finite difference method - national university of …bao/teach/ma5233/lect10.pdf · finite...
TRANSCRIPT
![Page 1: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/1.jpg)
Finite difference method
Poisson equation in 2D with Dirichelt BC Well-posedness – Existence – Uniqueness
![Page 2: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/2.jpg)
Finite difference method
![Page 3: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/3.jpg)
Finite difference method
Finite difference approximation: Dimension-by-dimension
Computation & error analysis – Solve the linear system efficiently – Error bound???
![Page 4: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/4.jpg)
Local truncation error
![Page 5: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/5.jpg)
Finite difference method
Order of accuracy: second order Error analysis -- maximum principle – Proof: Exercise!!
Efficient solver – Iterative solvers: CG, PCG, Gauss-Seidel, SOR, ….. – Direct Poisson solver
![Page 6: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/6.jpg)
Linear system
![Page 7: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/7.jpg)
Linear system
![Page 8: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/8.jpg)
Matrix form
![Page 9: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/9.jpg)
Matrix form
![Page 10: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/10.jpg)
Matrix form
Linear system: Iterative solvers – Rewrite the difference equations as
![Page 11: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/11.jpg)
SOR method – An example
![Page 12: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/12.jpg)
Fast Poisson solver via DST
For Poisson equation in 1D with Dirichlet BC Finite difference discretization Linear system:
![Page 13: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/13.jpg)
Fast Poisson solver
Homogeneous BC: In PDE level – sine transform
![Page 14: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/14.jpg)
Fast Poisson solver
Exact solution
![Page 15: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/15.jpg)
Fast Poisson solver
In discrtization level – Discrete since transform (DST)
![Page 16: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/16.jpg)
Fast Poisson solver
![Page 17: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/17.jpg)
Algorithm for fast Poisson solver via DST
![Page 18: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/18.jpg)
Fast Poisson solver
Inhomogeneous BC --- Homogenizing the BC
Introducing Plugging into the difference equations: – With
![Page 19: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/19.jpg)
Algorithm for fast Poisson solver via DST
![Page 20: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/20.jpg)
Fast Poisson solver in 2D
The equation: The finite difference discretization
![Page 21: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/21.jpg)
Fast Poisson solver in 2D
Discrete sine transform in 2D Plugging into the finite difference equations
![Page 22: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/22.jpg)
Algorithm for Fast Poisson solver in 2D
![Page 23: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/23.jpg)
Algorithm for Fast Poisson solver in 2D
![Page 24: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/24.jpg)
Comments on fast Poisson solver
Advantages – Direct solver & give exact solution to the linear system – Memory cost: O(M) & no extra memory is needed!! – Computational cost: O(M ln M) – Very efficient in 2D & 3D due to FST!!! – Can be extended to Neumann BC or periodic BC
Disadvantages: – The domain should be a rectangle in 2D & box in 3D – Uniform mesh in each direction is needed!! – The coefficients of the PDE must be constant!! – BC much be the same type in opposite edges!!!
![Page 25: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/25.jpg)
Extension to Neumann BC
The problem:
![Page 26: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/26.jpg)
Extension to Neumann BC
Discretization – at shifted grids points by half grid Direct Poisson solver via discrete cosine transform (DCT) – Exercise!!
Extension to periodic BC – discrete Fourier transform (DFT)
![Page 27: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/27.jpg)
Poisson equation in 2D on a disk or shell
The problem The ideas: Domain mapping or variable transform
![Page 28: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/28.jpg)
The discretization
![Page 29: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/29.jpg)
The discretization for a shell
![Page 30: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/30.jpg)
The discretization for a disk
![Page 31: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/31.jpg)
The discretization
Local truncation error: Order of accuracy: second order Error estimate –-- maximum principle
Solution of the linear system – Fast Poisson solver
• Discrete Fourier transform in transverse direction • Solve a linear system with tri-diagonal coefficient matrix in r-direction
![Page 32: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/32.jpg)
Extension
3D Poisson equation – In a box, sphere, shell, cylindrical cylinder, etc.
General linear elliptic equation Elliptic system – Navier system, ……
![Page 33: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/33.jpg)
More topics
Compact scheme --- high order methods with less grid points • 4th (6th ,…) order methods need 5 (7, …) grid points in 1D • Need use ``ghost’’ points near boundary
– Question: Can we design high order methods using less grid points??? – An example in 1D:
– Extension to 2D & 3D: Exercise!!!!
2 21 12 4 4
2
12 22 4 2 4
122 2
( ) 2 ( ) ( )( ) ( ) ( ) ( ) ( ) ( )
12 12
( ) ( ) ( ) ( ) ( )12 12
( ) ( ) (12
j j jx j xx j xxxx j xx xx j
xx j xx x j xx x j
xx j x x j
u x u x u x h hu x u x u x O h I u x O hh
h hu x I u x O h I u x O h
hu x I u x O
δ
δ δ
δ δ
+ −
−
−
− + = = ∂ + ∂ + = ∂ + ∂ +
∂ = + ∂ + = − ∂ +
∂ = + +
24 2 2 4
1 1 1 102
) ( ) ( ) ( )12
2 10, 1,2,..., 1, ,
12
x j x j
j j j j j jM
hh u x I f x O h
u u u f f fj M u u
h
δ δ
α β+ − + −
⇒ − = + +
− + + +
− = = − = =
![Page 34: Finite difference method - National University of …bao/teach/ma5233/lect10.pdf · Finite difference method Finite difference approximation: ... Direct Poisson solver via discrete](https://reader031.vdocuments.net/reader031/viewer/2022020100/5b8786d27f8b9aaf728b8d4d/html5/thumbnails/34.jpg)
More topics
General geometry in 2D & 3D – FEM or FVM or Boundary integral method
Adaptive mesh refinement (AMR) – Solution has sharp change locally – Refine the mesh adaptively based on the approximation
Nonlinear problem – Discretization, solve nonlinear system
…..