Teaching assistant ge-pde-ws2122

Class of Numerical methods for Partial Differential Equations (NumPDE) - WS 2021-2022

Global exercises (GEs): GE09GE10GE11GE12GE13GE14GE15GE15e

#TODO

• Linear Nonlinear Hyperbolic systems of conservation laws

  

  

  where unknown conserved quantity is defined as follows

  

  

• Cauchy's problems

• Riemann's problems

• Linear scalar advection equation

• Nonlinear viscous Burgers' inviscid Burgers' equations

• Hyperbolic systems of first order PDE:

  • (strictly) hyperbolicity: disctinct eigenvalues
  • diagonalizable matrix
  • Decomposition of system of first order PDEs into separated first order PDEs

• Shock solution vs Rarefaction solution

• Finite Volume Methods

• Conservation form

  

  

• Consistent numerical flux function

• Upwind-to-the-left (UWL):

  

  

• Upwind-to-the-right (UWR):

  

  

• Lax-Friedrichs (LF):

  

  

• Lax-Wendroff (LW):

• Lax-Friedrichs (LF) in conservation form:

  

  

  where the consistent numerical flux functions are defined as follows

  

  

  

  

• Lax-Wendroff (LW) in conservation form:

• Courant-Friedrichs-Lewy (CFL) condition

• Numerical Analysis: Lax Equivalence Theorem: Consistency Stability Convergence

  • Nonlinear numerical methods
  • Non-oscillatory behaviours of higher order numerical methods

• Exact Riemann solver: Godunov's scheme

• Approximate Riemann's solvers:

  • Roe's solver
  • Local Lax-Friedrichs (LLF) solver
  • Harten-Lax-van Leer (HLL) solver

• Structure of Finite Volume Method

  • Monotone -Contraction Total-Variation-Diminishing (TVD) Monotonicity-Preserving

• Nonlinear + higher order numerical scheme:

  • Harten theorem
  • Incremental form
  • TVD scheme

• A bit insight into Godunov's solver with graphical explanations

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Teaching assistant's extra materials and supplements

Unnoted materials

#TODO

• CW49/2021 Better open in web GE09
• CW50/2021 Better open in web GE10
• CW51/2021 Better open in web GE11
• CW02/2022 Better open in web GE12
• CW03/2022 Better open in web GE13
• CW03/2022 Better open in web GE14
• CW04/2022 Better open in web GE15
• CW05/2022 Better open in web GE15e

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Noted materials

#TODO

• CW49/2021 Better open in web GE09-offline
• CW50/2021 Better open in web GE10-offline
• CW51/2021 Better open in web GE11-offline
• CW02/2022 Better open in web GE12-noted-online
• CW03/2022 Better open in web GE13-noted-online
• CW04/2022 Better open in web GE14-noted-online
• CW05/2022 Better open in web GE15-noted-online