Ordinary Differential Equations JyunMing Chen 1 Contents Review
- Slides: 67
Ordinary Differential Equations Jyun-Ming Chen 1
Contents • Review • Euler’s method • 2 nd order methods – Midpoint – Heun’s • • Systems of ODE Stability Issue Implicit Methods Adaptive Stepsize • Runge-Kutta Method 2
Review • DE (Differential Equation) – An equation specifying the relations among the rate change (derivatives) of variables • ODE (Ordinary DE) vs. PDE (Partial DE) – The number of independent variables involved 3
Review (cont) • Solution of DE vs. Solution of Equation • Solution of an equation: • Geometrically, f(x) x 4
Review (cont) • Solution of an differential equation: Need additional conditions to specify a solution • Geometrically: x t 5
Review (cont) • Order of an ODE – The highest derivative in the equation • nth order ODE requires n conditions to specify the solution – IVP (initial value problem): All conditions specified at the same (initial) point – BVP (boundary value problem): otherwise 6
IVP VS. BVP Revisit Shooting Problem 7
IVP vs. BVP Physical meaning 8
Ode 2: solves 1 st and 2 nd order ODE Ic 1, ic 2, bc: setting conditions ‘ do not evaluate Maxima on ODE 9
Linear ODE • Linearity: – Involves no product nor nonlinear functions of y and its derivatives • nth order linear ODE 10
Focus of This Chapter • Solve IVP of nth order ODE numerically • e. g. , 11
ODE (IVP) • First order ODE (canonical form) • Every nth order ODE can be converted to n first order ODEs in the following method: 12
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Example 14
End of Review 15
The Canonical Problem This is Euler’s method 16
Example Compare with exact sol: 17
1 y Example (cont) y=e–x x 18
Error Analysis (Geometric Interpretation) Think in terms of Taylor’s expansion If the true solution were a straight line, then 19 Euler is exact
Error Analysis (From Taylor’s Expansion) Euler’s truncation error O(Dx 2) per step 1 st order method 20
y Cumulative Error x x=0 x=T Remark: Dx Error But computation time Number of steps = T/Dx Cumulative Err. = (T/Dx) O(Dx 2) = O(Dx) 21
Example (Euler’s) 22
Methods Improving Euler Motivated by Geometric Interpretation 23
Midpoint Method 24
Example (Midpoint) 25
Heun’s Method 26
Example (Heun’s) Note the result is the same as Midpoint!? 27
Remark • Comparison of Euler, Heun, midpoint – 1 st order: Euler – 2 nd order: Heun, midpoint • “order”: • All are special cases of RK (Runge-Kutta) methods Exact Euler (error) Midpoint (error) y(0. 1) 0. 905 0. 9 (0. 005) 0. 905 (0) y(0. 2) 0. 819 0. 81 (0. 009) 0. 819 (0) y(0. 3) 0. 741 0. 729 (0. 012) 0. 741 (0) 28
RK Methods 29
RK Methods (cont) 30
Taylor’s Expansion 31
RK st 1 Order 32
RK nd 2 Order 33
RK nd 2 Order (cont) 34
RK th 4 Order • Mostly commonly used one • Higher order … more evaluation, but less gain on accuracy Classical 4 th order RK 35
Classical th 4 order RK 36
System of ODE • Convert higher order ODE to 1 st order ODEs • All methods equally apply, in vector form 37
Example (Mass-Spring-Damper System) • Governing Equation • After setting the initial conditions x(0) and x’(0), compute the position and velocity of the mass for any t > 0 Initial Condition c k m x 38
Example (cont) 39
Example (cont) Assume m=1, c=1, k=1 (for ease of computation) set Dt=0. 1 40
Stability: Symptom 41
Symptom: Unstable Spring System Become unstable instantly … Start with this … Cause by stiff (k=4000) springs 42
Stability (cont) • Example Problem: Conditionally stable 43
Discussion • Different algorithm different stability limit – Check Midpoint Method • Different problem different stability limit – use the previous problem as benchmark 44
Review: Numerical Differentiation Taylor’s expansion: Forward difference Backward difference 45
Numerical Difference (cont) Central difference 46
Implicit Method (Backward Euler) Forward difference Backward difference 47
Example • Remark: – Always stable (for this problem) – Truncation error the same as Euler (only improve the stability) 48
Stiff Set of ODE Use the change of variable Get the following solution: Stability limit A stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is 49 taken to be extremely small
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Linear System of ODE with Constant Coefficients Do not really use (. . )-1. Solve linear system instead 51
Analysis Explicit and implicit Euler both in the form: represent x(0) in eigen basis x will converge if |li| 1 52
Analysis (cont) • Stiff equations have large eigenvalues • Explicit Euler requires small h to converge • Implicit Euler always converges (in this problem) 53
Example Explicit Implicit 54
Solving implicit methods by linearization is called a “semi-implicit” method Semi-Implicit Euler • Not guaranteed to be stable, but usually is Jacobian 55
About Jacobian Taylor’s expansion: Jacobian 56
Initial Condition c k m x Implicit Euler 57
Initial Condition c k Semi-Implicit Euler m x 58
Ex: Semi-Implicit Euler 59
Amazingly, this translates to… Very similar to Verlet integration formula… no wonder Verlet is pretty stable 60
Adaptive Stepsize • Solving ODE numerically … tracing the integral curve y(x) • what’s wrong with uniform step size – Uniformly small: waste effort – Uniformly large: might miss details 61
Step Doubling • Idea: – Estimate the truncation error by taking each step twice: one full step, two half steps – control the step size such that the estimated error is not too big. 1 (2 h 1) Desired h 0 2 (h 1) 62
Ex: RK nd 2 Order Overhead: # of f(x, y) evaluations 2 4– 2 = 6 63
Adaptive Step with RK 4 (NR) 64
GSL 65
Initialization 66
Iteration 67
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