Ordinary Differential Equations II
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1 Ordinary Differential Equations II CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations II 1 / 33
2 Almost Done! Last lecture on Wednesday! Homework 7: 6/3 (late days OK) Optional coding: 6/3 (no late days) Final exam review: 6/5 (if someone attends) Final exam: 6/8, 12:15pm-3:15pm (Hewlett 201; confirm make-up) CS 205A: Mathematical Methods Ordinary Differential Equations II 2 / 33
3 Study Tips Practice Especially on modeling problems! Do sanity checks Come up with a function and minimize it. Factor a matrix. Implement and experiment Iterate Look at solution, put it away, and try again. Time your practices Ask lots of questions! CS 205A: Mathematical Methods Ordinary Differential Equations II 3 / 33
4 Initial Value Problems Find f(t) : R R n Satisfying F [t, f(t), f (t), f (t),..., f (k) (t)] = 0 Given f(0), f (0), f (0),..., f (k 1) (0) CS 205A: Mathematical Methods Ordinary Differential Equations II 4 / 33
5 Most Famous Example F = m a Newton s second law CS 205A: Mathematical Methods Ordinary Differential Equations II 5 / 33
6 Reasonable Assumption Explicit ODE An ODE is explicit if can be written in the form f (k) (t) = F [t, f(t), f (t), f (t),..., f (k 1) (t)]. CS 205A: Mathematical Methods Ordinary Differential Equations II 6 / 33
7 d y After Reduction dt = F [ y] CS 205A: Mathematical Methods Ordinary Differential Equations II 7 / 33
8 Forward Euler y k+1 = y k + hf [ y k ] Explicit method O(h 2 ) localized truncation error O(h) global truncation error; first order accurate CS 205A: Mathematical Methods Ordinary Differential Equations II 8 / 33
9 Backward Euler y k+1 = y k + hf [ y k+1 ] Implicit method O(h 2 ) localized truncation error O(h) global truncation error; first order accurate CS 205A: Mathematical Methods Ordinary Differential Equations II 9 / 33
10 Trapezoid Method y k+1 = y k + h F [ y k] + F [ y k+1 ] 2 Implicit method O(h 3 ) localized truncation error O(h 2 ) global truncation error; second order accurate CS 205A: Mathematical Methods Ordinary Differential Equations II 10 / 33
11 Example y = A y CS 205A: Mathematical Methods Ordinary Differential Equations II 11 / 33
12 Example y = A y y k+1 = ( I n n ha 2 ) 1 ( I n n + ha 2 ) y k CS 205A: Mathematical Methods Ordinary Differential Equations II 11 / 33
13 Model Equation y = ay, a < 0 y k+1 = y k + h ay k + ay k+1 2 CS 205A: Mathematical Methods Ordinary Differential Equations II 12 / 33
14 Model Equation y = ay, a < 0 y k+1 = y k + h ay k + ay k+1 2 ( y k = ha ) k ha y 0 CS 205A: Mathematical Methods Ordinary Differential Equations II 12 / 33
15 Model Equation y = ay, a < 0 y k+1 = y k + h ay k + ay k+1 2 ( y k = ha ) k ha y 0 CS 205A: Mathematical Methods Ordinary Differential Equations II 12 / 33
16 Model Equation y = ay, a < 0 y k+1 = y k + h ay k + ay k+1 2 ( y k = ha ) k ha y 0 Unconditionally stable! CS 205A: Mathematical Methods Ordinary Differential Equations II 12 / 33
17 Model Equation y = ay, a < 0 y k+1 = y k + h ay k + ay k+1 2 ( y k = ha ) k ha y 0 Unconditionally stable! But this has nothing to do with accuracy. CS 205A: Mathematical Methods Ordinary Differential Equations II 12 / 33
