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Numerical methods for PDEs

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

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, .

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

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, ..

Numerical PDEs with Applications

- Computational Mathematics Scientific

computing/numerical analysis - Computational Physics
- Computational Chemistry
- Computational Biology
- Computational Fluid Dynamics
- Computational Enginnering
- Computational Materials Sciences
- ...

Different PDEs

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

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, .

Different PDEs

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

Different PDEs

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

Classifications

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

For a specific problem

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

For a specific problem

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

Model problems

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

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
- ..

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

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
- ..

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), ..

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, .

Finite difference method

- Choose

Finite difference method

- Finite difference

Finite difference method

- Finite differential

Finite difference method

- Order of approximation

Finite difference method

- Finite difference approximation
- Linear system

Finite difference method

- In matrix form
- With
- Solve the linear system obtain the approximate

solution

Finite difference method

- Question??

Finite difference method

- Local truncation error
- Order of accuracy second-order

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

Finite difference method

- For Neumann boundary condition
- Solvable condition
- Uniqueness condition

Finite difference method

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

Finite difference method

- In linear system

Finite difference method

- In matrix form
- With

Finite difference mehtod

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

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!!

Finite difference method

- For Poisson equation with variable coefficients
- Discretization Use type II finite difference

twice!!

Finite difference method

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