Inviscid Burgers's Equation
Burgers's equation is one of the simplest cases of a nonlinear, hyperbolic partial differential equation. It is useful as a test problem in computational fluid dynamics. The one-dimensional, inviscid (i.e., zero viscosity) form of the equation is given by[more]
where is the space coordinate, is time, and is some conserved physical quantity (e.g., momentum). In this Demonstration, solutions to this equation can be visualized for three different initial waveforms. The left plot shows the changing waveform with time and the right plot shows the solution in three dimensions. The time can be varied in 10 discrete steps.[less]
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.