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FDM for parabolic equations

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Consistency & Stability. Consider the general problem. It is a well-posed problem: ... Consistency. Unconditionally stable. From Lax equivalent theorem implies ... – PowerPoint PPT presentation

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Title: FDM for parabolic equations


1
FDM for parabolic equations
  • Consider the heat equation
  • where
  • Well-posed problem
  • Existence Uniqueness
  • Mass Energy decreasing

2
FDM for parabolic equations
3
CNFD
  • Crank-Nicolson 2nd order finite difference
  • Questions
  • How to solve the equations efficiently???
  • Convergence and order of accuracy???
  • Local truncation error Stability

4
Local truncation error
5
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

6
Matrix form
7
Solution algorithm
8
Convergence analysis
  • Convergence
  • Consistency Stability
  • Consider the general problem
  • It is a well-posed problem
  • Existence, uniqueness, continuously depend on
    initial data

9
Finite difference discretization
  • Time step
  • Mesh size
  • Index set of grid points
  • Exact solution at level n
  • Exact solution vector at level n on grid points
  • FDM approximation vector at level n
  • Norms
  • Maximum norm
  • 2-norm

10
Finite difference discretization
  • General form of finite difference scheme
  • Assume B1 is invertible, i.e. its representing
    matrix is non-singular
  • Formally it represents the differential equation
    in the limit
  • Uniformly well-conditioned

11
Convergence analysis
  • Truncation error
  • Consistency
  • Order of accuracy p-th order in time q-th
    order in space
  • Convergence
  • Order of convergence p-th order in time q-th
    order in space
  • Stability
  • two solutions have the same inhomogeneous terms
    but start with difference initial data

12
Convergence analysis
  • Stability condition
  • 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!!

13
Von Neumann method for stability
14
For CNFD
  • Plugging into CNFD
  • Amplification factor
  • Unconditionally stableno constraint for time
    step!!!!!
  • Energy method See details in class or as an
    exercise!!

15
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!!

16
Method of line approach
  • Discretize in space first

17
Method of line approach
18
Method of line approach
  • An ODE system
  • Discretize in time by ODE solver
  • Trapezoidal method
  • Forward Euler method
  • Backward Euler method
  • Runge-Kutta method, ..

19
Method of line approach
  • Discretize in time first

20
Method of linear approach
  • Discretize in space by finite difference
  • This is CNFD
  • Other discretization in space is possible

21
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

22
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

23
Extension
  • For Neumann BC
  • Discretization CNFD

24
Extension
  • Local truncation error 2nd order in space
    time
  • Consistency yes!!
  • Implicit method
  • Linear system -- exercise
  • Matrix form exercise
  • Stability unconditionally stable!!
  • Convergence

25
Extension
  • Variable coefficients
  • Discretization -- CNFD

26
Extension
  • Local truncation error 2nd order in space
    time
  • Consistency yes!!
  • Implicit method
  • Linear system -- exercise
  • Matrix form exercise
  • Stability unconditionally stable!!
  • Convergence

27
Extension
  • 2D heat equation
  • Discretization
  • Crank-Nicolson in time
  • Second order central difference in space

28
Discretization
29
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

30
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

31
Nonlinear parabolic PDEs
  • Allen-Cahn equation
  • Applications
  • Imaging science
  • Materials science
  • Geometry,

32
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

33
Time-splitting method
  • From , apply time-splitting technique
  • Step 1. Solve nonlinear ODE for
    half-stepintegrate exact!!!
  • Step 2. Solve a linear PDE for one step-- CNFD
  • Step 3. Solve nonlinear ODE for half-step
    Integrate exact!!!!
  • Accuracy in time second order!!!!
  • No need to solve nonlinear system!!!!

34
Implicit-explicit method
  • Ideas
  • Implicit for linear terms Explicit for
    nonlinear terms
  • Discretization
  • Method 1 for computing dynamics
  • Method 2 Convex-concave splitting
  • Method 3 for computing steady state

35
Integrate factor (IF) method
  • Rewrite
  • Multiply both side
  • Integrating over
  • Approximate in time via RK4 in space via FDM

36
Nonlinear parabolic PDEs
  • Sharp interface
  • Ginzburg-Landau equation (GLE)
  • General nonlinearity
  • System,.......
  • Compact scheme in space
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