Numerical Methods for Partial Differential Equations: an Overview.
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1 Numerical Methods for Partial Differential Equations: an Overview
2 PDEs are mathematical models of physical phenomena Heat conduction Wave motion
3 PDEs are mathematical models of 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 Biomolecular electrostatics: Poisson-Boltzmann equation Calcium dynamics, ion diffusion: Nernst-Planck equation Cell motion and interaction Blood flow: Navier-Stokes equation
4 PDEs are mathematical models of Engineering Fluid dynamics Euler equations Navier-Stokes Equations Electromagnetic Poisson equation Helmholtz s equation Maxwell equations Elasticity dynamics (structure of foundation) Navier system Material Sciences
5 PDEs are mathematical models of Semiconductor industry Drift-diffusion equations Euler-Poisson equations Schrodinger-Poisson equations Plasma physics Vlasov-Poisson equations Zakharov system Financial industry Black-Scholes equations Economics, Medicine, Life Sciences, Social Sciences
6 Numerical PDEs with Applications Computational Mathematics: Scientific computing/numerical analysis Computational Physics Computational Chemistry Computational Biology Computational Fluid Dynamics Computational Engineering Computational Materials Sciences Computational Social Sciences: Computational sociology
7 Different Types of PDEs Linear scalar PDE Poisson equation (Laplace equation) Heat equation Wave equation Helmholtz equation, Telegraph equation,
8 Different Types of 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,.
9 Different Types of PDEs Linear systems Navier system -- linear elasticity Stokes equations Maxwell equations
10 Different Types of PDEs Nonlinear systems Reaction-diffusion system System of conservation laws (Euler Equations ) Navier-Stokes equations
11 Classification of PDEs For scalar PDE Elliptic equations: Laplace equation, Poisson equation, Parabolic equations Heat equations, Hyperbolic equations Conservation laws,. For system of PDEs
12 For a specific problem Physical domains Boundary conditions (BC) Dirichlet boundary condition Neumann boundary condition Robin boundary condition Periodic boundary condition
13 For a specific problem Initial condition time-dependent problem For For Model problems Boundary-value problem (BVP)
14 Model problem Initial value problem Cauchy problem Initial boundary value problem (IBVP)
15 Major numerical methods for PDEs Finite difference method (FDM) Pros 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,. Easy to program Cons Not easy to deal with complex geometry Not easy for complicated boundary conditions
16 Major numerical methods for PDEs y j+1 y j u, u i, j+1 u, i 1 j i j u i+1, j Finite difference method (FDM) Pros Cons 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,. Easy to program Not easy to deal with complex geometry y j 1 Not easy for complicated boundary conditions xi 1 u i, j 1 xi x i+ 1
17 Major numerical methods for PDEs S D e m Φ D Finite element method (FEM) Pros Flexible to deal with problems with complex geometry and complicated boundary conditions Rigorous mathematical theory for analysis Widely used in mechanical structure analysis, heat transfer, electromagnetics Cons Need more mathematical knowledge to formulate a good and equivalent variational form Appears hard to program
18 Major numerical methods for PDEs Finite element method: complex geometry
19 Major numerical methods for PDEs Finite element method: adaptivity
20 Major numerical methods for PDEs Spectral method (Math676 Fall 2008) 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 Not easy to deal with complex geometry
21 Major numerical methods for PDEs Finite volume method (FVM) 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),..
22 At the end of the course, you shall be able to Generate your own code of standard 2D FD, FV and FEM Understand the pathway of algorithm formulation Understand the error analysis of FEM Understand various requirements of PDEs on numerical methods Read papers on numerical analysis of PDEs, carry out research
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