Lecture Notes on Discontinuous Galerkin Method and Numerical Fluxes

Key Concepts

• Discontinuous Galerkin Method (DG)

  • Involves clean formulation of discontinuous lurking formulation.
  • Introduction of new domains:
    • ( \tilde{\Omega} \gamma ): broken element or domain volume.
    • Mesh skeleton.

• New Operators Introduced

  • Average operator.
  • Jump operator (unintuitive definition).

Numerical Flux Selection

• Importance of selecting a numerical flux with specific properties:
  • Consistency
    • Definition: ( \hat{F}(\Phi, \Psi) = F(\Phi) )
    • Ensures that if the solution is single valued, the numerical flux equals the true flux.
  • Conservativity
    • Definition: Jump of ( f ) is zero.
    • A single-valued flux guarantees conservation properties in the solution.

Why Consistency and Conservativity Matter

• Consistency

  • If the method is consistent, it ensures convergence.
  • Guarantees that the true solution will satisfy the finite element formulation (variational consistency).
  • A consistent method will converge under mesh refinement.

• Conservativity

  • Ensures that the numerical flux is single-valued on both sides of the boundary, maintaining conservation properties.
  • A conservative flux simplifies the formulation, leading to cleaner numerical results.

Mathematical Implications

• If a flux is conservative:

  • The equation simplifies as the jump becomes zero.
  • This leads to clear relations in conservation laws for each element.

• Test Function Argument

  • Choosing a test function that is one inside a single element and zero elsewhere helps demonstrate conservation properties.
  • This is possible due to the discontinuous nature of the method; not feasible in continuous methods.

Final Thoughts

• Emphasis on ensuring that numerical fluxes are consistent and conservative to accurately reflect the physics of the relevant partial differential equations.
• Next steps include exploring different examples of numerical fluxes and analyzing their properties.