Abstract
In this chapter we shall first introduce the concepts of the Eulerian description for the advection/transport/wave equation and then present many different numerical approximations in the finite difference formulation for this one-dimensional partial differential equation. We shall present explicit, implicit, and predictor–corrector numerical methods, as well as the performance of some of the schemes with advecting a “Gaussian bell curve” and a discontinuous step function. While examining the performance of these numerical schemes we shall see apparent phase and amplitude errors, as well as what appears to be an unstable behavior of schemes.
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