Introduction to Numerical Methods for ODEs and PDEs

Methods of Approximation

Lecture 3: finite differences

Lecture 4: finite elements

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

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

Applications of finite differencing strategies

1. Time integration of canonical initial value problems (ODEs)

• Stability and accuracy; unconditional versus conditional stability

• Implicit vs. explicit schemes

2. Finite difference treatment of boundary value problems (steady state)

• Case study: 1D steady state advection-diffusion• Stabilization through upwinding

Applications of finite differencing strategies (cont.)

3. Finite difference treatment of initial/boundary value problems (time and space dependent)

• Semi-discrete approaches (method of lines)

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