This paper was a landmark; it introduced a new design main features of such a flow are recognizable in the numerical results presented. Subsequently, much effort was de-principle-total variation diminishing schemes-that led, in Harten's hands, and subsequently in the hands of others, voted by others to constructing variation-diminishing schemes for the calculation of two-dimensional flows. Such to an efficient, robust, highly accurate class of schemes for shock capturing free of oscillations. The citation index lists efforts are of course doomed to failure, since the total variation of exact solutions can blow up through the mecha-429 references to it, not only in journals of numerical analysis and computational fluid dynamics, but also in journals nism of focusing. Even in the scalar case, where no focusing takes place, Goodman and Leveque showed that there devoted to mechanical engineering, astronautics, astrophysics, geophysics, nuclear science and technology, space-are no TVD schemes. The only viable two-dimensional analogue of variation-diminishing is energy (or entropy) craft and rockets, plasma physics, sound and vibration, aerothermodynamics, hydraulics, turbo and jet engines, diminishing; it was shown in [3] how to combine some ideas of Friedrichs with those of Harten to construct such and computer vision and imaging.
The basic idea is a simple condition for a three-point schemes. The arguments are somewhat formal, but their soundness is demonstrated by numerical experiments. explicit scheme for a scalar quantity in one space variable to be variation diminishing, a condition that has been ex-The review paper [5] gives an overview of the shock capturing scheme in general and TVD schemes in par-tended to n point schemes in [1] and . The next step is the construction of a second-order five-point scheme that, ticular.
Harten originally called his schemes variation diminish-when interpreted as a three-point scheme, is variation diminishing. Even when the equation in question is linear, ing, abbreviated TVD; when Osher pointed out the usual meaning of these initials, the name was switched to total the scheme proposed is nonlinear, a surprising idea though one already present in Harten's master's and doctoral dis-variation nonincreasing (TVNI), but was eventually settled on the more euphonious TVD. sertations. It is precisely this nonlinearity that allows the escape from the class of monotone schemes, which are Peter Lax
In the last section the scheme is extended to two-dimen-Courant Institute of Mathematical Science sional problems, using Strang-type dimensional splitting