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← 03-40 Trapezoidal Stability Solution

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Showing Revision 2 created 10/24/2012 by Amara Bot.

  1. It turns out that all positive values for the time step are okay no matter how high.
  2. This is how similar to the backward Euler method
  3. and now we're going to look into why that is the case.
  4. The trapezoidal weir turns this differential equation into the following.
  5. The next value of x equals the current value of x plus the time step
  6. times the average of two rates of change, the current rate of change (-kx₁)
  7. and the next rate of change (-kx₂).
  8. I bring the x₂ to the left-hand side and combined this x₁ with that x₁
  9. and get (1+hk/2) * x2 = (1-hk/2) * x1.
  10. Let's check that, 1x₂, it's here. (hk/2)x₂ appears on the right-hand side with a - sign.
  11. 1x₁ appears here minus (hk/2)x₁ - (hk/2)x₁ appears here.
  12. Correct. And that's easy to solve for x₂. x₂ = (1-hk/2)/(1+hk/2)*x₁.
  13. If you look at what happens with the next step. We of course, get a second factor of that sort.
  14. The next next step, another factor of that sort. So what happens is that we get powers of this factor.
  15. The question is what's going to happen about powers of that factor.
  16. So we see that to get it from x₁ to x₂, we have to multiple by this factor
  17. and this is going to occur every step if we want to go from x₂ to x₃,
  18. we again multiply by that factor and so on.
  19. So in the end, we get the powers of this factor.
  20. So the question is as the nth power of this expression converts to 0,
  21. as it intends to infinity to check that, one can analyze three different cases.
  22. Let's first check what happens if the step size is moderate.
  23. For instance, if this expression hk/2 equals 0.1 and we get (1-0.1)/(1+0.1)
  24. meaning (0.9)/(1.1), this is a positive number, <1.
  25. If you take higher and higher powers of that, this number is going to become 0.
  26. The numerator becomes 0.9. The denominator becomes 1.1.
  27. The fraction is a number between 0 and 1 and if we take higher and higher powers of such a number,
  28. the results converge to 0, so this is stable.
  29. So in this case, we have stability.
  30. If hk/2 happens to be equal to 1, we get a fraction with a 0 in the numerator,
  31. that's always 0, no problem with that, so we have stability in that case.
  32. And if you use a really high value of h, for instance, such that hk/2=9,
  33. we get something like 1-9=-8 divided by 1+9=10.
  34. This is a negative number with absolute value less than 1.
  35. So, if we take higher and higher powers, the result converges to 0,
  36. we also have stability in this case but you see that this convergence comes along with an oscillation
  37. that's time changes from minus to plus to minus to plus as we form even powers and odd powers.
  38. The result has a positive sign. For odd powers, the result has a negative sign.
  39. So we see some kind of decaying oscillation here.