The advection equation describes the propagation of a substance or some conserved material through a moving fluid. When the matrix, A, has real eigenvalues and m linearly independent eigenvectors then the equation is an example of a hyperbolic PDE. Hyperbolic PDEs are often used to model wave phenomena: shock waves, acoustic waves, electromagnetic waves, seismic waves, etc.
When A is a one-by-one matrix (a constant), a, then the initial value problem of this kind of equation becomes quite easy to solve:
q(x,0) = u(x) \quad \Rightarrow \quad q(x,t) = u(x - at), \quad
Just differentiate both sides with respect to t and x to verify.
However, in the general case when m is greater than 1 (i.e. we are solving a system of advection equations) the problem becomes a little bit more difficult. Therefore, it's important to develop a fast and accurate means of solving the equation. I very much enjoy the way Randy's book outlines the pros and cons of applying several common discretization methods to the advection equation. The chapter on advection contains a section for each method describing the discretization, noting the speed and accuracy of the technique, and discussing the advantages / disadvantages of that method. Furthermore, he applies the same or similar methods to other types of PDEs, each with their own chapter.
For someone who spent most of their math education in number theory land, I appreciated the side-by-side comparison and simultaneous overview of the discretization techniques.
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