negative $\endgroup$ – Sara Borazjani Oct 22 '14 at 8:48
  • $\begingroup$ David,the initial condition is at phi=-0.1x, s=0.1 and the boundary condition is at x=0 $\endgroup$ – Sara Borazjani Oct 22 '14 at 10:31
  • $\begingroup$ It is much clearer now! Are you interested in solutions after characteristics cross? Do you have some basis for determining a meaningful weak solution after characteristics cross? $\endgroup$ – David Ketcheson Oct 23 '14 at 7:31
  • $\begingroup$ I am familiar with the analytical solution (self similar and method of characteristics) of first order hyperbolic equations and shock condition, but I never solved them numerically.do you think I can solve this equation by method of characteristics? $\endgroup$ – Sara Borazjani Oct 23 '14 at 12:13
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Since you have a scalar problem in one space dimension, it shouldn't be too difficult to come up with a semi-analytic solution using the method of characteristics. Here's something to get you started.

The problem can be written in a nicer form if you define $g=1/f$. Then you can solve for $s$ to find

$$s(g,x,\phi) = \frac{\alpha(x,\phi)}{\alpha(x,\phi) + g - 1}$$

where

$$\alpha(x,\phi) = \begin{cases} \frac{1+\frac{4}{10}\phi e^{-x}}{10} & \phi>0 \\ \frac{1}{10} & \phi \le 0.\end{cases}$$

Then your conservation law is simply

$$g_x - \left( s(g,x,\phi) g \right)_\phi = 0,$$

which looks like a traditional scalar hyperbolic PDE. The method of characteristics will allow you to find solutions up to the "time" when characteristics cross. You will probably need a Newton solver and an ODE solver to implement the method.

In case you want to compute an approximate solution by numerical discretization, I recommend the following:

  • If you know your solutions do not contain shocks, then I would just code up a simple finite difference discretization of the equation above (for instance, using Python or MATLAB and centered differences).
  • If you need to compute solutions with shocks, then more sophisticated methods are called for. You might consider using PyClaw (disclaimer: I'm a PyClaw developer).
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  • $\begingroup$ I am working on the semi analytic solution but do you know any package that can solve this numerically, to compare the result $\endgroup$ – Sara Borazjani Oct 27 '14 at 22:54
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This is just a small addition to David Ketcheson's comments about solving numerically.

If you want to obtain a numerical solution using MATLAB, Lawrence Shampine has written a MATLAB solver for hyperbolic PDEs that is designed to be (relatively) easy to use. If you look about half-way down this page:

http://faculty.smu.edu/shampine/current.html

you will see a section titled "Hyperbolic PDEs" that has links to his paper and MATLAB code for this capability. Because it is written entirely in MATLAB code, the "installation" is trivial.

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  • $\begingroup$ +1. But the difficulty with using this (or most other) software is that the problem has "initial values" given along a line that does not align with either of the coordinate axes. The easiest approach would probably be to rotate coordinates before using something like this. $\endgroup$ – David Ketcheson Nov 2 '14 at 4:57
  • $\begingroup$ yes David, you are right, I need to rotate the initial condition line $\endgroup$ – Sara Borazjani Nov 3 '14 at 23:28

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