User nicholas kinar - MathOverflow

Delta notation used for describing numerical stencil

2010-08-24T20:08:13Z

While reading some papers translated from the Russian literature, I've noticed that a delta symbol can be used to denote a FDTD stencil that discretizes a PDE. For example, in [1], a fourth order mixed partial derivative term is denoted by

$ 2\frac{{\partial ^4 u}}{{\partial ^2 x\partial ^2 y}} = \Delta _{xy}^4 u^{k + 1} _{i + 1,j + 1} + \Delta _{xy}^4 u^k _{i - 1,j - 1} $

where an example is given of

$\Delta _{xy}^4 u_{i + 1,j + 1} = \Delta _x^2 u_{i + 1,j + 2} - 2\Delta _x^2 u_{i + 1,j + 1} + \Delta _x^2 u_{i + 1,j}$

Notice that this example given in the paper does not have the $\{ k,k + 1\} $ superscipts.

Clearly ${i,j}$ are spatial indices and $k$ is the timestep. But what is being implied by the use of the delta symbol? I suspect that this is a differential, but I have never seen a differential with $u_{i,j}$ and $i,j$ indices. The author does not define the symbol in his paper, so I think that it should be implicitly understood. I am also unsure as to whether such a notation has also been used by other authors.

How would I write out $\Delta _{xy}^4 u_{i + 1,j + 1}$ and $\Delta _{xy}^4 u_{i - 1,j - 1}$ using a 5-point stencil or 7-point stencil? Are there any other papers which use similar notation?

[1] V. Saul'yev, “A difference method for solving parabolic equations of any order,” Computational Mathematics and Mathematical Physics, vol. 36(12), 1996, pp. 1697-1700.

Splitting wave equation for application of CPML

2010-08-03T21:16:53Z

A recent paper (Roden and Gedney, 2000) proposed the application of a Convolutional Perfectly Matched Layer (CPML) to approximate free-field conditions for Finite-Difference Time-Domain (FDTD) modelling in computational electromagnetics. On page 3 of this paper, the authors take the x-projection of Ampere's Law and show how the CPML can be nicely applied as a recursive convolution in the time domain.

I would like to apply this convolution to the wave equation

$ \frac{{\partial ^2 p}}{{\partial x^2 }} + \frac{{\partial ^2 p}}{{\partial y^2 }} = A\frac{{\partial ^2 p}}{{\partial t^2 }} + B\frac{{\partial p}}{{\partial t}} $

However, the wave equation must be re-written in terms of the x-projection and the y-projection. This leads me to my question:

In a similar fashion to the x-projection equation for Ampere's Law given in the paper by Roden and Gedney, is it possible to write an equation for the x-projection, and an equation for the y-projection of the wave equation given above? The terms "x-projection" and "y-projection" are used here in a similar fashion to the paper by Roden and Gedney.

Application of coordinate-stretching transformation for Perfectly Matched Layer

2010-07-24T20:12:25Z

A Perfectly Matched Layer (PML) is an absorbing boundary condition (ABC) which can be used to approximate free-field conditions for the numerical solution of wave equation problems.

PML note

The PML is normally applied to a PDE using the following transformation:

$ \frac{\partial }{{\partial x}} \to \frac{1}{{1 + i\frac{{\sigma (x)}}{\omega }}}\frac{\partial }{{\partial x}} $

In the above, $i = \sqrt { - 1}$ and $\sigma(x)$ is a function of position in the ABC. Now apparently

$ \frac{{\partial ^2 }}{{\partial x^2 }} \to \frac{1}{s}\frac{\partial }{{\partial x}}\left( {\frac{1}{s}\frac{\partial }{{\partial x}}} \right) $

$ s = 1 + i\frac{{\sigma (x)}}{\omega } $

But is it possible to apply a coordinate stretching in the following fashion: $ \frac{{\partial ^2 }}{{\partial x^2 }} \to \frac{1}{u}\frac{{\partial ^2 }}{{\partial x^2 }} $

The coordinate stretching performed in this fashion would be similar in function to the stretching performed by the transformation applied to $\partial /\partial x$.

Essentially what I would like to do is to apply the coordinate stretching directly to $\partial ^2 /\partial x^2$. I've looked in the PML literature for a very long time, and it seems that most interest is in the application of the coordinate stretching directly to $\partial /\partial x$.

Moreover, I can imagine the coordinate-stretching occurring in a similar fashion to stretching a rubber sheet. If the stretching is being done to the coordinates of $\partial /\partial x$, then what is happening to $\partial ^2 /\partial x^2$?

