Partial Differential Equations II
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1 Partial Differential Equations II CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Partial Differential Equations II 1 / 28
2 Almost Done! Homework 7: 12/2 (two days late!) Homework 8: 12/9 (optional) Section: 12/6 (final review) Final exam: 12/12, 12:15pm (Gates B03) Go to office hours! CS 205A: Mathematical Methods Partial Differential Equations II 2 / 28
3 Course Reviews On Axess! Additional comments: CS 205A: Mathematical Methods Partial Differential Equations II 3 / 28
4 Request for Help CS 205A notes your help! Textbook Review text Write reference implementations Solidify your CS205A knowledge CS 205A: Mathematical Methods Partial Differential Equations II 4 / 28
5 Final Exam Cumulative Similar format to midterm Two sheets of notes CS 205A: Mathematical Methods Partial Differential Equations II 5 / 28
6 This Week Couple relationships between derivatives. Pressure gradient determining fluid flow Image operators using x and y derivatives Partial Differential Equations (PDE) CS 205A: Mathematical Methods Partial Differential Equations II 6 / 28
7 Boundary Value Problems Dirichlet conditions: Value of f( x) on Ω Neumann conditions: Derivatives of f( x) on Ω Mixed or Robin conditions: Combination CS 205A: Mathematical Methods Partial Differential Equations II 7 / 28
8 Second-Order Model Equation ij a ij f x i x j + i b i f x i + cf = 0 ( A + b + c)f = 0 CS 205A: Mathematical Methods Partial Differential Equations II 8 / 28
9 Classification of Second-Order PDE ( A + b + c)f = 0 If A is positive or negative definite, system is elliptic. If A is positive or negative semidefinite, the system is parabolic. If A has only one eigenvalue of different sign from the rest, the system is hyperbolic. If A satisfies none of the criteria, the system is ultrahyperbolic. CS 205A: Mathematical Methods Partial Differential Equations II 9 / 28
10 Derivative Operator Matrix h 2 w = L 1 y Dirichlet CS 205A: Mathematical Methods Partial Differential Equations II 10 / 28
11 What About First Derivative? Potential for asymmetry at boundary Centered differences: Fencepost problem Possible resolution: Imitate leapfrog CS 205A: Mathematical Methods Partial Differential Equations II 11 / 28
12 Fencepost Problem CS 205A: Mathematical Methods Partial Differential Equations II 12 / 28
13 Big Idea Derivatives : Functions :: Matrices : Vectors CS 205A: Mathematical Methods Partial Differential Equations II 13 / 28
14 Elliptic PDE Lf = g L y = b CS 205A: Mathematical Methods Partial Differential Equations II 14 / 28
15 Elliptic PDE Lf = g L y = b Example: Laplace s equation on a line CS 205A: Mathematical Methods Partial Differential Equations II 14 / 28
16 Common Theme Elliptic PDE Positive definite matrix CS 205A: Mathematical Methods Partial Differential Equations II 15 / 28
17 Common Theme Elliptic PDE Positive definite matrix L = D D, D = CS 205A: Mathematical Methods Partial Differential Equations II 15 / 28
18 Common Theme Elliptic PDE Positive definite matrix L = D D, D = Review: Name two ways to solve. CS 205A: Mathematical Methods Partial Differential Equations II 15 / 28
19 Time Dependence Choice: 1. Treat t separate from x ( semidiscrete ) 2. Treat all variables democratically ( fully discrete ) CS 205A: Mathematical Methods Partial Differential Equations II 16 / 28
20 Semidiscrete Heat Equation f t = f xx CS 205A: Mathematical Methods Partial Differential Equations II 17 / 28
21 Semidiscrete Heat Equation f t = f xx f t = Lf CS 205A: Mathematical Methods Partial Differential Equations II 17 / 28
22 Semidiscrete Heat Equation f t = f xx f t = Lf Stability for elliptic spatial operator (parabolic PDE) CS 205A: Mathematical Methods Partial Differential Equations II 17 / 28
23 Semidiscrete Time Stepping Left with a multivariable ODE problem! Forward/backward Euler, RK, and friends CS 205A: Mathematical Methods Partial Differential Equations II 18 / 28
24 Semidiscrete Time Stepping Left with a multivariable ODE problem! Forward/backward Euler, RK, and friends Implicit vs. explicit (vs. symplectic) CS 205A: Mathematical Methods Partial Differential Equations II 18 / 28
25 Semidiscrete Time Stepping Left with a multivariable ODE problem! Forward/backward Euler, RK, and friends Implicit vs. explicit (vs. symplectic) Alternative: Eigenvector methods (low-frequency approximation) CS 205A: Mathematical Methods Partial Differential Equations II 18 / 28
26 Fully Discrete PDE Discretize x and t simultaneously Can create larger linear algebra problems Philosophical point: What is fully discrete? CS 205A: Mathematical Methods Partial Differential Equations II 19 / 28
27 Reminders Review Numerical PDEs Gradient Domain Inpainting Gradient Domain Inpainting CS 205A: Mathematical Methods Partial Differential Equations II 20 / 28 Fluids
28 Gradient Domain Pipeline for image I(x, y): 1. Compute gradient: v(x, y) = I(x, y) 2. Edit: v v 3. Reconstruct: g =? v CS 205A: Mathematical Methods Partial Differential Equations II 21 / 28
29 Gradient Domain Reconstruction min g Ω g v 2 2 da CS 205A: Mathematical Methods Partial Differential Equations II 22 / 28
30 Gradient Domain Reconstruction min g Ω g v 2 2 da 2 g = v Elliptic! CS 205A: Mathematical Methods Partial Differential Equations II 22 / 28
31 Incompressible Navier-Stokes ( v ρ t ) + v v = p + µ 2 v + f t [0, ): Time v(t) : Ω R 3 : Velocity ρ(t) : Ω R: Density p(t) : Ω R: Pressure f(t) : Ω R 3 : External forces (e.g. gravity) CS 205A: Mathematical Methods Partial Differential Equations II 23 / 28
32 Lagrangian vs. Eulerian Lagrangian: Track parcels of fluid Eulerian: Fluid flows past a point in space CS 205A: Mathematical Methods Partial Differential Equations II 24 / 28
33 Marker-and-Cell (MAC) Grid CS 205A: Mathematical Methods Partial Differential Equations II 25 / 28
34 Splitting for Incompressible Flow u = 0 (divergence-free) ρ t + u ρ = 0 (density advection) u t + u u + p ρ = g (velocity advection) CS 205A: Mathematical Methods Partial Differential Equations II 26 / 28
35 Steps for Flow (on board) 1. Adjust t 2. Advect velocity 3. Apply forces 4. Solve for pressure: p ρ divergence-free projection 5. Advect density = u; CS 205A: Mathematical Methods Partial Differential Equations II 27 / 28
36 Semilagrangian Advection ecmwf.int/newsevents/training/rcourse_notes/numerical_methods/numerical_methods/numerical_methods6.html Next CS 205A: Mathematical Methods Partial Differential Equations II 28 / 28
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