Despite its apparent simplicity this equation appears in a wide range of dis m heat 7 to financial er we will make extensive use of this equation, and several of the limiting cases contained therein, to illustrate the numerical techniques that will be presented

Background Developed over the last 25 years- Brandt (1973) published first paper with practical results Offers the possibility of solving a problem with work and storage proportional to the number of unknowns Well developed for linear elliptic problems application to other equations is still an active area of research

1 First Order ave Equation SLIDE 1 The simplest first order partial differential equation in two variables(a, t)is the linear wave equation. Recall that all first order PDE's are of hyperbolic type INITIAL BOUNDARY VALUE PROBLEM (IBVP) 0,x∈(0,1)

Motivation Consider a standard second order finite difference discretization

1 Background Brandt(1973)published first paper SLIDE 1 Offers the possibility of solving a problem with work and storage propor tional to the number of unknowns Well developed for elliptic proble

1 Motivation SLIDE 1 Consider a standard second order finite difference discretization of V-u= on a regular g 1.2. and 3 dimensions 1.1 1D Finite differences

Governing Equation Stability Analysis 3 Examples Relationship between σ and λh Implicit Time-Marching Scheme

Outline Governing Equation Stability Analysis 3 Examples Relationship between σ and λh Implicit Time-Marching Scheme

1 Finite difference formulas 1.1 Problem definition We have seen that one of the necessary ingredients in devising finite differe

Finite Difference Problem Definition Formulas