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Numerical methods for PDEs

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Computational Mathematics Scientific computing/numerical analysis ... Klein-Gordon equation, Nonlinear Schrodinger equation, Ginzburg-Landau equation, ... – PowerPoint PPT presentation

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Title: Numerical methods for PDEs


1
Numerical methods for PDEs
  • PDEs are mathematical models for
  • Physical Phenomena
  • Heat transfer
  • Wave motion

2
PDEs
  • Chemical Phenomena
  • Mixture problems
  • Motion of electron, atom Schrodinger equation
  • Chemical reaction rate Schrodinger equation
  • Semiconductor Schrodinger-Poisson equations
  • ..
  • Biological phenomena
  • Population of a biological species
  • Cell motion and interaction, blood flow, .

3
PDEs
  • Engineering
  • Fluid dynamics
  • Euler equations,
  • Navier-Stokes Equations, .
  • Electron magnetic
  • Poisson equation, Helmholtzs equation
  • Maxwell equations,
  • Elasticity dynamics (structure of foundation)
  • Navier system,
  • Material Sciences

4
PDEs
  • Semiconductor industry
  • Drift-diffusion equations,
  • Euler-Poisson equations
  • Schrodinger-Poisson equations,
  • Plasma physics
  • Vlasov-Poisson equations
  • Zakharov system, ..
  • Financial industry
  • Balck-Scholes equations, .
  • Economics, Medicine, Life Sciences, ..

5
Numerical PDEs with Applications
  • Computational Mathematics Scientific
    computing/numerical analysis
  • Computational Physics
  • Computational Chemistry
  • Computational Biology
  • Computational Fluid Dynamics
  • Computational Enginnering
  • Computational Materials Sciences
  • ...

6
Different PDEs
  • Linear scalar PDE
  • Poisson equation (Laplace equation)
  • Heat equation
  • Wave equation
  • Helmholtz equation, Telegraph equation,

7
Different PDEs
  • Nonlinear scalar PDE
  • Nonlinear Poisson equation
  • Nonlinear convection-diffusion equation
  • Korteweg-de Vries (KdV) equation
  • Eikonal equation, Hamilton-Jacobi equation,
    Klein-Gordon equation, Nonlinear Schrodinger
    equation, Ginzburg-Landau equation, .

8
Different PDEs
  • Linear systems
  • Navier system -- linear elasticity
  • Stokes equations
  • Maxwell equations
  • .

9
Different PDEs
  • Nonlinear systems
  • Reaction-diffusion system
  • System of conservation laws
  • Euler equations
  • Navier-Stokes equations, .

10
Classifications
  • For scalar PDE
  • Elliptic equations
  • Poisson equation,
  • Parabolic equations
  • Heat equations,
  • Hyperbolic equations
  • Conservation laws, .
  • For system of PDEs

11
For a specific problem
  • Physical domains
  • Boundary conditions (BC)
  • Dirichlet boundary condition
  • Neumann boundary condition
  • Robin boundary condition
  • Periodic boundary condition

12
For a specific problem
  • Initial condition time-dependent problem
  • For
  • For
  • Model problems
  • Boundary-value problem (BVP)

13
Model problems
  • Initial value problem Cauchy problem
  • Initial boundary value problem (IBVP)

14
Main numerical methods for PDEs
  • Finite difference method (FDM) this module
  • Advantages
  • Simple and easy to design the scheme
  • Flexible to deal with the nonlinear problem
  • Widely used for elliptic, parabolic and
    hyperbolic equations
  • Most popular method for simple geometry, .
  • Disadvantages
  • Not easy to deal with complex geometry
  • Not easy for complicated boundary conditions
  • ..

15
Main numerical methods
  • Finite element method (FEM) MA5240
  • Advantages
  • Flexible to deal with problems with complex
    geometry and complicated boundary conditions
  • Keep physical laws in the discretized level
  • Rigorous mathematical theory for error analysis
  • Widely used in mechanical structure analysis,
    computational fluid dynamics (CFD), heat
    transfer, electromagnetics,
  • Disadvantages
  • Need more mathematical knowledge to formulate a
    good and equivalent variational form

16
Main numerical methods
  • Spectral method MA5251
  • High (spectral) order of accuracy
  • Usually restricted for problems with regular
    geometry
  • Widely used for linear elliptic and parabolic
    equations on regular geometry
  • Widely used in quantum physics, quantum
    chemistry, material sciences,
  • Not easy to deal with nonlinear problem
  • Not easy to deal with hyperbolic problem
  • ..

17
Main numerical methods
  • Finite volume method (FVM) MA5250
  • Flexible to deal with problems with complex
    geometry and complicated boundary conditions
  • Keep physical laws in the discretized level
  • Widely used in CFD
  • Boundary element method (BEM)
  • Reduce a problem in one less dimension
  • Restricted to linear elliptic and parabolic
    equations
  • Need more mathematical knowledge to find a good
    and equivalent integral form
  • Very efficient fast Poisson solver when combined
    with the fast multipole method (FMM), ..

18
Finite difference method (FDM)
  • Consider a model problem
  • Ideas
  • Choose a set of grid points
  • Discretize (or approximate) the derivatives in
    the PDE by finite difference at the grid points
  • Discretize the boundary conditions when it is
    needed
  • Obtain a linear (or nonlinear) system
  • Solve the linear (or nonlinear) system and get an
    approximate solution of the original problem over
    the grid points
  • Analyze the error --- local truncation error,
    stability, convergence
  • How to solve the linear system efficiently Fast
    Poisson solver based on FFT, Multigrid, CG,
    GMRES, iterative methods, .

19
Finite difference method
  • Choose

20
Finite difference method
  • Finite difference

21
Finite difference method
  • Finite differential

22
Finite difference method
  • Order of approximation

23
Finite difference method
  • Finite difference approximation
  • Linear system

24
Finite difference method
  • In matrix form
  • With
  • Solve the linear system obtain the approximate
    solution

25
Finite difference method
  • Question??

26
Finite difference method
  • Local truncation error
  • Order of accuracy second-order

27
Finite difference method
  • Solution of the linear system
  • Thomas algorithm
  • Stability
  • No stability constraint
  • Error analysis
  • Proof See details in class or as an exercise

28
Finite difference method
  • For Neumann boundary condition
  • Solvable condition
  • Uniqueness condition

29
Finite difference method
  • Discretization
  • At shifted grid points by half grid
  • Use two ghost points
  • For the uniqueness condition

30
Finite difference method
  • In linear system

31
Finite difference method
  • In matrix form
  • With

32
Finite difference mehtod
  • Solution of the linear system
  • Compute approximation at grid points

33
Finite difference method
  • Local truncation error exercise!!
  • For the discrtization of the equation
  • For the discretization of boundary condition
  • Order of accuracy Second-order
  • Error analysis exercise!!
  • For Robin boundary condition -- exercise!!
  • For periodic boundary condition exercise!!

34
Finite difference method
  • For Poisson equation with variable coefficients
  • Discretization Use type II finite difference
    twice!!

35
Finite difference method
  • Discretization
  • Local truncation error exercise!!
  • Linear system exercise!!
  • Matrix form exercise!!
  • Error analysis exercise!!
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