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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, 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 PhysicsComputational ChemistryComputational BiologyComputational Fluid DynamicsComputational EnginneringComputational 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 domainsBoundary 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 problemIdeas– 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!!