View by Category

The presentation will start after a short

(15 second) video ad from one of our sponsors.

Hot tip: Video ads won’t appear to registered users who are logged in. And it’s free to register and free to log in!

(15 second) video ad from one of our sponsors.

Hot tip: Video ads won’t appear to registered users who are logged in. And it’s free to register and free to log in!

Loading...

PPT – Numerical methods for PDEs PowerPoint presentation | free to view - id: 10d49a-MjYwN

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

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!!

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Page of

CrystalGraphics Sales Tel: (800) 394-0700 x 1 or Send an email

Home About Us Terms and Conditions Privacy Policy Contact Us Send Us Feedback

Copyright 2016 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2016 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Numerical methods for PDEs" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!

Committed to assisting Nus University and other schools with their online training by sharing educational presentations for free