Splitting a second order PDE into a system of first order PDEs/ODEs

[nkinar]

In my research, I'm using a modified version of the wave equation:


$$\[ c^2 \left( {\frac{{\partial ^2 p}}{{\partial x^2 }} + \frac{{\partial ^2 p}}{{\partial y^2 }}} \right) = - \tau c^2 \left( {\frac{{\partial ^3 p}}{{\partial t\partial x^2 }} + \frac{{\partial ^3 p}}{{\partial t\partial y^2 }}} \right) + \frac{{\partial ^2 p}}{{\partial t^2 }} \]$$



I would like to take this PDE, and split the equation into a system of PDEs or ODEs. There is a PDF document on the internet which deals with this type of splitting on page 4, but I do not understand what is being mentioned when the author writes about an "auxiliary field."

Here is a link to the PDF:

http://math.mit.edu/~stevenj/18.369/pml.pdf

In this PDF, the author gives the source-free scalar wave equation:

$$\[ \nabla \cdot \left( {a\nabla u} \right) = \frac{1}{b}\frac{{\partial ^2 u}}{{\partial t^2 }} = \frac{{\ddot u}}{b} \]$$


The author then introduces an "auxiliary field", and re-writes the source-free scalar wave equation as the system of two coupled PDEs:

$$\[ \frac{{\partial u}}{{\partial t}} = b\nabla \cdot {\bf{v}} \]$$


$$\[ \frac{{\partial {\bf{v}}}}{{\partial t}} = a\nabla u \]$$


I would like to do the same for my modified version of the wave equation, but I am uncertain as how to deal with the mixed partial derivatives.

 

[matematikawan]

Thanks nkinar for the link. The note look interesting to read.

But I can't help with your problem. I'm myself trying to learn something about wave equation

 

[Marin]

Hi there!

Try out the following substitution from your pdf. file:

$$\frac{\partial v}{\partial t}=c^2\nabla p$$
$$\frac{\partial p}{\partial t}=\nabla v+\tau c^2\nabla^2p$$

which leads to the matrix form:

$$\frac{\partial}{\partial t}\left(\begin{array}{c}v\\p\end{array}\right)=\left(\begin{array}{cc}0&c^2\nabla\\\nabla&\tau c^2\nabla^2\end{array}\right)\left(\begin{array}{c}v\\p\end{array}\right)$$


In the pdf it's written that the matrix should be anti-hermitian in order the PDE to describe a wave. I am not sure but it seems to me that this one is not, you have to check it :)

the term 'ausxiliary field' sounds to me like an 'adequate substitution'. In this case it must be a vector field, since the Laplacian - nabla squared is div grad and the gradient field is a vector field.

 

[nkinar]

matematikawan: I am glad that you found the PDF interesting to read. That particular PDF discusses how to add Perfectly Matched Layer (PML) boundaries on the computational domain. Adding this type of boundary is useful when dealing with numerical physics problems which occur in the environmental sciences, where the computational domain is "unbounded." An example of this type of problem might be a numerical modeling problem of sound propagation in the ocean.

Hi Marin!

Thank you so much for your response, and for the substitution! I just independently verified that your substitution is indeed correct.

I checked the matrix to see if it was antihermitian; it does not appear to be antihermitian, but it may be possible to re-write it so that it is antihermitian.

Thank you so much for this, Marin!