2. Partial Differential Equations and the Method of Characteristics

This chapter is dedicated to partial differential equations of first and second orders, treated with the method of characteristics. Examples are given for the numerical treatment of linear damped waves and quasi-linear undamped waves. Preliminaries are set for the successive analysis of methods of orthogonal collocations developed in Chap. 3. Partial differential equations of second order are discussed with reference to the problem of existence, uniqueness, und continuity of the solutions in the context of the Hadamard conditions of well-posedness. The equations of characteristics are introduced also for the partial differential equations of second order and are elucidated with numerical examples. Numerical solutions of the Laplace equation with three numerical techniques are reviewed, and relative advantages or drawbacks are discussed.