COPYRIGHT 2008, PRINCETON UNIVERSITY PRESS

468 chapter 17T8060position402000time10203020temperature010080100–10010t80x400temperature2010–10 0time10 0203010080604020position0Figure 17.14 A numerical calculation of the temperature versus position and versus time, withisotherm contours projected onto the horizontal plane on the left and with a red-blue colorscale used to indicate temperature on the right (the color is visible on the figures on the CD).17.17.3 Von Neumann Stability AssessmentWhen we solve a PDE by converting it to a difference equation, we hope that thesolution of the latter is a good approximation to the solution of the former. If thedifference-equation solution diverges, then we know we have a bad approximation,but if it converges, then we may feel confident that we have a good approximationto the PDE. The von Neumann stability analysis is based on the assumption thateigenmodes of the difference equation can be written asT m,j = ξ(k) j e ikm∆x , (17.72)where x = m∆x and t = j∆t, but i = √ −1 is the imaginary number. The constantk in (17.72) is an unknown wave vector (2π/λ), and ξ(k) is an unknown complexfunction. View (17.72) as a basis function that oscillates in space (the exponential)with an amplitude or amplification factor ξ(k) j that increases by a power of ξ for eachtime step. If the general solution to the difference equation can be expanded in termsof these eigenmodes, then the general solution will be stable if the eigenmodes arestable. Clearly, for an eigenmode to be stable, the amplitude ξ cannot grow in timej, which means |ξ(k)| < 1 for all values of the parameter k [Pres 94, Anc 02].Application of a stability analysis is more straightforward than it might appear.We just substitute the expression (17.72) into the difference equation (17.71):[ξ j+1 e ikm∆x = ξ j+ e ikm∆x + η ξ j e ik(m+1)∆x + ξ j+ e ik(m−1)∆x − 2ξ j+ e ikm∆x] .After canceling some common factors, it is easy to solve for ξ:ξ(k)=1+2η[cos(k∆x) − 1]. (17.73)−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 468