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~ Ordinary Differential Equations ~ Stiffness and Multistep Methods Chapter 26

~ Ordinary Differential Equations ~ Stiffness and Multistep Methods Chapter 26. Credit: Prof. Lale Yurttas, Chemical Eng., Texas A&M University. Stiffness.

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~ Ordinary Differential Equations ~ Stiffness and Multistep Methods Chapter 26

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  1. ~ Ordinary Differential Equations ~Stiffness and Multistep MethodsChapter 26 Credit: Prof. Lale Yurttas, Chemical Eng., Texas A&M University

  2. Stiffness • A stiff system is the one involving rapidly changing components together with slowly changing ones. Stiff ODEs have both fast and slow components • Both individual and systems of ODEs can be stiff: • If y(0)=0, the analytical solution is developed as:

  3. Insight into the step size required for stability of such a solution can be gained by examining the homogeneous part of the ODE: • The solution starts at y(0)=y0 and asymptotically approaches zero. • If Euler’s method is used to solve the problem numerically: The stability of this formula depends on the step size h: solution

  4. Thus, for transient part of the equation, the step size must be < 2/1000 = 0.002 to maintain stability. • While this criterion maintains stability, an even smaller step size would be required to obtain an accurate solution. • Rather than using explicit approaches,implicit methods offer an alternative remedy. An implicit form of Euler’s method can be developed by evaluating the derivative at a future time. Backward or implicitEuler’s method: The approach is called unconditionally stable regardless of the step size 

  5. Figure 26.2 Solution by • Explicit • Implicit Euler

  6. Multistep Methods The Non-Self-Starting Heun Method • Heun method uses Euler’s method as a predictor and trapezoidal rule as a corrector • Predictor is the weak link in the method because it has the greatest error, O(h2) • One way to improve Heun’s method is to develop a predictor that has a local error of O(h3).

  7. Multistep Methods General Representation of Non-Self-Starting Heun Method

  8. Multistep Methods Example 26.2 Integratey’ = 4e0.8x- 0.5y from x=0 to x=4 (Using h=1 andInitial conditions: x-1= -1, y-1= -0.393 and x0= 0, y0= 2) Solution: which represents a relative error of -5.73% (True value is 6.1946). We can apply the Multistep formula iteratively to improve this result: If we continue the iterations, it converges to y=6.36086 when ea=-2.68%. Then, for the second step: With the new method, we get better error rates and faster convergence (compared to the Heun method)

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