Editing Appendix/Ramblings/ForCohlHoward (section)

===Our Choice, Historically===

Over the past four decades, the nonrelativistic fluid simulations that have been performed by LSU's astrophysics group have been carried out using various renditions of an ''explicit'' <b>Finite-Volume Method</b>.  The explicit FVM method was chosen primarily because &hellip;
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It was the method to which I was introduced by my [[Appendix/Ramblings/MyDoctoralStudents#Years_1976_-_1978|doctoral dissertation advisors]], Peter Bodenheimer and David Black, that was considered appropriate to the type of fluid-flows problems that were the focus of my dissertation research.  Note that Bodenheimer is the first author of the book that is [[#Joel%27s_Opening_Comments|referenced above]] as [[Appendix/References#BLRY07|<font color="red">BLRY07</font>]].
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Time-integration is fairly straightforward because all terms (except the time derivative) are specified entirely in terms of the current (as opposed to advanced) time.
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The nonlinear ''advection'' term on the LHS is written in a way that guarantees global momentum conservation, if the ''source'' terms on the RHS of the Euler equation sum to zero.
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The amount of computer time that is required to take each step forward in time is typically much smaller when using an ''explicit'' as opposed to ''implicit'' integration scheme.
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A major disadvantage of our decades-old, explicit-FVM approach is that, in order for the scheme to be numerically stable, roughly speaking each integration time-step <math>\Delta t</math> must be smaller than the smallest value of the ratio, <math>\Delta x/c_s</math>, as determined across the entire grid, where <math>\Delta x</math> is the grid spacing and <math>c_s</math> is the fluid sound-speed.  (Note that <math>\Delta t</math> is essentially the sound-crossing time across the smallest grid cell.)  This constraint on the integration time-step is computationally burdensome when attempting to employ a high-resolution (small <math>\Delta x</math>) grid and/or when attempting to model environments where the fluid is relatively hot and therefore characterized by a high sound speed.  Even worse, <b>it is ''impossible'' to model the evolution of ''incompressible'' fluids using an explicit-FVM scheme</b> because, in effect in such fluids, sound travels at an infinite speed; that is, <math>c_s = \infty</math> so <math>\Delta t</math> goes to zero.