https://github.com/matnoble/pde_coding
FDM, FEM
https://github.com/matnoble/pde_coding
Last synced: about 2 months ago
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FDM, FEM
- Host: GitHub
- URL: https://github.com/matnoble/pde_coding
- Owner: MatNoble
- Created: 2019-06-21T14:28:17.000Z (almost 7 years ago)
- Default Branch: master
- Last Pushed: 2020-09-22T02:15:11.000Z (over 5 years ago)
- Last Synced: 2025-03-05T17:16:06.704Z (over 1 year ago)
- Language: MATLAB
- Homepage:
- Size: 14 MB
- Stars: 3
- Watchers: 1
- Forks: 1
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
Awesome Lists containing this project
README
## Some Results
### Landau–Lifshitz equation
### Marker and Cell Methond for Stokes Equations
## Using FDM or FEM solve poisson and heat equations in 1D and 2D
### Basic equations
poisson equation:
heat equation:
### Finite Difference Methods
>
- Replacing the dervatives by finite differences(to do finite difference scheme)
-2&space;u\left(x_{i},&space;y_{j}\right)+u\left(x_{i+1},&space;y_{j}\right)}{h_{x}^{2}}&space;\\&space;u_{y&space;y}&space;&&space;\approx&space;\frac{u\left(x_{i},&space;y_{j-1}\right)-2&space;u\left(x_{i},&space;y_{j}\right)+u\left(x_{i},&space;y_{j+1}\right)}{h_{y}^{2}}&space;\end{aligned} "\begin{aligned} u_{x x} & \approx \frac{u\left(x_{i-1}, y_{j}\right)-2 u\left(x_{i}, y_{j}\right)+u\left(x_{i+1}, y_{j}\right)}{h_{x}^{2}} \\ u_{y y} & \approx \frac{u\left(x_{i}, y_{j-1}\right)-2 u\left(x_{i}, y_{j}\right)+u\left(x_{i}, y_{j+1}\right)}{h_{y}^{2}} \end{aligned}")
- Taylor series(to do error estimate)
=\frac{f\left(x_{0}\right)}{0&space;!}+\frac{f^{\prime}\left(x_{0}\right)}{1&space;!}\left(x-x_{0}\right)+\ldots+\frac{f^{(n)}\left(x_{0}\right)}{n&space;!}\left(x-x_{0}\right)^{n}+R_{n}(x) "f(x)=\frac{f\left(x_{0}\right)}{0 !}+\frac{f^{\prime}\left(x_{0}\right)}{1 !}\left(x-x_{0}\right)+\ldots+\frac{f^{(n)}\left(x_{0}\right)}{n !}\left(x-x_{0}\right)^{n}+R_{n}(x)")
=\mathrm{o}\left[\left(x-x_{0}\right)^{n}\right] "R_{n}(x)=\mathrm{o}\left[\left(x-x_{0}\right)^{n}\right]")
#### Example 1
[for the 1D heat problem](https://github.com/MatNoble/PDE_coding/tree/master/FD/1D):
\left(x_{1}<x<x_{2}\right)(t>0)}&space;\\&space;{u(x,&space;0)=g_{t}(x)\left(x_{1}<x<x_{2}\right)}&space;\\&space;{u\left(x_{1},&space;t\right)=g_{1}(t),&space;u\left(x_{2},&space;t\right)=g_{2}(t)(t>0)}\end{array}\right. "\left\{\begin{array}{c}{u_{t}- u_{x x}=f(x, t)\left(x_{1}<x<x_{2}\right)(t>0)} \\ {u(x, 0)=g_{t}(x)\left(x_{1}<x<x_{2}\right)} \\ {u\left(x_{1}, t\right)=g_{1}(t), u\left(x_{2}, t\right)=g_{2}(t)(t>0)}\end{array}\right.")
with specific parameters:
=10&space;\frac{\cos&space;\left(\left(\frac{x-x_{1}}{2}\right)^{2}\right)&space;\sin&space;\left(\left(x-x_{2}\right)^{3}\right)}{x^{\frac{4}{5}}}\\ "u(x, 0)=10 \frac{\cos \left(\left(\frac{x-x_{1}}{2}\right)^{2}\right) \sin \left(\left(x-x_{2}\right)^{3}\right)}{x^{\frac{4}{5}}}\\")
=u\left(x_{2},&space;t\right)=0 "u\left(x_{1}, t\right)=u\left(x_{2}, t\right)=0")
=0 "f(x, t)=0")
have the flowing result:
### Finite Element Methods
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- Integration by part
- Variational
- Weak formula
- Garlekin formula
- Finite dimensional space approximates infinite dimensional space
### Approximate functions
Gibbs phenomenon
