partial differential equation, PDE by vxj14228

VIEWS: 123 PAGES: 17

									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