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Introduction to Numerical Methods for ODEs and PDEsPowerPoint Presentation

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)
- Case study: 1D steady state advection-diffusion
- 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

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