# Introduction to Numerical Methods for ODEs and PDEs - PowerPoint PPT Presentation

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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.

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Introduction to Numerical Methods for ODEs and PDEs

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## 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

• 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)

• Stabilization through upwinding

### Applications of finite differencing strategies (cont.)

• 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