# Numerical methods for PDEs PDEs are mathematical models for –Physical Phenomena Heat transfer Wave motion

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<ul><li> Slide 1 </li> <li> Numerical methods for PDEs PDEs are mathematical models for Physical Phenomena Heat transfer Wave motion </li> <li> Slide 2 </li> <li> 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, . </li> <li> Slide 3 </li> <li> 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 </li> <li> Slide 4 </li> <li> 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, .. </li> <li> Slide 5 </li> <li> Numerical PDEs with Applications Computational Mathematics Scientific computing/numerical analysis Computational Physics Computational Chemistry Computational Biology Computational Fluid Dynamics Computational Enginnering Computational Materials Sciences ... </li> <li> Slide 6 </li> <li> Different PDEs Linear scalar PDE: Poisson equation (Laplace equation) Heat equation Wave equation Helmholtz equation, Telegraph equation, </li> <li> Slide 7 </li> <li> 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, . </li> <li> Slide 8 </li> <li> Different PDEs Linear systems Navier system -- linear elasticity Stokes equations Maxwell equations . </li> <li> Slide 9 </li> <li> Different PDEs Nonlinear systems Reaction-diffusion system System of conservation laws Euler equations Navier-Stokes equations, . </li> <li> Slide 10 </li> <li> Classifications For scalar PDE Elliptic equations: Poisson equation, Parabolic equations Heat equations, Hyperbolic equations Conservation laws, . For system of PDEs </li> <li> Slide 11 </li> <li> For a specific problem Physical domains Boundary conditions (BC) Dirichlet boundary condition Neumann boundary condition Robin boundary condition Periodic boundary condition </li> <li> Slide 12 </li> <li> For a specific problem Initial condition time-dependent problem For Model problems Boundary-value problem (BVP) </li> <li> Slide 13 </li> <li> Model problems Initial value problem Cauchy problem Initial boundary value problem (IBVP) </li> <li> Slide 14 </li> <li> 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 .. </li> <li> Slide 15 </li> <li> 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 </li> <li> Slide 16 </li> <li> 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 .. </li> <li> Slide 17 </li> <li> 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), .. </li> <li> Slide 18 </li> <li> 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, . </li> <li> Slide 19 </li> <li> Finite difference method Choose </li> <li> Slide 20 </li> <li> Finite difference method Finite difference </li> <li> Slide 21 </li> <li> Finite difference method Finite differential </li> <li> Slide 22 </li> <li> Finite difference method Order of approximation </li> <li> Slide 23 </li> <li> Finite difference method Finite difference approximation Linear system </li> <li> Slide 24 </li> <li> Finite difference method In matrix form With Solve the linear system & obtain the approximate solution </li> <li> Slide 25 </li> <li> Finite difference method Question?? </li> <li> Slide 26 </li> <li> Finite difference method Local truncation error: Order of accuracy: second-order </li> <li> Slide 27 </li> <li> 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 </li> <li> Slide 28 </li> <li> Finite difference method For Neumann boundary condition Solvable condition Uniqueness condition </li> <li> Slide 29 </li> <li> Finite difference method Discretization At shifted grid points by half grid Use two ghost points For the uniqueness condition </li> <li> Slide 30 </li> <li> Finite difference method In linear system </li> <li> Slide 31 </li> <li> Finite difference method In matrix form With </li> <li> Slide 32 </li> <li> Finite difference mehtod Solution of the linear system Compute approximation at grid points </li> <li> Slide 33 </li> <li> 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!! </li> <li> Slide 34 </li> <li> Finite difference method For Poisson equation with variable coefficients Discretization: Use type II finite difference twice!! </li> <li> Slide 35 </li> <li> Finite difference method Discretization Local truncation error exercise!! Linear system exercise!! Matrix form exercise!! Error analysis exercise!! </li> </ul>