Numerical methods for PDEs

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Numerical methods for PDEs

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

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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, .

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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

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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, ..

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Numerical PDEs with Applications

• Computational Mathematics Scientific
  computing/numerical analysis
• Computational Physics
• Computational Chemistry
• Computational Biology
• Computational Fluid Dynamics
• Computational Enginnering
• Computational Materials Sciences
• ...

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Different PDEs

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

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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, .

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Different PDEs

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

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Different PDEs

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

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Classifications

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

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For a specific problem

• Physical domains
• Boundary conditions (BC)
• Dirichlet boundary condition
• Neumann boundary condition
• Robin boundary condition
• Periodic boundary condition

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For a specific problem

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

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Model problems

• Initial value problem Cauchy problem
• Initial boundary value problem (IBVP)

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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
• ..

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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

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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
• ..

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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), ..

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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, .

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Finite difference method

• Choose

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Finite difference method

• Finite difference

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Finite difference method

• Finite differential

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Finite difference method

• Order of approximation

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Finite difference method

• Finite difference approximation
• Linear system

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Finite difference method

• In matrix form
• With
• Solve the linear system obtain the approximate
  solution

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Finite difference method

• Question??

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Finite difference method

• Local truncation error
• Order of accuracy second-order

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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

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Finite difference method

• For Neumann boundary condition
• Solvable condition
• Uniqueness condition

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Finite difference method

• Discretization
• At shifted grid points by half grid
• Use two ghost points
• For the uniqueness condition

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Finite difference method

• In linear system

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Finite difference method

• In matrix form
• With

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Finite difference mehtod

• Solution of the linear system
• Compute approximation at grid points

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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!!

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Finite difference method

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

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Finite difference method

• Discretization
• Local truncation error exercise!!
• Linear system exercise!!
• Matrix form exercise!!
• Error analysis exercise!!