By K. W. Morton
This moment variation of a hugely winning graduate textual content offers a whole advent to partial differential equations and numerical research. Revised to incorporate new sections on finite quantity equipment, changed equation research, and multigrid and conjugate gradient equipment, the second one variation brings the reader updated with the newest theoretical and commercial advancements. First version Hb (1995): 0-521-41855-0 First version Pb (1995): 0-521-42922-6
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Additional info for Numerical solution of partial differential equations
Indeed, as we shall show in Chapter 6 where a sharper error analysis based on the maximum principle is presented, the error remains O (∆x)2 despite this O(∆x) truncation error near the boundary. 42 Parabolic equations in one space variable An alternative, and more widely used, approach is to keep the ﬁrst n grid point at x = 0 but to introduce a ﬁctitious value U−1 outside the domain so that we can use central diﬀerences to write n U1n − U−1 = αn U0n + g n . 114) Then the usual diﬀerence approximation is also applied at x = 0 so that n U−1 can be eliminated.
69) to calculate the values of Uj in succession may in general be numerically unstable, and lead to increasing errors. 68) are suﬃcient to guarantee this (see Exercise 4). 19). Thus it takes about twice as long for each time step. The importance of the implicit method is, of course, that the time steps can be much larger, for, as we shall see, there is no longer any stability restriction on ∆t. We shall give a proof of the convergence of this implicit scheme in the next section, as a particular case of a more general method.
This would normally be regarded as unacceptable. When µ = 1 the maximum principle holds, and the numerical values all lie between 0 and 1, as required. However, at the ﬁrst time level the 38 Parabolic equations in one space variable numerical solution shows two peaks, one each side of the mid-point; the exact solution of the problem will have only a single maximum for all t. These results correspond to a rather extreme case, and the unacceptable behaviour only persists for a few time steps; thereafter the solution becomes very smooth in each case.