Read e-book online Generalized Riemann Problems in Computational Fluid Dynamics PDF

By Ben-Artzi M., Falcovitz J.

This monograph supplies a scientific presentation of the GRP method, ranging from the underlying mathematical rules, via simple scheme research and scheme extensions (such as reacting circulation or two-dimensional flows regarding relocating or desk bound boundaries). An array of instructive examples illustrates the variety of functions, extending from (simple) scalar equations to computational fluid dynamics. history fabric from mathematical research and fluid dynamics is equipped, making the publication available to either researchers and graduate scholars of utilized arithmetic, technology, and engineering.

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31) 26 2. Scalar Conservation Laws (d) The Lax–Wendroff scheme U n+1 − U nj j t = −a U nj+1 − U nj−1 2 x + a2 U nj+1 − 2U nj + U nj−1 x2 t . 32) Observe that in the first three schemes the left-hand side approximates the derivative u t while the right-hand side approximates −au x . In the Lax–Wendroff scheme the left-hand side approximates u t + 12 t u tt , while the right-hand side approximates −au x + 12 t a 2 u x x . Note that for smooth solutions the relation u tt = a 2 u x x follows from u t = −au x by differentiation with respect to t.

20), ˜ tn+1 )| ≤ sup |u(x, ˜ tn )| = sup U nj = Mn . 13) We shall not develop here the mathematical theory of numerical schemes. The reader is referred to Godlewski and Raviart [54] for an extensive survey. We emphasize that there are almost no results concerning the convergence of schemes approximating systems of equations, and this is particularly true for the system of equations governing compressible inviscid flow, at which this monograph is primarily aimed. 1)], the existing proofs pertain mostly to the case of a first-order scheme (such as the Godunov scheme), whereas we are concerned with the GRP method, which is of second-order accuracy.

Let 1 , 2 be the two parts of the boundary of , separated by C. 18) −u(x, t) d x + f (u(x, t)) dt + 2 −C −u 2 (x, t) d x + f (u 2 (x, t)) dt = 0. 18) we have denoted by u 1 , u 2 the boundary values of u, as attained by approaching the discontinuity curve C from 1 , 2 , respectively. Observe that the closed curves 1 ∪ C and 2 ∪ (−C) are traversed counterclockwise. 1. 17) is valid when applied to the boundary = 1 ∪ 2 of . Thus, we have also −u(x, t) d x + f (u(x, t)) dt = 0. 19) we conclude [u 2 (x, t) − u 1 (x, t)] d x − [ f (u 2 (x, t)) − f (u 1 (x, t))] dt = 0.

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