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Which one of the following methods used for solution of ordinary differential equations is conditionally stable?

Which one of the following methods used for solution of ordinary differential equations is conditionally stable? Correct Answer Euler’s method

Concept:

  • A stable problem, i.e., a problem for which small changes in the initial conditions elicit only small changes in the solution, there are two basic notions of numerical stability.
  • The first notion of stability is concerned with the behaviour of the numerical solution for a fixed value t > 0 as h → 0.
  • A second notion of stability is concerned with the behaviour of the solution as t → ∞ with a fixed step size h. This notion of stability is often referred to as absolute stability, and it is important when dealing with stiff ODEs.
  • An absolutely stable numerical method is one for which the numerical solution of a stable problem behaves also in this controlled fashion.

 

Explanation:

Considering the Euler’s method,

Yn+1 = yn + h f(tn, yn)

Euler’s method is only conditionally stable, i.e., the step size has to be chosen sufficiently small to ensure stability.

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