18 Is Stability Useful Here? R = y k+1 y k CS 205A: Mathematical Methods Ordinary Differential Equations II 13 / 33
19 Is Stability Useful Here? R = y k+1 y k Oscillatory behavior! CS 205A: Mathematical Methods Ordinary Differential Equations II 13 / 33
20 Example Useful for mid-size time steps CS 205A: Mathematical Methods Ordinary Differential Equations II 14 / 33
21 Pattern Convergence as h 0 is not the whole story nor is stability. Often should consider behavior for fixed h > 0. CS 205A: Mathematical Methods Ordinary Differential Equations II 15 / 33
22 Pattern Convergence as h 0 is not the whole story nor is stability. Often should consider behavior for fixed h > 0. Qualitative and quantitative analysis needed! CS 205A: Mathematical Methods Ordinary Differential Equations II 15 / 33
23 Observation y k+1 = y k + tk +h t k F [ y(t)] dt CS 205A: Mathematical Methods Ordinary Differential Equations II 16 / 33
24 Alternative Derivation of Trapezoid Method y k+1 = y k + h 2 (F [ y k] + F [ y k+1 ]) + O(h 3 ) Implicit method CS 205A: Mathematical Methods Ordinary Differential Equations II 17 / 33
25 Heun s Method y k+1 = y k + h 2 (F [ y k] + F [ y k + hf [ y k ]]) + O(h 3 ) Replace implicit part with lower-order integrator! CS 205A: Mathematical Methods Ordinary Differential Equations II 18 / 33
26 Stability of Heun s Method y k+1 = y = ay, a < 0 (1 + ha + 12 ) h2 a 2 y k CS 205A: Mathematical Methods Ordinary Differential Equations II 19 / 33
27 Stability of Heun s Method y k+1 = y = ay, a < 0 (1 + ha + 12 ) h2 a 2 y k = h < 2 a CS 205A: Mathematical Methods Ordinary Differential Equations II 19 / 33
28 Stability of Heun s Method y k+1 = y = ay, a < 0 (1 + ha + 12 ) h2 a 2 y k = h < 2 a Forward Euler: h < 2 a CS 205A: Mathematical Methods Ordinary Differential Equations II 19 / 33
29 Runge-Kutta Methods Apply quadrature to: y k+1 = y k + tk+1 t k F [ y(t)] dt CS 205A: Mathematical Methods Ordinary Differential Equations II 20 / 33
30 Runge-Kutta Methods Apply quadrature to: y k+1 = y k + tk+1 t k Use low-order integrators F [ y(t)] dt CS 205A: Mathematical Methods Ordinary Differential Equations II 20 / 33
31 RK4: Simpson s Rule y k+1 = y k + h 6 ( k k k 3 + k 4 ) where k 1 = F [ y k ] k2 = F [ y k + 12 ] h k 1 k3 = F [ y k + 12 ] h k 2 k4 = F [ y k + h ] k 3 CS 205A: Mathematical Methods Ordinary Differential Equations II 21 / 33
32 Using the Model Equation y A y = y(t) e At y 0 CS 205A: Mathematical Methods Ordinary Differential Equations II 22 / 33
33 Exponential Integrators y = A y + G[ y] CS 205A: Mathematical Methods Ordinary Differential Equations II 23 / 33
34 Exponential Integrators y = A y + G[ y] First-order exponential integrator: y k+1 = e Ah y k + A 1 (e Ah I n n )G[ y k ] CS 205A: Mathematical Methods Ordinary Differential Equations II 23 / 33
35 Returning to Our Reduction y (t) = v(t) v (t) = a(t) a(t) = F [t, y(t), v(t)] CS 205A: Mathematical Methods Ordinary Differential Equations II 24 / 33
36 Returning to Our Reduction y (t) = v(t) v (t) = a(t) a(t) = F [t, y(t), v(t)] y(t k + h) = y(t k ) + h y (t k ) + h2 2 y (t k ) + O(h 3 ) Don t need high-order estimators for derivatives. CS 205A: Mathematical Methods Ordinary Differential Equations II 24 / 33
