PDEs and Examples of Phenomena Modeled n n

PDEs and Examples of Phenomena Modeled n n Ordinary differential equation: equation containing derivatives of a function of one variable Partial differential equation: equation containing derivatives of a function of two or more variables Models: n Air flow over an aircraft wing n Blood circulation in human body n Water circulation in an ocean n Bridge deformations as its carries traffic n Evolution of a thunderstorm n Oscillations of a skyscraper hit by earthquake n Strength of a toy n Financial Markets

Model of Sea Surface Temperature in Atlantic Ocean Courtesy MICOM group at the Rosenstiel School of Marine and Atmospheric Science, University of Miami

Solving PDEs Finite element method n Finite difference method (our focus) u Converts PDE into matrix equation t Linear system over discrete basis elements u Result is usually a sparse matrix u Matrix-based algorithms represent matrices explicitly u Matrix-free algorithms represent matrix values implicitly (our focus) n

FINITE DIFFERENCE In numerical analysis, two different approaches are commonly used: The finite difference and the finite element methods. In heat transfer problems, the finite difference method is used more often and will be discussed here. The finite difference method involves: Ø Establish nodal networks Ø Derive finite difference approximations for the governing equation at both interior and exterior nodal points Ø Develop a system of simultaneous algebraic nodal equations Ø Solve the system of equations using numerical schemes

Finite Difference Methods: Outline Solving ordinary and partial differential equations n Finite difference methods (FDM) vs Finite Element Methods (FEM) n Vibrating string problem n Steady state heat distribution problem n

Class of Linear Second-order PDEs n Linear second-order PDEs are of the form where A - H are functions of x and y only n Elliptic PDEs: B 2 - AC < 0 (steady state heat equations) n Parabolic PDEs: B 2 - AC = 0 (heat transfer equations) n Hyperbolic PDEs: B 2 - AC > 0 (wave equations)

Difference Quotients

Formulas for 1 st, 2 d Derivatives

Vibrating String Problem Vibrating string modeled by a hyperbolic PDE

The Nodal Networks

Finite Difference Approximation

Finite Difference Approximation cont.

Finite Difference Approximation cont.

A System of Algebraic Equations

Matrix Form

Numerical Solutions

Iteration

Example

Example (cont. )

Example (cont. )

Summary of nodal finite-difference relations for various configurations: Case 1 Interior Node

Case 2 Node at an internal corner with convection

Case 3 Node at a plane surface with convection

Case 4 Node at an external corner with convection

Case 5 Node at a plane surface with uniform heat flux