1 / 6

Introduction to Numerical Methods for ODEs and PDEs

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.

juanwilson
Download Presentation

Introduction to Numerical Methods for ODEs and PDEs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related