First-Order Scalar Equation in One Dimension

Abstract

In fluid dynamics, convective transport described by first-order derivatives in the governing equations plays an important role. The Maxwell equations in electromagnetics are of first order. First-order differential operators which seem very simple, are in fact difficult to deal with numerically. For the solution of first-order differential equations, the classic Galerkin method or the central difference method fails miserably. In this chapter we use a very simple one-dimensional model problem to explain why the classic Galerkin method generates oscillatory solutions, and why the least-squares method does not need upwinding and is perfectly suitable for the solution of first-order differential equations.