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Dr. Jie Zou PHY 33201 Chapter 9 Ordinary Differential Equations: Initial-Value Problems Lecture (II) 1 1 Besides the main textbook, also see Ref.: “Applied Numerical Methods with MATLAB for Engineers and Scientists”, Steven Chapra, 2nd ed., Ch. 20, McGraw Hill, 2008.

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Dr. Jie Zou PHY 33202 Outline Error Analysis for and Stability of Euler’s Method Improvements of Euler’s method (I) Heun’s Method (II) The Midpoint Method (next lecture)

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Dr. Jie Zou PHY 33203 Error analysis for Euler’s method Two types of errors involved in the numerical solution of an ODE: (1) Truncation errors: Errors due to the nature of the numerical techniques used to approximate values of y. Two parts: (I) Local truncation error: Error due to an application of the numerical method over a single step. (II) Propagated truncation error: Error due to the approximations produced during the previous steps. Global truncation error (total error): The sum of (I) and (II). (2) Roundoff errors: Errors due to the limitation of the computer to keep an infinite number of significant digits.

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4 Analysis of the magnitude and properties of the truncation error Derive Euler’s method from the Taylor series expansion: Taylor series expansion of y(t) about a point t i : h = t i+1 – t i : step size; R n = y (n+1) ( )h n+1 /(n+1)!: the reminder term; lies somewhere between t i and t i+1. Evaluating y(t) at point t i+1, we have: Using the general form of an ODE, dy/dt = f(t, y), we have: O(h n+1 ) : For the nth-order approximation, the local truncation error h n+1.

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Dr. Jie Zou PHY 33205 True and approximate local truncation error Comparing the following two equations: (1) Taylor series expansion (2) Euler’s method True local truncation error Approximate local truncation error When h is sufficiently small, the higher- order terms are usually negligible.

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Dr. Jie Zou PHY 33206 Global truncation error It can be shown that the global truncation error is O(h), i.e., step size h. Note: The global error can be reduced by decreasing the step size h. If the solution of the ODE is linear, there is no error involved in Euler’s method – Euler’s method is a first-order method.

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Dr. Jie Zou PHY 33207 Stability of Euler’s method “Unstable”: A numerical solution is said to be unstable if errors grow exponentially for a problem for which there is a bounded solution. The stability of a particular numerical method depends on three factors: (1) Differential equation (2) Numerical method (3) Step size An example: Let’s look at a simple ODE dy/dt = -ay, y(0) = y 0 Calculus: y = y 0 e -at Euler’s method: y i+1 = y i + (dy i /dt)h = y i (1 – ah) If h > 2/a, then |1 – ah| > 1 and |y i | as i - Euler’s method is conditionally stable.

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Dr. Jie Zou PHY 33208 Improvement of Euler’s method: (I) Heun’s method Heun’s method: a predictor-corrector approach (1) Predicator: Use the slope at the beginning of an interval to predict that at the end of the interval. (2) Corrector: Use the averaged slope to obtain an improved estimate of y at the end of the interval. Predictor: y i+1 0 = y i m + f(t i,y i )h Corrector: y i+1 j = y i m + {[(f(t i,y i m ) + f(t i+1, y i+1 j- 1 )]/2}h, for j = 1, 2, … The corrector equation is iterative; j indicates the jth iteration. Stopping criterion: | a | = |(y i+1 j – y i+1 j-1 )/y i+1 j | x 100% (a) Predictor (b) Corrector

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Dr. Jie Zou PHY 33209 Example: Heun’s method Example 20.2 (Ref.): Use Heun’s method with iteration to integrate y’ = 4e 0.8t – 0.5y from t = 0 to 4 with a step size of 1. The initial condition at t = 0 is y = 2. Employ a stopping criterion of 0.00001% to terminate the corrector iterations. (1) By hand. (2) Write an M-file. A copy of the code will be handed out later.

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Dr. Jie Zou PHY 332010 Results ty true y Euler | t | (%) y Heun | t | (%) y Heun | t | (%) 02.00000 -----2.00000-----2.00000---- 16.194635.0000019.286.701088.186.360872.68 214.8439211.4021623.1916.319789.9415.302243.09 333.6771725.5132124.2437.19992510.4634.743283.17 475.3389656.8493124.5483.3377710.6277.735103.18 Without Iteration With Iteration Table 20.2 (Ref. by Chapra)

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