High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws
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...For some other discussions of TVD conditions, see [180], [349], [429], [435], [465]....
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...This graphical analysis of φ was first presented by Sweby [429], who analyzed a wide class of flux-limiter methods (for nonlinear conservation laws as well as the advection equation)....
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...In particular we will see how this relates to fluxlimiter methods of the type studied by Sweby [429], who used the algebraic total variation diminishing (TVD) conditions of Harten [179] to derive conditions that limiter functions should satisfy for more general nonlinear conservation laws....
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"High Resolution Schemes Using Flux ..." refers result in this paper
...There are two main differences between the approach adopted here and that of Boris and Book [ I ] (and later Zalesak [24])....
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...Although others (e.g., Boris and Book [I], Zalesak [24] and Le Roux [9]) have proposed schemes involving forms of flux limiters, they do not fall into the framework considered here, not being expressible as functions only of the ratio r....
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"High Resolution Schemes Using Flux ..." refers background or methods in this paper
...…any second order scheme relying only on the points (ukP2, ukPl, uk, u ~ + ~ ) must be a weighted average of the Lax-Wendroff scheme and the Warming and Beam upwind scheme (cf. Van Leer's [22] approach of using Fromm's scheme, the arithmetic average of these two schemes, as a starting point), i.e....
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...Like Roe [14], Van Leer [22] before him and more recently Chakravarthy and Osher [2] we take the limiter to be a function of consecutive gradients (in the linear case), i.e., qk= q(rk) where We now seek to choose the function q ( r ) in such a way that the limited antidiffusive flux (3.5) is…...
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