Partial Differential Equations

Slides:

Advertisements

Similar presentations

Review of accuracy analysis Euler: Local error = O(h 2 ) Global error = O(h) Runge-Kutta Order 4: Local error = O(h 5 ) Global error = O(h 4 ) But there’s.

Advertisements

1cs542g-term Notes  No extra class tomorrow.

Total Recall Math, Part 2 Ordinary diff. equations First order ODE, one boundary/initial condition: Second order ODE.

PART 7 Ordinary Differential Equations ODEs

ECE602 BME I Partial Differential Equations in Biomedical Engineering.

AE/ME 339 Computational Fluid Dynamics (CFD) K. M. Isaac Professor of Aerospace Engineering.

MANE 4240 & CIVL 4240 Introduction to Finite Elements Introduction to differential equations Prof. Suvranu De.

PDEs: General classification

PARTIAL DIFFERENTIAL EQUATIONS

1 Chapter 9 NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS.

Ordinary Differential Equations S.-Y. Leu Sept. 21,28, 2005.

Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.

Numerical Methods for Partial Differential Equations

Types of Governing equations

COMPUTATIONAL MODELING FOR ENGINEERING MECN 6040 Professor: Dr. Omar E. Meza Castillo Department.

III Solution of pde’s using variational principles

CISE301: Numerical Methods Topic 9 Partial Differential Equations (PDEs) Lectures KFUPM (Term 101) Section 04 Read & CISE301_Topic9.

SE301: Numerical Methods Topic 9 Partial Differential Equations (PDEs) Lectures KFUPM Read & CISE301_Topic9 KFUPM.

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

Scientific Computing Partial Differential Equations Explicit Solution of Wave Equation.

COMPUTATIONAL MODELING FOR ENGINEERING MECN 6040 Professor: Dr. Omar E. Meza Castillo Department.

1 Numerical Methods and Software for Partial Differential Equations Lecturer:Dr Yvonne Fryer Time:Mondays 10am-1pm Location:QA210 / KW116 (Black)

Hyperbolic PDEs Numerical Methods for PDEs Spring 2007 Jim E. Jones.

CIS888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering Partial Differential Equations - Background Physical problems are governed.

Partial Differential Equations Introduction –Adam Zornes, Deng Li Discretization Methods –Chunfang Chen, Danny Thorne, Adam Zornes.

S.S. Yang and J.K. Lee FEMLAB and its applications POSTEC H Plasma Application Modeling Lab. Oct. 25, 2005.

Scientific Computing Partial Differential Equations Poisson Equation.

Ordinary Differential Equations

A conservative FE-discretisation of the Navier-Stokes equation JASS 2005, St. Petersburg Thomas Satzger.

Math 3120 Differential Equations with Boundary Value Problems

Numerical Methods for Solving ODEs Euler Method. Ordinary Differential Equations  A differential equation is an equation in which includes derivatives.

+ Numerical Integration Techniques A Brief Introduction By Kai Zhao January, 2011.

Phase Separation and Dynamics of a Two Component Bose-Einstein Condensate.

Lecture 15 Solving the time dependent Schrödinger equation

Solving an elliptic PDE using finite differences Numerical Methods for PDEs Spring 2007 Jim E. Jones.

Akram Bitar and Larry Manevitz Department of Computer Science

Introduction to PDE classification Numerical Methods for PDEs Spring 2007 Jim E. Jones References: Partial Differential Equations of Applied Mathematics,

Ch 1.3: Classification of Differential Equations

Engineering Analysis – Computational Fluid Dynamics –

CIS/ME 794Y A Case Study in Computational Science & Engineering 2-D Conservation of Mass uu dx dy vv (x,y)

Introduction to Numerical Methods for ODEs and PDEs Lectures 1 and 2: Some representative problems from engineering science and classification of equation.

Engineering Analysis – Computational Fluid Dynamics –

Numerical methods 1 An Introduction to Numerical Methods For Weather Prediction by Mariano Hortal office 122.

Discretization Methods Chapter 2. Training Manual May 15, 2001 Inventory # Discretization Methods Topics Equations and The Goal Brief overview.

Partial Derivatives bounded domain Its boundary denoted by

Partial Derivatives Example: Find If solution: Partial Derivatives Example: Find If solution: gradient grad(u) = gradient.

1 Chapter 1 Introduction to Differential Equations 1.1 Introduction The mathematical formulation problems in engineering and science usually leads to equations.

Math 445: Applied PDEs: models, problems, methods D. Gurarie.

Partial Differential Equations Introduction – Deng Li Discretization Methods –Chunfang Chen, Danny Thorne, Adam Zornes CS521 Feb.,7, 2006.

Dr. Mujahed AlDhaifallah ( Term 342)

Differential Equations

9 Nov B - Introduction to Scientific Computing1 Sparse Systems and Iterative Methods Paul Heckbert Computer Science Department Carnegie Mellon.

