Improvements to SOIL: An Eulerian hydrodynamics code Page: 7 of 20
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Ui + V2 “
i + 1
This is a two-step process that is stable but produces a lot of
artificial diffusion. Other first-order schemes such as those due to
MacCormick (1971)5 and Rusanov (1962)6 have been applied with various degrees
of success. SOIL uses the upwind scheme described in Section II which is
marginally second order.
In this discussion of advection schemes, we will assume an ideal gas
equation of state, i e.
e = P/(7 - 1) ,
in order to simplify the equations. The first order schemes we will consider
1. Lax-Wendroff (two-step),
2. General upwind methods, and
In the Lax-Wendroff scheme, the first step is a provisional one with Un
based on the un+l/2 from the first step. The step 1+1/2, i-1/2 is called a
leapfrog step. There is some built in numerical viscosity but additional
viscosity may be needed to smooth out ripples behind the shock front. Further
discussions of Lax-Wendroff schemes can be found in the literature (Richmyer
and Morton for instance).8
In the last decade, an interesting scheme was developed by Boris and Book
at NRL called flux corrected transport (FCT). A quick look at the explicit
version will help to understand this method. First, a highly diffusive step
~P (t + 5t) = PjCO + 6tpj(t + St/2) ,
•'i + V»('i * i(t)
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Davis, C.G. Improvements to SOIL: An Eulerian hydrodynamics code, report, April 1, 1988; New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc1064690/m1/7/: accessed April 18, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.