By Randall D.
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A ﬁnite-difference scheme for which the discretization error can be made small for any initial condition is called a convergent ﬁnite difference scheme. 80) is a necessary condition for convergence when the upstream scheme is used. 80). 75). 81) if c < 0 . This is bad, though, because if-tests can cause slow execution on certain types of computers, and besides, if-tests are ugly. 82) 28 n+1 n n n n Basic Concepts n u j – u j – 1 uj + 1 – uj – uj uj - + c - ---------------------- = 0 .
Unfortunately, it isn’t necessarily true. Letting u ( x, t ) denote the (exact) solution of the differential equation, we see that u ( j∆x, n∆t ) is the value of this exact solution at the discrete point ( j∆x, n∆t ) on the grid n shown in Fig. 3, while u j is the “exact” solution of a ﬁnite-difference equation, at the n same point. In general, u j ≠ u ( j∆x, n∆t ) . We wish that they were equal! A measure of the accuracy of the ﬁnite-difference scheme can be obtained by substituting the solution of the differential equation into the ﬁnite-difference equation.
Discretization error measures the accuracy with which the solution of the differential equation has been approximated. Minimizing the truncation error is usually easy. Minimizing the discretization error can be much harder. 84) where c∆t µ ≡ -------- . 85) This scheme has the form of either an interpolation or an extrapolation, depending on the value of µ . To see this, refer to Fig. 6. 87) For 0 ≤ µ ≤ 1 we have interpolation. For µ < 0 or µ > 1 we have extrapolation. Note that for the case of interpolation, u jn + 1 will be intermediate in value between u jn– 1 and u jn .