PDEs: General classification

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Presentation transcript:

PDEs: General classification “Elliptic” Typical: LaPlace’s Eq. steady-state gravity, electrostatics “Parabolic” Typical: Heat Eq. conduction “Hyperbolic” Typical: Wave Eq. vibration, propagation

PDEs: discretization and computational molecules Forward difference Backward difference Symmetric difference Second difference Mixed partials Discrete Laplacian Example: Marching algorithm for heat equation

PDEs: Where do they come from? Sketch of derivations: Heat (diffusion) equation Wave equation solution: traveling waves boundary conditions eigenmodes characteristics  Dan Russell's excellent site

PDEs (wrapping up; segue to Fourier analysis, DSP, image processing, etc.) More on numerical solution von Neumann stability analysis Consistency Example of inconsistency Nonlinear vibration Fermi-Pasta-Ulam-Tsingou and the idea of a numerical experiment Fourier modes, nonlinear PDEs, solitons

Upcoming themes Surprise in science Fourier (modal) analysis The Fourier transform becomes an algorithm The FFT, O(n log n) vs. n2, changes life Sound and images become digital