Boyce/DiPrima 9 th ed, Ch1.3: Classification of Differential Equations Elementary Differential Equations and Boundary Value Problems, 9 th edition, by.

Solving Partial Differential Equation Numerically Pertemuan 13 Matakuliah: S0262-Analisis Numerik Tahun: 2010.

Ordinary Differential Equations (ODEs). Objectives of Topic  Solve Ordinary Differential Equations (ODEs).  Appreciate the importance of numerical methods.

Ch. 12 Partial Differential Equations

Introduction to Differential Equations

MESB374 System Modeling and Analysis

Christopher Crawford PHY

Scientific Computing Lab

Introduction to Partial Differential Equations

525602:Advanced Numerical Methods for ME

Paul Heckbert Computer Science Department Carnegie Mellon University

Finite Difference Method

ME/AE 339 Computational Fluid Dynamics K. M. Isaac Topic2_PDE

Scientific Computing Lab

Brief introduction to Partial Differential Equations (PDEs)

Partial Differential Equations

Introduction to Ordinary Differential Equations

Introduction to Fluid Mechanics

Presentation transcript:

Partial Differential Equations Paul Heckbert Computer Science Department Carnegie Mellon University 7 Nov. 2000 15-859B - Introduction to Scientific Computing

Differential Equation Classes 1 dimension of unknown: ordinary differential equation (ODE) – unknown is a function of one variable, e.g. y(t) partial differential equation (PDE) – unknown is a function of multiple variables, e.g. u(t,x,y) number of equations: single differential equation, e.g. y’=y system of differential equations (coupled), e.g. y1’=y2, y2’=-g order nth order DE has nth derivative, and no higher, e.g. y’’=-g 7 Nov. 2000 15-859B - Introduction to Scientific Computing

Differential Equation Classes 2 linear & nonlinear: linear differential equation: all terms linear in unknown and its derivatives e.g. x’’+ax’+bx+c=0 – linear x’=t2x – linear x’’=1/x – nonlinear 7 Nov. 2000 15-859B - Introduction to Scientific Computing

PDE’s in Science & Engineering 1 Laplace’s Equation: 2u = uxx +uyy +uzz = 0 unknown: u(x,y,z) gravitational / electrostatic potential Heat Equation: ut = a22u unknown: u(t,x,y,z) heat conduction Wave Equation: utt = a22u wave propagation 7 Nov. 2000 15-859B - Introduction to Scientific Computing

PDE’s in Science & Engineering 2 Schrödinger Wave Equation quantum mechanics (electron probability densities) Navier-Stokes Equation fluid flow (fluid velocity & pressure) 7 Nov. 2000 15-859B - Introduction to Scientific Computing

2nd Order PDE Classification We classify conic curve ax2+bxy+cy2+dx+ey+f=0 as ellipse/parabola/hyperbola according to sign of discriminant b2-4ac. Similarly we classify 2nd order PDE auxx+buxy+cuyy+dux+euy+fu+g=0: b2-4ac < 0 – elliptic (e.g. equilibrium) b2-4ac = 0 – parabolic (e.g. diffusion) b2-4ac > 0 – hyperbolic (e.g. wave motion) For general PDE’s, class can change from point to point 7 Nov. 2000 15-859B - Introduction to Scientific Computing

Example: Wave Equation utt = c uxx for 0x1, t0 initial cond.: u(0,x)=sin(x)+x+2, ut(0,x)=4sin(2x) boundary cond.: u(t,0) = 2, u(t,1)=3 c=1 unknown: u(t,x) simulated using Euler’s method in t discretize unknown function: 7 Nov. 2000 15-859B - Introduction to Scientific Computing

Wave Equation: Numerical Solution for t = 2*dt:dt:endt u2(2:n) = 2*u1(2:n)-u0(2:n) +c*(dt/dx)^2*(u1(3:n+1)-2*u1(2:n)+u1(1:n-1)); u0 = u1; u1 = u2; end k+1 k k-1 j-1 j j+1 7 Nov. 2000 15-859B - Introduction to Scientific Computing

15-859B - Introduction to Scientific Computing Wave Equation Results dx=1/30 dt=.01 7 Nov. 2000 15-859B - Introduction to Scientific Computing

15-859B - Introduction to Scientific Computing Wave Equation Results 7 Nov. 2000 15-859B - Introduction to Scientific Computing

15-859B - Introduction to Scientific Computing Wave Equation Results 7 Nov. 2000 15-859B - Introduction to Scientific Computing

Poor results when dt too big dx=.05 dt=.06 Euler’s method unstable when step too large 7 Nov. 2000 15-859B - Introduction to Scientific Computing

15-859B - Introduction to Scientific Computing PDE Solution Methods Discretize in space, transform into system of IVP’s Discretize in space and time, finite difference method. Discretize in space and time, finite element method. Latter methods yield sparse systems. Sometimes the geometry and boundary conditions are simple (e.g. rectangular grid); Sometimes they’re not (need mesh of triangles). 7 Nov. 2000 15-859B - Introduction to Scientific Computing