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parabolicstability

parabolicstability - F1.4ZH2 Numerical Solution of PDEs...

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F1.4ZH2 Numerical Solution of PDEs Page 15 LTEs (continued) The first term on the right of the LTE for the FTCS scheme is referred to as the leading term of the local truncation error (LTE): for this scheme it is first order accurate in time and 2nd order in space . Definition: consistency/order If the LTE of a scheme 0 as Δ x, Δ t 0, the scheme is said to be consistent . This is the minimum requirement for any numerical scheme. Furthermore, if the LTE is of order O x p , Δ t q ), the scheme is said to be of order p in space and q in time. If r is fixed so we can write the LTE as O x p ) as in the above expression for the FTCS scheme, we say more generally that the scheme is p th order. So the FTCS scheme for the heat equation is 2nd order if r 6 = 1 6 , and 4th order if r = 1 6 . Exercise: Check that all the 3rd order terms in the LTE for the FTCS scheme vanish. 2.4 Matrix version of FTCS scheme We have seen that if Δ x, Δ t are small, then the LTE for the FTCS scheme 0 as Δ x, Δ t 0. This is saying that the error is small after one time step, starting with the exact solution. However this is not enough to guarantee that the scheme gives good results after a large number of time steps. An example we have previously referred to briefly is the problems that occur in the FTCS scheme if r > 0 . 5. Suppose we take the example we looked at above, with J = 4, but this time take “triangular” initial conditions F ( x ) = 2 x, x 0 . 5 2(1 - x ) , x > 0 . 5 (This choice of an IC with a discontinuous derivative, rather than a sin( πx ), results in the problems becoming apparent at smaller times. Similar things happen for sin( πx ) but take longer to manifest themselves). With r = 0 . 4 (left) and r = 0 . 6 (right) we get the two graphs shown below. t 0 t 1 t 2 t 3 t 4 t 5 x 0 j=0 x 1 j=1 x 2 j=2 x 3 j=3 x 4 j=4 w j n t 0 t 1 t 2 t 3 t 4 t 5 x 0 j=0 x 1 j=1 x 2 j=2 x 3 j=3 x 4 j=4 w j n Note that when r = 0 . 4 the numerical solution decreases smoothly towards zero, in line with what we would expect in the physical model. But when r = 0 . 6 the solution develops spatial

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F1.4ZH2 Numerical Solution of PDEs Page 16 oscillations which are increasing in amplitude. If run for further times (try it!), these oscillations become unbounded, i.e. w n j → ∞ as n → ∞ , in contrast to the exact solution which can be shown to satisfy 0 as t → ∞ . We would regard this as bad. Even at small times we find the temperature is becoming negative at some points, physically contradicting the laws of thermodynamics! Technically if this sort of bad behaviour occurs we say the scheme unstable . We study one way of analysing this behaviour in this section. To do this we write the FTCS scheme in matrix form. Set w n = w n 1 w n 2 . . . w n J - 1 , the vector of values of the numerical solution at the internal spatial grid points at time level t n . We know from the initial conditions that w 0 = u 0 = F ( x 1 ) F ( x 2 ) .
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parabolicstability - F1.4ZH2 Numerical Solution of PDEs...

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