Review on characterization of partial differential equation, discretization, and solution
Автор: Energy_Harmony
Загружено: 2025-12-31
Просмотров: 34
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Video Keypoints Summary
Simplification Methods: Transform PDEs to ODEs via similarity variables e.g., η = x / √(4ατ) for transient heat equation or symmetry, reducing variables for self-similar problems like boundary layers or Couette flow.
PDE Characterization: Classify using discriminant B² - 4AC: hyperbolic positive, waves/shocks), parabolic =0, diffusion, elliptic negative, steady-state; guides marching, stability, and boundary conditions.
Discretization Techniques: Finite Difference for simple geometries, Finite Element with flexible meshes, Finite Volume using conservative form, versatile for fluxes; steady vs. unsteady stencils with central/upwind schemes.
Solution Algorithms: SIMPLE or pressure-velocity coupling, segregated with sequential scalars, coupled using block matrices; time-marching like Crank-Nicolson for transients.
Boundary Layers & Turbulence: Prandtl equations simplified for thin layers; non-uniform grids, conjugate heat transfer; RANS:k-ε, k-ω, LES/DES for modeling mean/fluctuating flows.
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