Upload
phungthuan
View
242
Download
2
Embed Size (px)
Citation preview
Finite difference method
Poisson equation in 2D with Dirichelt BC Well-posedness – Existence – Uniqueness
Finite difference method
Finite difference method
Finite difference approximation: Dimension-by-dimension
Computation & error analysis – Solve the linear system efficiently – Error bound???
Local truncation error
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
Linear system
Linear system
Matrix form
Matrix form
Matrix form
Linear system: Iterative solvers – Rewrite the difference equations as
SOR method – An example
Fast Poisson solver via DST
For Poisson equation in 1D with Dirichlet BC Finite difference discretization Linear system:
Fast Poisson solver
Homogeneous BC: In PDE level – sine transform
Fast Poisson solver
Exact solution
Fast Poisson solver
In discrtization level – Discrete since transform (DST)
Fast Poisson solver
Algorithm for fast Poisson solver via DST
Fast Poisson solver
Inhomogeneous BC --- Homogenizing the BC
Introducing Plugging into the difference equations: – With
Algorithm for fast Poisson solver via DST
Fast Poisson solver in 2D
The equation: The finite difference discretization
Fast Poisson solver in 2D
Discrete sine transform in 2D Plugging into the finite difference equations
Algorithm for Fast Poisson solver in 2D
Algorithm for Fast Poisson solver in 2D
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!!!
Extension to Neumann BC
The problem:
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)
Poisson equation in 2D on a disk or shell
The problem The ideas: Domain mapping or variable transform
The discretization
The discretization for a shell
The discretization for a disk
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
Extension
3D Poisson equation – In a box, sphere, shell, cylindrical cylinder, etc.
General linear elliptic equation Elliptic system – Navier system, ……
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+ − + −
⇒ − = + +
− + + +
− = = − = =
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
…..