ABSTRACT
This chapter presents a brief introduction to some techniques available for approximating the solution to partial differential equations (PDEs). The implicit method, now to be described, overcomes the stability requirement by being unconditionally stable. It discusses methods for the numerical solution of the one-dimensional PDEs of hyperbolic type. For general use, iterative techniques often represent the best approach to the solution of such systems of equations. Nonlinear PDE problems are encountered in many fields of engineering and sciences and have applications to many physical systems including fluid dynamics, porous media, gas dynamics, traffic flow, shock waves, and many others. Numerical methods typically involve approximations in a finite-dimensional setting. There are many techniques available for deriving these approximations. Unfortunately, most basis functions lead to full matrices, which must be inverted to obtain the approximation. Moreover, to obtain better and better approximations, one is forced to choose more basis functions which in turn leads to even larger matrices.
