Transcription of NUMERICAL STABILITY; IMPLICIT METHODS
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NUMERICAL STABILITY; IMPLICIT METHODS
NUMERICAL stability ; IMPLICIT METHODS . When solving the initial value problem Y 0 (x) = f (x, Y (x)), x0 x b Y (x0 ) = Y0. we know that small changes in the initial data Y0 will result in small changes in the solution of the differential equation. More precisely, consider the perturbed problem Y 0 (x) = f (x, Y (x)), x0 x b Y (x0 ) = Y0 + . Then assuming f (x, z) and f (x, z)/ z are continuous for x0 x b, < z < , we have max |Y (x) Y (x)| c | |. x0 x b for some constant c > 0. We would like our NUMERICAL METHODS to have a similar property. Consider the Euler method yn+1 = yn + hf (xn , yn ) , n = 0, 1, .. y0 = Y0. and then consider the perturbed problem . yn+1 = yn + hf (xn , yn ) , n = 0, 1, .. y0 = Y0 + . We can show the following: max |yn yn | cb | |. x0 xn b for some constant cb > 0 and for all sufficiently small values of the stepsize h.
NUMERICAL STABILITY; IMPLICIT METHODS When solving the initial value problem Y0(x) = f(x;Y(x)); x 0 x b Y(x 0) = Y 0 we know that small changes in the initial data Y 0 will result in small changes in the solution of the di erential equation.
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