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Note that u j is a weighted mean of u j and u j – 1 . 80)], we may write n+1 uj n n ≤ uj ( 1 – µ ) + uj – 1 µ . 89) 30 n+1 max ( j ) u j Basic Concepts n ≤ max ( j ) u j , provided that 0 ≤ µ ≤ 1. 90) n We have shown that for 0 ≤ µ ≤ 1 the solution u j remains bounded for all time. Therefore, 0 ≤ µ ≤ 1 is a sufficient condition for stability. For this scheme, the condition for stability has turned out to be the same as the condition for convergence. In other words, if the scheme is convergent it is stable, and vice versa.

56) k = –∞ ∞ ∑ 2 c k ( δy ) k = 2! 59) c k ( δx ) k ( δy ) k = 0 . 57), it is clear that c k is of order δ , where δ denotes δx or δy . 60) or of order one. 57), to obtain first-order accuracy. 60) involve only six equations, and so six grid points are needed. To get second-order (or higher) accuracy, we will need to add more points, unless we are fortunate enough to use a highly symmetrical grid that permits the conditions for higherorder accuracy to be satisfied automatically. 63) k = –∞ ∞ ∑ 3 c k ( δy ) k = 0 .

Recall that β = 0 for explicit schemes. 2 Non-iterative schemes. 47 scheme can be made at least as high as l + 1 . Later we refer back to these rules of thumb. With the approach outlined above, schemes of higher order accuracy are made possible by bringing in more time levels. It is also possible to obtain schemes of higher accuracy in other ways. This will be explained later. We now survey a number of time-differencing schemes, without specifying f . In this analysis, we can determine the order of accuracy of each scheme.

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