Numerical Methods for Generalized Zakharov System

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Numerical Methods for Generalized Zakharov System Powered By Docstoc
					       FDM for parabolic equations
Consider the heat equation




– where
Well-posed problem
– Existence & Uniqueness
– Mass & Energy decreasing
FDM for parabolic equations
                                  CNFD
Crank-Nicolson + 2nd order finite difference




Questions
– How to solve the equations efficiently???
– Convergence and order of accuracy???
    • Local truncation error & Stability
Local truncation error
                     Linear system

Order of accuracy: 2nd in space and time
Consistency: yes!!!

Linear system

– With
Implicit scheme!!!
– At each time step, we need solve a linear system
Matrix form
Solution algorithm
                   Convergence analysis

Convergence
– Consistency & Stability
Consider the general problem
 u( x , t )
              L( u )     (0, T )   L  differential operator
    t
 g (u )  g 0             
 u( x ,0)  u0 ( x )    
It is a well-posed problem:
– Existence, uniqueness, continuously depend on initial data
          Finite difference discretization
Time step: k  t tn  n k , n  0,1, 2, , N
Mesh size: h  x
Index set of grid points: J 
Exact solution at level n: u n ( x ) : u( x , tn )  u n ( x ) 2 :  |u n ( x ) |2 d x
Exact solution vector at level n on grid points:                     


          un : {u( x j , tn ), j  J }
FDM approximation vector at level n:              U n : {un , j  J }
                                                                j

Norms
 – Maximum norm     U n : max {| u n |, j  J  }
                                    j
                       
 – 2-norm
             U n :  | u n |2 V j
                          j          with V j   weights
                      2
                             jJ 
           Finite difference discretization

  General form of finite difference scheme
B1 U n 1  B0 U n  F n       with      B1 & B0  difference operators
   – Assume B1 is invertible, i.e. its representing matrix is non-singular
                   n 1      1
               U           B [ B0 U  F ]
                             1
                                          n        n


  – Formally it represents the differential equation in the limit
     n 1
                           h ,k 0 u
 B1 U  [ B0 U  F ] 
                 n    n
                                        L(u)  0  B1  O(1 / k )
                                    t
  – Uniformly well-conditioned
              B11  K1 t              for all h  h0 & k  k0
                 Convergence analysis
Truncation error:    T n : B1 u n 1  [ B0 u n  F n ]   with   u n  exact solution

Consistency: Tjn  0 when k  t & h  x  0 for all j  J 
Order of accuracy: p-th order in time & q-th order in space
 | Tjn | C[k p  hq ] when k  t & h  x  0 for all j  J 
Convergence U n  u n h , 0 0 with n k  t  (0, T ]
                          k



Order of convergence: p-th order in time & q-th order in space
     U n  un  C[k p  h q ] when k & h  0
Stability:   V n W n  K V 0 W 0        for all nk  T
 – two solutions have the same inhomogeneous terms but start with
   difference initial data
              Convergence analysis

Stability condition:
     ( B11B0 )n  K   for all nk  T
 – von Neumann method – based on Fourier transform
 – Energy method
Lax Equivalence Theorem: For a consistent difference
approximation to a well-posed linear evolutionary problem, which
is uniformly well-conditioned, the stability of the scheme is
necessary and sufficient for the convergence.
Proof: See details in class or as an exercise!!
Von Neumann method for stability
                       For CNFD

Plugging into CNFD



Amplification factor



Unconditionally stable—no constraint for time step!!!!!
Energy method – See details in class or as an exercise!!
             Convergence analysis
Convergence of CNFD
– Consistency
– Unconditionally stable
– From Lax equivalent theorem implies convergence!!!
Convergence rate

Other methods for analysis
– Energy method –-- Exercise!!
– Based on maximum principle — Exercise!!
          Method of line approach

Discretize in space first
Method of line approach
           Method of line approach

An ODE system


Discretize in time by ODE solver
–   Trapezoidal method
–   Forward Euler method
–   Backward Euler method
–   Runge-Kutta method, ……..
           Method of line approach

Discretize in time first
         Method of linear approach

Discretize in space by finite difference




– This is CNFD
– Other discretization in space is possible
    Other discrtization for heat equation

Forward Euler finite difference method




–   Local truncation error:
–   Explicit method & direct marching in time
–   Consistency: yes!!
–   Stability condition:
–   Under stability condition, it converges
  Other discrtization for heat equation
Backward Euler finite difference method



