Linear Advection

In this problem we create a planar density pulse in a region of uniform pressure p₀ and velocity u₀, with the velocity normal to the pulse plane. The density pulse is defined via

r(s) = r₁f(s/w) + r₀[1-f(s/w)] ,

where s is the distance of a point from the pulse midplane, w is the characteristic width of the pulse, and the pulse shape function f is, for a square pulse,

f_SP(x) =

ě
í
î

|x| < 1

|x| > 1

and for a Gaussian pulse,

f_GP(x) = e^-x² .

For these initial conditions the Euler equations reduce to a single wave equation with wave speed u₀; hence the density pulse should move across the computational volume at this speed without changing shape. Advection problems similar to this were first proposed by Boris and Book (1973) and Forester (1977).

To demonstrate the performance of FLASH 1.0 on the advection problem, we have performed tests of both the square and Gaussian pulse profiles with the pulse normal parallel to the x-axis (q=0^o) and at an angle to the x-axis (q=45^o) in two dimensions. The square pulse used r₁=1, r₀=10^-3, p₀=10^-6, u₀=1, and w=0.1. With six levels of refinement in the domain [0,1] x [0,1], this value of w corresponds to having about 52 zones across the pulse width. The Gaussian pulse tests used the same values of r₁, r₀, p₀, and u₀, but with w=0.015625. This value of w corresponds to about 8 zones across the pulse width at six levels of refinement. For each test we performed runs at two, four, and six levels of refinement to examine the quality of the numerical solution as the resolution of the advected pulse improves. The runs with q=0^o used zero-gradient (outflow) boundary conditions, while the runs performed at an angle to the x-axis used periodic boundaries.

Plots

Comparison of the density for 2, 4, and 6 levels of refinement with the analytical solution at t=0.4
(GIF, Postscript)

References
Boris, J. P. and Book, D. L., 1973, J. Comp. Phys., 11, 38
Forester, C. K. 1977, J. Comp. Phys., 23, 1
Zalesak, S. T. 1987, in Advances in Computer Methods for Partial Differential Equations VI, eds. Vichnevetsky, R. and Stepleman, R. S. (IMACS), 15

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This file was last modified on 10 October 1999.