we can approximate the advection equation dfdt udfdx 0 as

we can approximate the advection equation df/dt + udf/dx = 0 as follows Show that this method is consistent and find its truncation error

Solution

The first expression given equal to 0 is nothing but

f1^n(t)

Hence the given expression

ft = -ufx gives

f^n(t) = -u^n fx -u^(n-1) f\'x-u^n-2 f\"x-.... (using product rule)

= 3fi^n-4fi-1^b _ fi-2^n/2Delta x + E1

where E1 is a small quantity.

AS E1 is negligible this is a consistent method and

Truncation error = difference between estimated and actual

= E1