For example, a combined advective-diffusion equation, such as ∂u ∂2 u ∂u = −v +D 2 ∂t ∂x ∂x () might profitably use an explicit scheme for the advective term combined with a Crank-Nicholson or other implicit scheme for the diffusion term. The alternating-direction implicit (ADI) method, equation (), is an example of operator splitting with a slightly different twist. Let us reinterpret () | Fourier and Cyclic Reduction Methods 857 For example a combined advective-diffusion equation such as du du d2 u dt dx dx2 might profitably use an explicit scheme for the advective term combined with a Crank-Nicholson or other implicit scheme for the diffusion term. The alternating-direction implicit ADI method equation is an example of operator splitting with a slightly different twist. Let us reinterpret to have a different meaning Let U1 now denote an updating method that includes algebraically all the pieces of the total operator L but which is desirably stable only for the L1 piece likewise U2 . Um. Then a method of getting from un to un 1 is un 1 m Ui un At m un 2 m U2 un 1 m At m un 1 Um un m-1 m At m The timestep for each fractional step in is now only 1 m of the full timestep because each partial operation acts with all the terms of the original operator. Equation is usually though not always stable as a differencing scheme for the operator L. In fact as a rule of thumb it is often sufficient to have stable U s only for the operator pieces having the highest number of spatial derivatives the other Uj s can be unstable to make the overall scheme stable It is at this point that we turn our attention from initial value problems to boundary value problems. These will occupy us for the remainder of the chapter. CITED REFERENCES AND FURTHER READING Ames . 1977 Numerical Methods for Partial Differential Equations 2nd ed. New York Academic Press . Fourier and Cyclic Reduction Methods for Boundary Value Problems As discussed in most boundary value problems elliptic equations for example reduce to solving large sparse linear systems of the form Au b Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 either once for boundary value equations that are linear or iteratively for boundary value equations that are nonlinear. 858 Chapter 19. Partial Differential .
