Chapter 8 starts out with a physics motivation, as well as a mathematical statement of the problems that will be tackled in later sections. Starting from differential-equation initial-value problems, the text introduces both explicit and implicit methods, like backward Euler and the fourth-order Runge-Kutta method. Emphasis is placed on the interplay between method stability and problem conditioning (stiffness). The chapter then discusses boundary-value problems, first, via a combination of the earlier machinery on initial-value problems along with root-finding techniques and, second, via a finite-difference/matrix approach, which converts the problem to a linear system of equations. Next, the chapter tackles eigenvalue problems, again, via either rootfinding plus earlier tools, or a finite-difference approach; this time, the latter turns into a matrix eigenvalue problem. The second edition discussesfinite-difference approaches to solving the diffusion equation. The chapter is rounded out by a physics project, on Poisson’s equation in two dimensions, and a problem set. The physics project introduces and uses the two-dimensional fast Fourier transform, as part of a spectral method applied to the solution of a partial differential equation.

The application of finite-difference methods to initial-value problems, with emphasis on parabolic equations, is considered using the one- and two-dimensional unsteady diffusion equations as model problems.Single-step methods are introducted for ordinary differential equations, and more general explicit and implicit methods are articulated for partial differential equations.Numerical stability analysis is covered using the matrix method, von Neumann method, and the modified wavenumber method.These numerical methods are also applied to nonlinear convection problems.A brief introduction to numerical methods for hyperbolic equations is provided, and parallel computing is discussed.