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The (1+1)D Advection Equation

Perhaps the simplest hyperbolic partial differential equation imaginable is the so-called scalar advection or one-way wave equation in (1+1)D, defined by

$\displaystyle \frac{\partial i}{\partial t}+\alpha\frac{\partial i}{\partial x}=0$ (3.47)

where $ \alpha$ is a real constant [176]. It is complemented by the initial condition

$\displaystyle i(x,0) = i_{0}(x),\hspace{1.0in} -\infty<x<\infty$ (3.48)

Here, the solution $ i(x,t)$ is assumed continuously differentiable (though it need not be% latex2html id marker 80698
\setcounter{footnote}{2}\fnsymbol{footnote}), and is defined over the entire $ x$-axis, and for $ t\geq 0$. The solution is simply

$\displaystyle i(x,t) = i_{0}(x-\alpha t)$ (3.49)

That is, the initial data travels to the left or right (depending on the sign of $ \alpha$) with speed $ \vert\alpha\vert$. Despite its simplicity, it is often used as a model for numerical schemes [95].


Stefan Bilbao 2002-01-22