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# Advanced Numerical Methods for PDEs MATH 652

CSU

GPA 3.78

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This 22 page Class Notes was uploaded by Melvina Keeling on Monday September 21, 2015. The Class Notes belongs to MATH 652 at Colorado State University taught by Yongcheng Zhou in Fall. Since its upload, it has received 38 views. For similar materials see /class/210078/math-652-colorado-state-university in Mathematics (M) at Colorado State University.

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Date Created: 09/21/15

Numerical Methods for Partial Differential Equations an Overview math652Spring2009colorstate PDEs are mathematical models of physical phenomena Heat conduction ut H um 0 lt at lt L t gt 0 u10 u0c 0 g a 3 L7 u0t 17 21L7 t 0 t2 0 uac7 t temperature at position a and time t Ii heat conductivity coe icient no given function Wave motion utt a2 u f3t 0 lt a lt L7 tgt 07 uc O now utm0 u1z 0 S c g L 0115 91 uL7 t 92t7 t2 ua t displacement at position x and time t a constant no 711 91 and 92 given functions PDEs are mathematical models of Chemical Phenomena Mixture problems 0 Motion of electron atom Schrodinger equation 0 Chemical reaction rate Schrodinger equation 0 Semiconductor SchrodingerPoisson equations Biological Phenomena Population of a biological species 0 Biomolecular electrostatics PoissonBoltzmann equation 0 Calcium dynamics ion diffusion Nernst Planck equation 0 Cell motion and interaction 0 Blood ow Navier Stokes equation PDEs are mathematical models of Engineering Fluid dynamics J Euler equations J Navier Stokes Equations Electromagnetic J Poisson equation J Helmholtz s equation J Maxwell equations 0 Elasticity dynamics structure of foundation J Navier system 0 Material Sciences PDEs are mathematical models of Semiconductor industry Driftdiffusion equations EulerPoisson equations 0 SchrodingerPoisson equations Plasma physics 0 VlasovPoisson equations Zakharov system Financial industry BlackScholes equations Economics Medicine Life Sciences Social Sciences Numerical PDEs with Applications 0 Computational Mathematics Scienti c computingnumerical analysis 0 Computational Physics 0 Computational Chemistry 0 Computational Biology 0 Computational Fluid Dynamics 0 Computational Engineering 0 Computational Materials Sciences 0 Computational Social Sciences Computational sociology Different Types of PDEs Linear scalar PDE Poisson equation Laplace equation 1 A u a fx um Heat equation ut AUZO lit 120 Wave equation UH AU 2 0 utt um Helmholtz equation Telegraph equation Different Types of PDEs Nonlinear scalar PDE Nonlinear Poisson equation Aufu7 umfu Nonlinear convectiondiffusion equation at fuz 111 V gt 0 Kortewegde Vries KdV equation utuuxumx 0 Eikonal equation HamiltonJacobi equation KleinGordon equation Nonlinear Schrodinger equation GinzburgLandau equation Different Types of PDEs Linear systems NaVier system linear elasticity u A u A ugraddiv u 0 Stokes equations ut Vp 1 A 11 V u 0 Maxwell equations VXEOtB 0 VxB OtE 2 J VE p VB 0 Different Types of PDEs Nonlinear system s Reactiondiffusion system 111 Au 2 System of conservation laws Euler Equations ut div Fu 0 NavierStokes equations Classi cation of PDEs For scalar PDE Elliptic equations Laplace equation Poisson equation Parabolic equations Heat equations Hyperbolic equations Conservation laws For system of PDEs For a specific problem Physical domains uxz Boundary conditions BC Dirichlet boundary condition ua a ub Neumann boundary condition u39a a Robin boundary condition u39a mum 1 Periodic boundary condition W1 715 alt33ltb For a speci c problem Initial condition timedependent problem For at ux0 u0x For utt 39 39 39 11090 x nibquot Z Model problems Boundaryvalue problem BVP um x a lt 76 lt b ua a ub B Model problem Initial value problem 7 Cauchy problem utauO or ufugcO7 ooltmltoo ux0 ugI7 00 lt 6 lt 00 Initial boundary value problem IBVP u rcvugcuum7 altzlcltb7 Vgt0 uat 91137 ub t 92257 t2 0 uL 7 O uo17 a S x g b Maj or numerical methods for PDEs 0 Finite difference method FDM Pros a Simple and easy to design the scheme a Flexible to deal with the nonlinear problem a Widely used for elliptic parabolic and hyperbolic equations a Most popular method for simple geometry 1 Easy to program Cons a Not easy to deal With complex geometry 1 Not easy for complicated boundary conditions I Iu I IL39IZ1l I I yjl I Ma or numerlcal methods for PDEs I I I y Juan Lug luau I I I I I I 0 Finite difference method FDM yAF 139 5171 T39 Pros El Simple and easy to design the scheme xi xi39 El Flexible to deal with the nonlinear problem El Widely used for elliptic parabolic and hyperbolic equations El Most popular method for simple geometry El Easy to program Cons El Not easy to deal With complex geometry El Not easy for complicated boundary conditions Maj or numerical methods for PDEs 0 Finite element method Pros a Flexible to deal With problems With complex geometry and complicated boundary conditions a Rigorous mathematical theory for analysis a Widely used in mechanical structure analysis heat transfer electromagnetics Cons 1 Need more mathematical knowledge to formulate a good and equivalent variational form 1 Appears hard to program Maj or numerical methods for PDES Finite element method complex geometry 39 4615quot r 53933quot quot 3 i v v V Maj or numerical methods for PDEs Finite element method adaptivity Maj or numerical methods for PDEs 0 Spectral method Math676 Fall 2008 El El El 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 Maj or 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 0 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 ef cient fast Poisson solver When combined With the fast multipole method FMM 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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