Roache, . 1976, Computational Fluid Dynamics (Albuquerque: Hermosa). [7] Woodward, P., and Colella, P. 1984, Journal of Computational Physics, vol. 54, pp. 115–173. [8] Rizzi, A., and Engquist, B. 1987, Journal of Computational Physics, vol. 72, pp. 1–69. [9] | Diffusive Initial Value Problems 847 Roache . 1976 Computational FluidDynamics Albuquerque Hermosa . 7 Woodward P. and Colella P. 1984 JournalofComputational Physics vol. 54 pp. 115-173. 8 Rizzi A. and Engquist B. 1987 Journal ofComputational Physics vol. 72 pp. 1-69. 9 Diffusive Initial Value Problems Recall the model parabolic equation the diffusion equation in one space dimension du @t d_ dx where D is the diffusion coefficient. Actually this equation is a flux-conservative equation of the form considered in the previous section with F -d@u dx the flux in the . --direction. We will assume D 0 otherwise equation has physically unstable solutions A small disturbance evolves to become more and more concentrated instead of dispersing. Don t make the mistake of trying to find a stable differencing scheme for a problem whose underlying PDEs are themselves unstable Even though is of the form already considered it is useful to consider it as a model in its own right. The particular form of flux and its direct generalizations occur quite frequently in practice. Moreover we have already seen that numerical viscosity and artificial viscosity can introduce diffusive pieces like the right-hand side of in many other situations. Consider first the case when D is a constant. Then the equation du t @2u dt dx2 can be differenced in the obvious way - _ r . - 2un un-1 At _ Ax 2 This is the FTCS scheme again except that it is a second derivative that has been differenced on the right-hand side. But this makes a world of difference The FTCS scheme was unstable for the hyperbolic equation however a quick calculation shows that the amplification factor for equation is e i - 4DAt 2 kAx 777 sin ---- Ax 2 V 2 J The requirement 1 leads to the stability criterion 2DAt Ax 2 - Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 848 Chapter 19. Partial Differential .
