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, Helmholtz’s 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!!