The numerical treatment of partial differential equations is, by itself, a vast subject. Partial differential equations are at the heart of many, if not most, computer analyses or simulations of continuous physical systems, such as fluids, electromagnetic fields, the human body, and so on. | Chapter 19. Partial Differential Equations Introduction The numerical treatment of partial differential equations is by itself a vast subject. Partial differential equations are at the heart of many if not most computer analyses or simulations of continuous physical systems such as fluids electromagnetic fields the human body and so on. The intent of this chapter is to give the briefest possible useful introduction. Ideally there wouldbe an entire second volume of Numerical Recipes dealing with partial differential equations alone. The references 1 -4 provide of course available alternatives. In most mathematics books partial differential equations PDEs are classified into the three categories hyperbolic parabolic and elliptic on the basis of their characteristics or curves of information propagation. The prototypical example of a hyperbolic equation is the one-dimensional wave equation -2 S where v constant is the velocity of wave propagation. The prototypical parabolic equation is the diffusion equation du dt d_ dx where D is the diffusion coefficient. The prototypical elliptic equation is the Poisson equation S 5 xy Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 where the source term p is given. If the source term is equal to zero the equation is Laplace s equation. From a computational point of view the classification into these three canonical types is not very meaningful or at least not as important as some other essential distinctions. Equations and both define initial value or Cauchy problems If information on u perhaps including time derivative information is 827 828 Chapter 19. Partial Differential Equations o o o o o o o boundary o o o o o o o conditions o o o o o o o . initial values a o boundary values o o o o o o o o o o o o o o o b Figure . Initial value problem a and boundary value problem b are contrasted. In a initial values are given on one time slice and it
