Chapter 4 Numerical Techniques for Parabolic, Elliptic ...· Page 4-1 Chapter 4 Numerical Techniques

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Transcript of Chapter 4 Numerical Techniques for Parabolic, Elliptic ...· Page 4-1 Chapter 4 Numerical Techniques

  • r Parabolic,DEs

    (4-1)

    (4-2)

    ui 1n+ )

    tx)2---------

    12---

    Page 4-1

    Chapter 4 Numerical Techniques foElliptic, and Hyperbolic P

    4-1 Parabolic PDEs in 1D

    (1) Explicit Methods

    (a) FTCS

    (b) The DuFort-Frankel Scheme

    CTCS but in the diffusion term is replaced by

    tu

    x2

    2

    u

    =

    uin 1+ ui

    n tx( )2

    -------------- ui 1+n 2ui

    n(+=

    O t( ) x( )2,[ ]

    Stability Requires(

    -----i-1 i i+1

    n

    n+1

    uin

    uin

    uin 1+ ui

    n 1+

    2---------------------------------= ?????

  • (4-3)

    (4-4)

    called time linear extrapolation?

    t)] t( ) O t( )2[ ]+

    )2]

    1-----

    1+ ui 1n+ )

    Page 4-2

    Digression:

    Accordingly,

    or

    uin

    uin 1+ ui

    n 1+

    2-----------------------------------= ui

    n 1+ uin ui

    n uin 1= ???

    uin 1+ ui

    nt

    u

    i

    nO t( )2[ ]+ + ui

    nui

    n uin 1( )

    t------------------------------- O ([++= =

    uin 1+ ui

    n uin ui

    n 1 O t([+=

    uin 1+ ui

    n 1

    2 t( )----------------------------------

    ui 1+n 2

    uin 1+ ui

    n 1+

    2----------------------------------- ui

    n+

    x( )2------------------------------------------------------------------------=

    12 t( )x( )2

    ------------------+ uin 1+ 1

    2 t( )x( )2

    ------------------ uin 1 2 t( )

    x( )2------------------ ui

    n(+=

  • (4-5)

    e rate, such that

    ol. to start with and 2

    (4-3)/(4-4) gives

    x)2 O x( )4[ ]+t

    n

    tx

    ---- 0

    t2

    2

    u

    A Hyp. Eq.=

    Page 4-3

    Discussions:

    (a)

    (b)Unconditionally stable(c)Consistency (??)

    Note: If and were to approach zero at the sam

    (d) Not a self-starting method, need 2 time level stime level storage space

    O t( )2 x( )2 tx------ 2, ,

    uin 1+ in time about ui

    n uin 1 in time about ui

    n

    ui 1+n in space about ui

    n ui 1n in space about ui

    n

    into

    tu

    t2

    2

    u t

    x------ 2 1

    6---

    t3

    3

    u

    t( )2 O t( )3[ ]+ ++ t2

    2

    u

    12------

    x4

    4

    u (+=

    4 5( )[ ]x 0t 0

    lim PDE Only if --

    x 0t 0

    lim

    t x

    tx------

    x 0t 0

    lim K then 4 5( ) becomest

    u Kt2

    2

    u

    +

  • icit Scheme)

    (4-6)

    (4-7)ui

    n 1+ uin

    1

    t( )2[ ]

    Page 4-4

    (2) Implicit Methods

    (a) BTCS (Fully/Euler Implicit Scheme)

    (b) Crank-Nicolson Scheme (Trapezoidal Impl

    (c) One-Parameter Formulation

    then if

    NOTE: If

    tu

    F r u , ,( )= e.g., Fx

    2

    2

    u=

    uin

    t----------

    tu

    i

    n 1+1 ( )

    tu

    i

    n+= where

    uin =

    0

    1 Euler Implicit= O t( )[ ]

    12--- Crank-Nicolson Implicit= O

    0 Euler Explicit= O t( )[ ]12--- 1 Unconditionally Stable

    0 12--- Conditionally Stable

  • (4-8)

