# partial differential equation, PDE by vxj14228

VIEWS: 123 PAGES: 17

• pg 1
```									4.

partial differential equation, PDE
1                PDE                              ;

x
f
f
x
f (t x/c)

1
•       Finite difference method PDE
t

x
(1) FTCS (Forward in Time and Central difference Scheme       ).

t

x
t

Domain of dependence
x

Euler
Domain of dependence
(1) FTCS (Forward in Time and Central difference Scheme   ).

(2) Lax (- Friedrich) scheme                         .

(3) Leap-Flog scheme

(4) Lax-Wendroff scheme

(5) 1st order upwind scheme 1
• Lax, Lax-Wendroff, 1           FTCS scheme +.

cf)

τ   h
• Lax, Lax-Wendroff, 1                            FTCS scheme +.

(1)                 (FTCS :                                ).

(2) Lax (- Friedrich)       .

(3) Lax-Wendroff        .

(4) 1                   1st order upwind scheme    (               )

(                c                             )
τ   h

τ
PDE                                    ODE
•                 Local truncation error       :
PDE
ex.) One-step method.

•                Global discretization error

Definitions: (          Consistency)

Definitions: (          Convergence)
Definitions: (             Zero-Stability h        0                  )

Definitions: (             -h

Remark:
τ0(h)

C0(h)                  n

Theorem: (Lax’s equivalence theorem)
Convergence      Zero (or Absolute) - stability.
Zero (or Absolute) - stability and Consistency         Convergence.
PDE

j–1/2           j+1/2

j–1               j             j+1

j–1/2, j+1/2 ,                 flux   PDE

j-1/2, j+1/2…

Euler, Lax, Lax-Wendroff, 1st order upwind
(1)         Euler :
(1)         Euler :

(2) Lax :

(3) Lax-Wendroff :

(4) 1st order upwind :
von Neumann stability analysis.
•                                                PDE

PDE                    one-step formula      h
C0

t=0
von Neumann stability analysis.

PDE                  f (t x/c)

0
zero stability       :

absolute stability       :

PDE
1
N        N

l2       Perceval
4-1) Lax-Wendroff        2

4-2) FTCS, Lax, Lax-Wendroff, 1st order upwind,
PDE

(                            )

4-3) FTCS, Lax, Lax-Wendroff, 1st order upwind,

```
To top