With the advent of faster computers, numerical simulation of physical phenomena is becoming more practical and more common. Computational prototyping is becoming a significant part of the design process for engineering systems. With ever-increasing computer performance the outlook is even brighter, and computer simulations are expected to replace expensive physical testing of design prototypes.

This book is an outgrowth of my lecture notes for a course in computational mathematics taught to first-year engineering graduate students at Stanford. The course is the third in a sequence of three quarter-courses in computational mathematics. The students are expected to have completed the first two courses in the sequence: numerical linear algebra and elementary partial differential equations. Although familiarity with linear algebra in some depth is essential, mastery of the analytical tools for the solution of partial differential equations (PDEs) is not; only familiarity with PDEs as governing equations for physical systems is desirable. There is a long tradition at Stanford of emphasizing that engineering students learn numerical analysis (as opposed to learning to run canned computer codes). I believe it is important for students to be educated about the fundamentals of numerical methods. My first lesson in numerics includes a warning to the students not to believe, at first glance, the numerical output spewed out from a computer.