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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)

    • 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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