Answer by Nicholas Kinar for Application of coordinate-stretching transformation for Perfectly Matched Layer

2010-07-25T21:27:00Z

The complex contour $\tilde x$ is given by

$ \tilde x(x) = x + if(x) $

In the above, $x$ is the real part of the contour, $i = \sqrt { - 1}$, and $\frac{{\partial f}}{{\partial x}} = \frac{{\sigma (x)}}{\omega }$

Then

$ \frac{{\partial \tilde x(x)}}{{\partial x}} = 1 + i\frac{{\partial f(x)}}{{\partial x}} $

$ \frac{{\partial ^2 \tilde x(x)}}{{\partial x^2 }} = i\frac{{\partial ^2 f(x)}}{{\partial x^2 }} $

Now applying this to the differential along the deformed contour, the transformation becomes:

$ \frac{{\partial ^2 }}{{\partial x^2 }} \to \frac{1}{{i\frac{{\partial ^2 f(x)}}{{\partial x^2 }}}}\frac{{\partial ^2 }}{{\partial x^2 }} $

Answer by Nicholas Kinar for creating high quality figures of surfaces

2010-07-25T20:11:06Z

Perhaps VTK (the Visualization Toolkit) from Kitware? You can set up interactive windows to easily shift camera position of 3D surfaces.

VTK

Another suggestion could very well be Paraview:

Paraview

Boundary conditions of wave equation near infinity

2010-07-24T01:55:40Z

For the following wave equation

$ \frac{{\partial ^2 p}}{{\partial ^2 x}} + \frac{{\partial ^2 p}}{{\partial ^2 y}} = A\frac{{\partial ^2 p}}{{\partial ^2 t}} + B\frac{{\partial p}}{{\partial t}} $

is there a way to show that there are boundary conditions at or near positive and negative infinity, for both non-zero B and B=0 conditions, and for {A,B} as rational numbers? I believe that this should follow from Sommerfeld's condition of radiation, and should perhaps be similar to conditions for the ordinary wave equation. What are these boundary conditions? Ideally, I think that the boundary conditions should involve both time and spatial derivatives.

By "positive and negative infinity" I mean that I am interested in what happens when $x \to \pm \infty $ and $y \to \pm \infty$. I've been working on a problem where I would like to computationally solve the wave equation with boundary conditions that approximate infinity. So I suppose that this would be an imposed compatibility condition.

Comment by Nicholas Kinar

2012-08-01T14:20:11Z

@Douglas: OK, thanks for pointing me in the right direction.

Comment by Nicholas Kinar

2012-07-31T14:03:47Z

@DouglasZare: The gist of the matter is that I am searching for an operation on $P\exp(x)$ to turn it into $P\exp(kx)$ for an applied math research project that I'm working on this summer. I am wondering if such an operation exists. Perhaps the question does not belong on this site. Could the question be moved to math.stackexchange or deleted from this site?

Comment by Nicholas Kinar

2012-07-31T05:10:53Z

I had to edit this question since I had forgotten the $P$. Could this question be re-opened? It is really due to a typing error that I had forgotten the $P$.

Comment by Nicholas Kinar

2010-08-25T03:05:13Z

It also appears that $C_1 = (1 + 7r)d$ and $d = (1 - 7r)^{ - 1}$

Comment by Nicholas Kinar

2010-08-25T02:40:43Z

Thank you Federico. It is somewhat strange, but with the definition that you give, Equation 17 of the paper seems to be consistent. Another possibility is that since the paper was translated from Russian into English, there was an error made by the typesetter. I've tried to request the original version of the paper for comparison, since there is a possibility that something was left out. Moreover, I strongly suspect that the $u_{i,j + 2}^k$ term of Equation 17 should be $u_{i,j - 2}^k$, and that the $\{ C_2 , \ldots C_6 \}$ coefficients are of the wrong sign.

Comment by Nicholas Kinar

2010-08-04T02:20:01Z

Using the properties of convolution, I applied the Roden and Gedney CPML directly to the scalar wave equation given in my original posting. I found that when applied to my scalar wave equation, the CPML will absorb waves traveling in the x-direction and y-direction (perpendicular to the CPML). Waves arriving at other angles of incidence were reflected. This may be the reason why the CPML in the Roden and Gedney paper was applied only to the x-projection and the y-projection equations.

Comment by Nicholas Kinar

2010-08-03T21:58:16Z

That's a good question, Willie. I would like to do something similar to Roden and Gedney, but all that I have is a scalar wave equation. Might there be a way to start with a similar (vector) equation and then take an x-projection and a y-projection?

Comment by Nicholas Kinar

2010-07-26T01:23:52Z

Thank you for pointing this out; this allows me to better understand what I should do next and how I should structure my further research into problems involving the PML.

Comment by Nicholas Kinar

2010-07-25T21:44:23Z

@Willie: Okay, then could you follow a similar transformation and post an alternate answer?

Comment by Nicholas Kinar

2010-07-25T20:15:10Z

@Willie: I think that you are right; I'll edit the question above in an attempt to be a little more clear. Thank you for suggesting this.

Comment by Nicholas Kinar

2010-07-25T15:21:41Z

@Willie: I agree that the restriction to odd dimensions is indeed interesting. Thanks for pointing this out.

Comment by Nicholas Kinar

2010-07-24T23:45:26Z

Thanks Piero; that is a fine paper.

Comment by Nicholas Kinar

2010-07-24T19:35:01Z

Sure, number (2) is exactly what I am looking for; so thank you! Could you clarify what is meant by "infinity" as per Willie's comment above? Could you suggest a reference (book/paper/monograph) dealing with application of Kelvin transforms?