Introduction to Numerical Methods for ODEs and PDEs

1. Introduction to Numerical Methods for ODEs and PDEs Methods of Approximation Lecture 3: finite differences Lecture 4: finite elements

2. Prevalent numerical methods in engineering and the sciences We will introduce in some detail the basic ideas associated with two classes of numerical methods • Finite Difference Methods (in which the strong form of the boundary value problem, introduced in the model problems, is directly approximated using difference operators) • Finite Element Methods (in which the weak form of the boundary value problem, derived through integral weighting of the BVP, is approximated instead) ….while skipping a third class of methods which are quite prevalent Boundary Element Methods (BEM) • Predominantly for linear problems; based on reciprocity theorems and Green’s function solutions

3. Finite Difference Methods Rely on direct approximation of governing differential equations, using numerical differentiation formulas • Ordinary derivative approximations • Forward difference approximations • Backward difference approximations • Central difference operators • Partial derivative approximations

4. Applications of finite differencing strategies • Time integration of canonical initial value problems (ODEs) • Stability and accuracy; unconditional versus conditional stability • Implicit vs. explicit schemes • Finite difference treatment of boundary value problems (steady state) • Case study: 1D steady state advection-diffusion • Stabilization through upwinding

5. Applications of finite differencing strategies (cont.) • Finite difference treatment of initial/boundary value problems (time and space dependent) • Semi-discrete approaches (method of lines)

6. Finite Element Methods Using the 1D rod problem (elliptic) as a template: • Development of weak form (variational principle) • Galerkin approximation versus other weighting approaches • Development of discrete equations for linear shape function case