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Stability2010

# Stability2010 - Stability A brief introduction We have...

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11/3/2010 1 Stability A brief introduction We have solved this equation (advection equation, one way wave equation, transport problem) with discretisation in space and time: Forward in time 2 Centered in space

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11/3/2010 2 Euler method, FTCS Forward in Time Centered in Space 3 Euler method, FTCS y Explicit scheme and easy to derive. y Needs little storage and executes fast. y Big disadvantage: FTCS-Method is basically useless! y Why? y Algorithm is numerical unstable. Leonard Euler 1707-1783 4
11/3/2010 3 What is numerical stability? sum = 0 Say we have to add 100 numbers of array a[i] using a computer with only 2 significant digits. for i = 1 to 100 sum = sum + a[i] end - Looks reasonable, doesn’t it? - But imagine a[0]=1.0 and all other a[i]=0.01 5 - Our two-digit computer gets: sum=1.0 - Better algorithm: Sort first a[i] by absolute values - Two-digit comp gets: sum=2.0, which is a much better approximation of the true solution 1.99 Can we check if a numerical scheme is stable without computation?

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Stability2010 - Stability A brief introduction We have...

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