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Types of differential equations Discontinuous Galerkin Method What is this ?

Basic Concepts & Physics. Types of differential equations Discontinuous Galerkin Method What is this ? Why do we use it ? How it differs from Continuous Galerkin method ? Where and when is it applicable ?. Understanding Discontinuous Galerkin. (Input). Unknowns.

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Types of differential equations Discontinuous Galerkin Method What is this ?

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  1. Basic Concepts & Physics • Types of differential equations • Discontinuous Galerkin Method • What is this ? • Why do we use it ? • How it differs from Continuous Galerkin method ? • Where and when is it applicable ?

  2. Understanding Discontinuous Galerkin

  3. (Input) Unknowns tn-1 tn yhand vh are allowed to be discontinuous at the nodes Discretization strategy of DG

  4. x Xi-1 Xi Xi+1 y ym ym- ym+ t tm tm-1 Discretization of continuous and DG Continuous Galerkin Discontinuous Galerkin

  5. V = {v : v is continuous on the intervals In} Let v(t) V be an arbitrary function and y(t) is solution Integrating by parts Formulation

  6. Local and Global view DG with piecewise polynomials

  7. DGM(0), DGM(1),DGM(2),DGM(3) DGM(0)

  8. DGM(1)

  9. DGM(1)....

  10. DGM(2)

  11. DGM(2)….

  12. DGM(2)….

  13. Observations for Calculations

  14. Graphical representation for results

  15. A - stability and Pade Approximation

  16. Conclusion • Galerkin methods for ordinary differential equations • are A-stable. • With piecewise polynomial spaces of degree q = 0,1,2 • the order is p = 2k+1

  17. Tasks Ahead • Solving all types of differential equations with RKDG and DG • Space discretization with LDG and comparing it with Galerkin • matrix perturbation methods • SGM methods for convection dominated problems

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