Inviscid Burgers's Equation

The inviscid Burgers's equation is a flux-conservative partial differential equation, and is thus amenable to finite-volume methods, which are often used for numerically solving nonlinear, hyperbolic partial differential equations. Hyperbolic problems are represented in the flux-conserving form

,

where is the flux function for some conserved quantity . The inviscid Burgers's equation is easily represented in this form if the flux function is considered to be The finite-volume method takes the following form with forward-Euler time-stepping

,

where is the space index, is the time index, is the flux at the right boundary enclosing the volume around point , and is the flux at the left boundary enclosing the volume around point . There are several different finite-volume methods, which differ in their determination of and In this Demonstration, the Harten-Lax-van Leer (HLL) method is used because it is good at capturing shocks in an evolving waveform, which is a feature often found in solutions to Burgers's equation.

Reference: R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge, UK: Cambridge University Press, 2002.