    (4-9)

    er/nonlinear in u):

    linearization, e.g.,

    nO t( )2[ ]+

    E AuE

    and Bu

    F= =

    n Bnun O t( )2[ ]+ +

    Page 4-5

    Accordingly

    e.g.,

    d) Delta () Formulation

    Define

    For the following problem (fluxes E and F may be lin

    uin 1+ ui

    n

    t-------------------------- Fi

    n 1+ 1 ( )Fin+=

    uin 1+ ui

    n

    t--------------------------

    x2

    2

    u

    i

    n 1+

    1 ( )x

    2

    2

    u

    i

    n

    +=

    un un 1+ un=

    tu

    xE

    x2

    2

    F

    += where E and F = fn(t,x,u, ... ) then

    En 1+ En tt

    E n O t( )2[ ]+ + En t

    uE

    tu += =

    n 1+ En t An un

    t---------- O t( )[ ]+ O t( )2[ ]+ += where

    En 1+ En Anun O t( )2[ ]+ += similarly Fn 1+ F=

  • (4-10)

    (4-11)

    x

    Enx

    Fn+

    x

    Fn+

    Fn

    )x

    x

    un

    u

    2 O t( )2[ ]

    Page 4-6

    e.gs.,

    e) Two-Parameter Formulation

    un 1+ unt

    -------------------------- x

    En Anunx

    Fn Bnun+( )+ + 1 ( )+=

    x

    Anunx

    Bnun( )+x

    Enx

    Fn+ +=

    un t( )x

    Anunx

    Bnun( )+ t( )x

    En=

    1 t( )x

    Anx

    Bn+ un t( )

    x

    Enx

    +=

    tu

    x2

    2

    u

    = 1 t( )x

    x

    un t(=

    tu

    uxu

    +tu

    x u2

    2----- += 0= E u

    2

    2-----= A =

    1t1 2+--------------

    tu

    n 1+un+

    1 1( )t1 2+

    -------------------------t

    u

    n 21 2+--------------un 1 1

    12---

    + +=

    O t( )2[ ] if 1 212---+=

  • 4

    (4-12)

    Accuracy in Time

    O t( )[ ]

    O t( )2[ ]

    O t( )2[ ]

    O t( )[ ]

    O t( )2[ ]

    Page 4-7

    -2 Multi-Dimensional Parabolic PDEs

    e.g.,

    Scheme

    0 0 Euler Explicit Scheme

    0 - 0.5 Explicit Leapfrog Scheme

    0.5 0 Implicit Trapezoidal Scheme (C-N)

    1 0 Euler Implicit Scheme

    1 0.5 Three-Point Backward Implicit Scheme

    1 2

    tu

    x2

    2

    u

    y2

    2

    u

    +

    =

  • (4-13)i j, 1n

    ---------------

    i

    j

    n

    x

    y

    ty( )2

    --------------

    12--- or d

    14---

    Page 4-8

    (1) Explicit: FTCS

    (a)

    (b) Stability:

    Note:

    Inefficient and unattractive

    Implicit Methods

    ui j,n 1+ ui j,

    n

    t-----------------------------

    ui 1 j,+n 2ui j,

    n ui 1 j,n+

    x( )2-----------------------------------------------------------

    ui j, 1+n 2ui j,

    n u+

    y( )2--------------------------------------------+=

    yx

    i-1,j i,j i+1,j

    i,j-1

    i,j+1

    x

    y

    t

    O t( ) x( )2 y( )2, ,[ ]

    dx dy+( )12--- dx

    tx( )2

    --------------= dy =

    if x y The diffusion number dx dy d 2d = = =

  • (4-14)

    (4-15)

    (4-16)

    ed Matrix

    j1+ ui j, 1

    n 1++

    )2------------------------------

    i j 1+,n 1+ ui j,

    n=

    1+1 fi j,=

    .......................

    ........................

    ............................ .

    .

