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**Ordinary Differential Equations**

Jyun-Ming Chen

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**Contents Review Euler’s method 2nd order methods Runge-Kutta Method**

Midpoint Heun’s Runge-Kutta Method Systems of ODE Stability Issue Implicit Methods Adaptive Stepsize

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**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

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**Solution of DE vs. Solution of Equation**

Review (cont) Solution of DE vs. Solution of Equation Solution of an equation: Geometrically, f(x) x

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**Need additional conditions to specify a solution**

Review (cont) Solution of an differential equation: Geometrically: t x

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**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

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**IVP VS. BVP Revisit Shooting Problem**

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IVP vs. BVP Physical meaning

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**Maxima on ODE Ode2: solves 1st and 2nd order ODE**

Ic1, ic2, bc: setting conditions ‘ do not evaluate Maxima on ODE

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**Linear ODE Linearity: nth order linear ODE**

Involves no product nor nonlinear functions of y and its derivatives nth order linear ODE

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Focus of This Chapter Solve IVP of nth order ODE numerically e.g.,

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**ODE (IVP) First order ODE (canonical form)**

Every nth order ODE can be converted to n first order ODEs in the following method:

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Example

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End of Review

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The Canonical Problem This is Euler’s method

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Example Compare with exact sol:

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Example (cont) 1 x y y=e–x

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**Error Analysis (Geometric Interpretation)**

Think in terms of Taylor’s expansion If the true solution were a straight line, then Euler is exact

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**Error Analysis (From Taylor’s Expansion)**

Euler’s Euler’s truncation error O(Dx2) per step 1st order method

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**Cumulative Error y x Remark: Dx Error But computation time x = 0**

x = T Number of steps = T/Dx Cumulative Err. = (T/Dx) O(Dx2) = O(Dx)

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Example (Euler’s)

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**Methods Improving Euler**

Motivated by Geometric Interpretation

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Midpoint Method

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Example (Midpoint)

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Heun’s Method

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**Note the result is the same as Midpoint!?**

Example (Heun’s) Note the result is the same as Midpoint!?

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**Remark Comparison of Euler, Heun, midpoint “order”:**

1st order: Euler 2nd 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) y(0.2) 0.819 0.81 (0.009) y(0.3) 0.741 0.729 (0.012)

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RK Methods

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RK Methods (cont)

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Taylor’s Expansion

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RK 1st Order

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RK 2nd Order

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RK 2nd Order (cont)

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**RK 4th Order Mostly commonly used one**

Higher order … more evaluation, but less gain on accuracy Classical 4th order RK

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Classical 4th order RK

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**System of ODE Convert higher order ODE to 1st order ODEs**

All methods equally apply, in vector form

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**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 c k Initial Condition x m

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Example (cont)

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**Example (cont) set Dt=0.1 Assume m=1,c=1, k=1**

(for ease of computation) set Dt=0.1

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Stability: Symptom

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**Symptom: Unstable Spring System**

Become unstable instantly … Start with this … Cause by stiff (k=4000) springs

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Stability (cont) Example Problem: Conditionally stable

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**Discussion Different algorithm different stability limit**

Check Midpoint Method Different problem different stability limit use the previous problem as benchmark

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**Review: Numerical Differentiation**

Taylor’s expansion: Forward difference Backward difference

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**Numerical Difference (cont)**

Central difference

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**Implicit Method (Backward Euler)**

Forward difference Backward difference

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**Example Remark: Always stable (for this problem)**

Truncation error the same as Euler (only improve the stability)

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**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 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

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**Analysis Explicit and implicit Euler both in the form:**

represent x(0) in eigen basis x will converge if |li| 1

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**Analysis (cont) Stiff equations have large eigenvalues**

Explicit Euler requires small h to converge Implicit Euler always converges (in this problem)

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Example Explicit Implicit

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**Semi-Implicit Euler Not guaranteed to be stable, but usually is**

Solving implicit methods by linearization is called a “semi-implicit” method Semi-Implicit Euler Not guaranteed to be stable, but usually is Jacobian

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About Jacobian Taylor’s expansion: Jacobian

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c k Initial Condition x m Implicit Euler

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c k Initial Condition x m Semi-Implicit Euler

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**Ex: Semi-Implicit Euler**

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**Amazingly, this translates to…**

Very similar to Verlet integration formula… no wonder Verlet is pretty stable

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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

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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(2h1) 2(h1) Desired h0

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Ex: RK 2nd Order Overhead: # of f(x,y) evaluations 24–2 = 6

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**Adaptive Step with RK4 (NR)**

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GSL

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Initialization

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Iteration

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