Really great video! I have a question on the 'enclosed boundary' part of the video: given a finite-difference stencil for the Laplacian using points up-down-left-right, is it necessary to define the corners of your square enclosing boundary? I would think that, since the internal points do not 'see' these points in the corner, that they would not need to be included (i.e. the map function could be zero in the corners of the square?) Thanks again!
Great lecture. I am wondering about how one would go about incorporating periodic boundaries into this problem formulation. Can you point me in the right direction?
That is pretty easy. I recommend learning how the finite-difference method is being implemented here. That is taught in Computational Methods at the following link: empossible.net/academics/emp4301_5301/ Work through Lectures 6f - 6k and then Lectures 7a - 7i. You will even see periodic boundary conditions taught and incorporated into the derivative matrices. With this background, this lecture on Laplace's equation will be a lot easier to understand and customize.
@@empossible1577 Hi, I am back (again) :D I took me some time to get to it, but now I implemented the finite difference method for several boundary conditions. I wanted to compare with the example in this video as a first test of my code. Please correct me if I am wrong, but the first example in the video is using Neuman boundary conditions right? However, when I try to compute this example with Neuman BCs with my code I get lines with all zeros in the L matrix, meaning the inverse does not exist. I wrote down the 2nd derrivative matrices D2x and D2y, and it seems even in the derivation I am getting lines with all zeros due to the Neuman BCs when summing up D2x and D2y. I have checked the derivation and my code several times now, and I can't find a mistake. At this point I feel rather stupid and would appreciate if you took a look. I uploaded a plot of my derivative matrices for neuman boundary conditions (without the constant factors) for a grid with Nx=Ny=6 here (i.postimg.cc/VLw0zV7G/Dxy-neuman-Nx6-Ny6.png). Is there an error or am I missing something?
Really great video!
I have a question on the 'enclosed boundary' part of the video: given a finite-difference stencil for the Laplacian using points up-down-left-right, is it necessary to define the corners of your square enclosing boundary? I would think that, since the internal points do not 'see' these points in the corner, that they would not need to be included (i.e. the map function could be zero in the corners of the square?)
Thanks again!
I don't think those are needed. The finite-difference equations themselves only reach horizontally and vertically across the grid, not diagonally.
Great lecture. I am wondering about how one would go about incorporating periodic boundaries into this problem formulation. Can you point me in the right direction?
That is pretty easy. I recommend learning how the finite-difference method is being implemented here. That is taught in Computational Methods at the following link:
empossible.net/academics/emp4301_5301/
Work through Lectures 6f - 6k and then Lectures 7a - 7i. You will even see periodic boundary conditions taught and incorporated into the derivative matrices. With this background, this lecture on Laplace's equation will be a lot easier to understand and customize.
@@empossible1577 Thank you for the fast reply.
@@empossible1577 Hi, I am back (again) :D
I took me some time to get to it, but now I implemented the finite difference method for several boundary conditions. I wanted to compare with the example in this video as a first test of my code. Please correct me if I am wrong, but the first example in the video is using Neuman boundary conditions right? However, when I try to compute this example with Neuman BCs with my code I get lines with all zeros in the L matrix, meaning the inverse does not exist. I wrote down the 2nd derrivative matrices D2x and D2y, and it seems even in the derivation I am getting lines with all zeros due to the Neuman BCs when summing up D2x and D2y.
I have checked the derivation and my code several times now, and I can't find a mistake. At this point I feel rather stupid and would appreciate if you took a look. I uploaded a plot of my derivative matrices for neuman boundary conditions (without the constant factors) for a grid with Nx=Ny=6 here (i.postimg.cc/VLw0zV7G/Dxy-neuman-Nx6-Ny6.png). Is there an error or am I missing something?
good tutorials!
Thank you!
Thank you! You save my day : )
Glad I could help!