37 New Formulae v k+1 = v k + tk+1 t k a(t) dt CS 205A: Mathematical Methods Ordinary Differential Equations II 25 / 33
38 New Formulae v k+1 = v k + y k+1 = y k + h v k + tk+1 t k tk+1 t k a(t) dt (t k+1 t) a(t) dt CS 205A: Mathematical Methods Ordinary Differential Equations II 25 / 33
39 New Formulae v k+1 = v k + y k+1 = y k + h v k + tk+1 t k tk+1 t k a(t) dt (t k+1 t) a(t) dt a(τ) = (1 γ) a k + γ a k+1 + a (τ)(τ hγ t k ) + O(h 2 ) for τ [a, b] CS 205A: Mathematical Methods Ordinary Differential Equations II 25 / 33
40 Newmark Class y k+1 = y k + h v k + ( ) 1 2 β h 2 a k + βh 2 a k+1 v k+1 = v k + (1 γ)h a k + γh a k+1 a k = F [t k, y k, v k ] CS 205A: Mathematical Methods Ordinary Differential Equations II 26 / 33
41 Newmark Class y k+1 = y k + h v k + ( ) 1 2 β h 2 a k + βh 2 a k+1 v k+1 = v k + (1 γ)h a k + γh a k+1 a k = F [t k, y k, v k ] Parameters: β, γ (can be implicit or explicit!) CS 205A: Mathematical Methods Ordinary Differential Equations II 26 / 33
42 Constant Acceleration: β = γ = 0 y k+1 = y k + h v k h2 a k v k+1 = v k + h a k Explicit, exact for constant acceleration, linear accuracy CS 205A: Mathematical Methods Ordinary Differential Equations II 27 / 33
43 Constant Implicit Acceleration: β = 1 /2, γ = 1 y k+1 = y k + h v k h2 a k+1 v k+1 = v k + h a k+1 Backward Euler for velocity, midpoint for position; globally first-order accurate CS 205A: Mathematical Methods Ordinary Differential Equations II 28 / 33
44 Trapezoid: β = 1 /4, γ = 1 /2 x k+1 = x k h( v k + v k+1 ) v k+1 = v k h( a k + a k+1 ) Trapezoid for position and velocity; globally second-order accurate CS 205A: Mathematical Methods Ordinary Differential Equations II 29 / 33
45 Central Differencing: β = 0, γ = 1 /2 v k+1 = y k+2 y k 2h a k+1 = y k+1 2 y k+1 + y k h 2 Central difference for position and velocity; globally second-order accurate CS 205A: Mathematical Methods Ordinary Differential Equations II 30 / 33
46 Newmark Schemes Generalizes large class of integrators Second-order accuracy (exactly) when γ = 1 /2 Unconditional stability when 4β > 2γ > 1 (otherwise radius is function of β, γ) CS 205A: Mathematical Methods Ordinary Differential Equations II 31 / 33
47 Staggered Grid y k+1 = y k + h v k+ 1/2 v k+ 3/2 = v k+ 1/2 + h a k+1 [ a k+1 = F t k+1, x k+1, 1 ] 2 ( v k+1/2 + v k+ 3/2) CS 205A: Mathematical Methods Ordinary Differential Equations II 32 / 33
48 Staggered Grid y k+1 = y k + h v k+ 1/2 v k+ 3/2 = v k+ 1/2 + h a k+1 [ a k+1 = F t k+1, x k+1, 1 ] 2 ( v k+1/2 + v k+ 3/2) Leapfrog: a not a function of v; explicit CS 205A: Mathematical Methods Ordinary Differential Equations II 32 / 33
49 Staggered Grid y k+1 = y k + h v k+ 1/2 v k+ 3/2 = v k+ 1/2 + h a k+1 [ a k+1 = F t k+1, x k+1, 1 ] 2 ( v k+1/2 + v k+ 3/2) Leapfrog: a not a function of v; explicit Otherwise: Symmetry in v dependence CS 205A: Mathematical Methods Ordinary Differential Equations II 32 / 33
50 Leapfrog Integration y 0 y 1 y 2 y 3 y 4 v 0 v1/2 v3/2 v5/2 v7/2 a 0 a 1 a 2 a 3 a 4 +t Next CS 205A: Mathematical Methods Ordinary Differential Equations II 33 / 33
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