FDM for parabolic equations

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FDM for parabolic equations

• Consider the heat equation
• where
• Well-posed problem
• Existence Uniqueness
• Mass Energy decreasing

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FDM for parabolic equations
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CNFD

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

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Local truncation error
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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

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Matrix form
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Solution algorithm
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Convergence analysis

• Convergence
• Consistency Stability
• Consider the general problem
• It is a well-posed problem
• Existence, uniqueness, continuously depend on
  initial data

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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

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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

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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

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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!!

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Von Neumann method for stability
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For CNFD

• Plugging into CNFD
• Amplification factor
• Unconditionally stableno constraint for time
  step!!!!!
• Energy method See details in class or as an
  exercise!!

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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!!

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Method of line approach

• Discretize in space first

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Method of line approach
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Method of line approach

• An ODE system
• Discretize in time by ODE solver
• Trapezoidal method
• Forward Euler method
• Backward Euler method
• Runge-Kutta method, ..

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Method of line approach

• Discretize in time first

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Method of linear approach

• Discretize in space by finite difference
• This is CNFD
• Other discretization in space is possible

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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

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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

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Extension

• For Neumann BC
• Discretization CNFD

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Extension

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

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Extension

• Variable coefficients
• Discretization -- CNFD

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Extension

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

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Extension

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

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Discretization
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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

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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

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Nonlinear parabolic PDEs

• Allen-Cahn equation
• Applications
• Imaging science
• Materials science
• Geometry,

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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

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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!!!!

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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

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Integrate factor (IF) method

• Rewrite
• Multiply both side
• Integrating over
• Approximate in time via RK4 in space via FDM

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Nonlinear parabolic PDEs

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