    .

    imax - 1 spaces

    Page 4-9

    (1) Implicit Methods:

    (a) Fully Implicit (Euler Implicit)

    or

    [(imax-1)2 x (jmax-1)

    2] Pentadiagonal band

    Inefficient and impreactical,

    ADI (Alternating-Direction-Implicit),

    AF (Approximate-Factorization),

    Fractional Step,

    etc.

    ui j,n 1+ ui j,

    n

    t-----------------------------

    ui 1 j,+n 1+ 2ui j,

    n 1+ ui 1 j,n 1++

    x( )2-----------------------------------------------------------------

    ui j, 1+n 1+ 2ui,

    n

    y(-----------------------------------+=

    dxui 1 j,+n 1+ dxui 1 j,

    n 1+ 2dx 2dy 1+ +( )ui j,n 1+ dyui j 1,

    n 1+ dyu+ ++

    ai j, ui 1 j,+n 1+ bi j, ui 1 j,

    n 1+ ci j, ui j,n 1+ di j, ui j 1,

    n 1+ ei j, ui j,n ++ + + +

    ..........

    .

    .

    .

  • (4-17)

    i j, ui j, 1+ )n

    y)2-----------------------------------

    i j, ui j, 1+ )n 1+

    y)2-----------------------------------------

    n+1

    n+1/2

    n

    t/2

    t/2

    Page 4-10

    (b) ADI (Alternating-Direction-Implicit)

    Discussions:

    (a)(b) Unconditionally Stable(c) Two sweeps, one along x, another

    along y, each with tridiagonal matrix(d) BC for n+1/2 ? if time dependent.

    ui j,n

    12---+

    ui j,n

    t2-----

    ----------------------------- ui 1 j,+ 2ui j, ui 1 j,+( )

    n12---+

    x( )2---------------------------------------------------------------------------

    ui j, 1+ 2u(

    (-------------------------------+=

    Solve for ui j,n

    12---+

    and then

    ui j,n 1+ ui j,

    n12---+

    t2-----

    ----------------------------------- ui 1 j,+ 2ui j, ui 1 j,+( )

    n12---+

    x( )2---------------------------------------------------------------------------

    ui j, 1+ 2u(

    (---------------------------------+=

    to get ui j,n 1+( )

    O t( )2 x( )2 y( )2, ,[ ]

  • (4-18)

    (4-19)

    AF

    tu

    i j,

    n

    tx)2------------

    ty)2------------

    j dyy2ui j,

    n+ )

    Page 4-11

    (c) AF (Approximate-Factorization)

    Recall

    Define

    then for , we have

    ADI AF and LU Decomposition

    ui j,n

    t-------------

    tu

    i j,

    n 1+1 ( )

    tu

    i j,

    n+=

    ui j,n 1+ t( )

    tu

    i j,

    n 1+ ui j,

    n 1 ( ) t( )+=

    x2ui j, ui 1+ j, 2ui j, ui 1 j,+= dx (

    --=

    y2ui j, ui j 1+, 2ui j, ui j 1,+= dy (

    --=

    tu

    x2

    2

    u

    y2

    2

    u

    +

    =

    ui j,n 1+ dxx

    2ui j,n 1+ dyy

    2ui j,n 1++( ) ui j,

    n 1 ( ) dxx2ui,

    n(+=

  • (4-20)

    (4-21)

    (4-22)

    quence

    (4-23)

    (4-24)

    (4-25)

    ui j,*

    i j,n 1+

    Page 4-12

    or

    where

    By AF, (4-20) becomes

    (4-22) is solved according to the following se

    Discussions:(a) (4-24) into (4-23), gives

    1 dxx2 dyy

    2+( )[ ]ui j,n 1+ RHS( )i j,

    n=

    RHS( )i j,n 1 1 ( ) dxx

    2 dyy2+( )+[ ]ui j,

    n=

    1 dxx2( ) 1 dyy

    2( )ui j,n 1+ RHS( )i j,

    n=

    1 dxx2( )ui j,

    * RHS( )i j,n= Solve for

    1 dyy2( )ui j,

    n 1+ ui j,*= Solve for u

    1 dxx2( ) 1 dyy

    2( )ui j,n 1+ RHS( )i j,

    n=