– Local truncation error:
– Implicit method:
    • At each step, the linear system can be solved by Thomas algorithm
– Consistency: yes!!
– Unconditionally stable!!!
– It converges and has convergence rate
                  Extension

For Neumann BC



Discretization: CNFD
                        Extension

Local truncation error: 2nd order in space & time

Consistency: yes!!
Implicit method
Linear system -- exercise
Matrix form – exercise
Stability: unconditionally stable!!
Convergence:
                    Extension

Variable coefficients

Discretization -- CNFD
                        Extension

Local truncation error: 2nd order in space & time

Consistency: yes!!
Implicit method
Linear system -- exercise
Matrix form – exercise
Stability: unconditionally stable!!
Convergence:
                     Extension

2D heat equation




Discretization
– Crank-Nicolson in time
– Second order central difference in space
Discretization
                        Extension

Local truncation error: 2nd order in space & time

Consistency: yes!!
Implicit method
Linear system – At every step, use direct Poisson solver
Matrix form – exercise
Stability: unconditionally stable!!
Convergence:
                         More topics
With Rabin or periodic BCs
2D heat equation in a disk or a shell
3D heat equation in a box, spehere, ….
More general case




ADI (alternating direction implicit) for 2D & 3D
Compact scheme
Nonlinear equation & system of heat equations
            Nonlinear parabolic PDEs
Allen-Cahn equation
     t u   u   u (1  u 2 )     (0, T )
    u( x , t )  1                   
   u( x ,0)  u0 ( x )             
Applications
– Imaging science
– Materials science
– Geometry,……
              Numerical methods

Standard finite difference methods
– Crank-Nicolson finite difference
– Forward Euler finite difference
– Backward Euler finite difference
Special techniques
– Time-splitting (split-step) method
– Implicit-explicit method
– Integration factor method
                Time-splitting method
From [tn , tn 1 ] , apply time-splitting technique
 – Step 1. Solve nonlinear ODE for half-step—integrate exact!!!
            t u   u (1  u 2 )
 – Step 2. Solve a linear PDE for one step-- CNFD
                t u   u
 – Step 3. Solve nonlinear ODE for half-step — Integrate exact!!!!
             t u   u (1  u 2 )
Accuracy in time: second order!!!!
No need to solve nonlinear system!!!!
                                Implicit-explicit method

          Ideas:
           – Implicit for linear terms & Explicit for nonlinear terms
          Discretization
           – Method 1 – for computing dynamics
u n 1  u n 1                                                                                   3 n 1 n 1
             [ h u n 1   h u n ]   u n 1/2 (1  (u n 1/2 )2 ),            u n 1/2 :     u  u
      k      2                                                                                   2    2
           – Method 2          – Convex-concave splitting
    u n 1  u n 1 1                            
                    [ h u n 1   h u n 1 ]  (u n 1  u n 1 )   u n   u n (1  (u n ) 2 ,    0
          2k        2                            2
           – Method 3 – for computing steady state
         u n 1  u n
                        h u n 1   u n 1  u n [   (1  (u n ) 2 )]              parameter
               k
                 Integrate factor (IF) method

Rewrite  t u   u   u (1  u )
                                                                                                        2


Multiply both side
   t ( e  t u )  [ t u   u ]e  t   e  t u (1  u 2 )
Integrating over [tn , tn 1 ]
                                                                tn 1

                                               u ( tn )        
            tn 1                    tn 
       e              u(tn 1 )  e                                     e  t u(t ) (1  u 2 (t ))dt
                                                                 tn
                                                  tn 1

       u(tn 1 )  e u(tn )                      e
                            k                            ( tn 1 t ) 
                                                                           u(t ) (1  u 2 (t ))dt
                                                   tn


Approximate in time via RK4 & in space via FDM
                         un1  ekh un   S (ekh un (1  (un )2 ))
                          j           j              j        j
         Nonlinear parabolic PDEs
Sharp interface
                   1
    t u   u            u (1  u 2 )   0       1
                      2
Ginzburg-Landau equation (GLE)
                  1
  t u   u          u (1 | u |2 )     u:    d
                                                    
                  2

General nonlinearity
       t u   u  f (u )
System,.......
Compact scheme in space
1 | u |2 )    u: d 
               2
General nonlinearity
       t u   u  f (u )
System,.......
Compact scheme in space

				
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