# Numerical Methods for Generalized Zakharov System by tyndale

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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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