My contribution to a final year mathematics group honours project titled Numerical Methods for Solving the Advection Equation. This project, conducted with two others, culminated in a 50 page report of our findings and a 30 minute presentation given to peers and professors.

The advection equation is given as $\frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = 0$, where $a$ (the wave speed) is some constant. The exact solution to this equation, given periodic spacial boundaries and an initial condition $u(0, x) = f(x)$, is given as $u(t, x) = f(x - at)$.

This Python package contains code for numerically solving the 1-D advection equation. It implements various finite difference schemes and flux limiters. The contents of this package were intended to show the properties of various numerical methods. Hence, some methods are unstable and not meant to be used practically.

In the following examples, we use the initial condition shown in the figure below over the spacial interval $[0, 1]$.

The Lax-Wendroff scheme gives the below results (orange) overlayed with the exact solution (blue). The temporal index is denoted as $n$.

The van Leer flux limiter gives the below results (orange) overlayed with the exact solution (blue). The temporal index is denoted as $n$.

The Sweby diagram can be used to find flux limiter functions which yield second-order accurate, TVD schemes.

[1] R. J. LeVeque, Finite difference methods for ordinary and partial differential equations, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2007.

[2] R. J. LeVeque, Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, England, 2002.

[3] P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM Journal on Numerical Analysis 21 (1984), no. 5, 995–1011, available at https://doi.org/10.1137/0721062.

[4] A. Harten, High resolution schemes for hyperbolic conservation laws, Journal of Com- putational Physics 49 (1983), no. 3, 357–393, available at https://doi.org/10.1016/0021-9991(83)90136-5.