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Hybrid Discontinuous Galerkin Methods for Serre-Green-Naghdi Water Wave Models

This paper explores the application of hybrid discontinuous Galerkin methods to solve Serre-Green-Naghdi water wave models. The motivations include studying dispersive waves away from the coast and the governing equations of the water wave problem. The research delves into the Dirichlet-Neumann operator, dispersive properties, and asymptotic expansion of water wave models. Specific focus is given to the Korteweg-de Vries and Green-Naghdi equations, discussing solution procedures, stability, accuracy, nonlinear solvers, and examples. The study also presents numerical results for both types of equations.

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Hybrid Discontinuous Galerkin Methods for Serre-Green-Naghdi Water Wave Models

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  1. Hybrid Discontinuous Galerkin Methods for Serre-Green-Naghdi Water Wave Models Ali Sami and Clint Dawson

  2. Acknowledgments

  3. Motivation

  4. Motivation: Long waves away from the coast

  5. Motivation: Dispersive waves

  6. Motivation: Dispersive waves

  7. The Water Wave Problem: Assumptions

  8. Shallow Water Regime

  9. Shallow Water Regime

  10. Shallow Water Regime

  11. Shallow Water Regime

  12. The Water Wave Problem: Governing equations

  13. The Water Wave Problem: Governing equations

  14. The Water Wave Problem: Governing equations

  15. The Water Wave Problem: Dirichlet Neumann Operator

  16. The Water Wave Problem: Dirichlet Neumann Operator

  17. The Water Wave Problem: Dispersive properties

  18. The Water Wave Problem: Dispersive properties

  19. The Water Wave Problem: Nondimensionalization

  20. Water Wave Problem: Asymptotic expansion

  21. Water Wave Problem: Asymptotic expansion

  22. Asymptotic models:

  23. Outline

  24. Korteweg de Vries Equation

  25. Korteweg de Vries Equation: Solution to linear equation

  26. Korteweg de Vries Equation: Solution procedure using HDG

  27. Korteweg de Vries Equation: Solution procedure using HDG

  28. Korteweg de Vries Equation: Stability of the method

  29. Korteweg de Vries Equation: Accuracy of the results

  30. Korteweg de Vries Equation: Nonlinear solver

  31. Korteweg de Vries Equation: Nonlinear solver

  32. Korteweg de Vries Equation: Nonlinear accuracy

  33. Korteweg de Vries Equation: Nonlinear examples

  34. Korteweg de Vries Equation: Nonlinear examples

  35. Korteweg de Vries Equation: Nonlinear examples

  36. Outline

  37. NSWE: Solution approach

  38. NSWE: Solution approach

  39. NSWE: Solution approach

  40. NSWE: Numerical results

  41. NSWE: Numerical results

  42. NSWE: Numerical results

  43. NSWE: Numerical results

  44. NSWE: Numerical results

  45. Outline

  46. Asymptotic models:

  47. Green-Naghdi Equation: Solution approach

  48. Green-Naghdi Equation: Strang splitting

  49. Green-Naghdi Equation: Solving the dispersive part

  50. Green-Naghdi Equation: Solving the dispersive part

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