By Ben-Artzi M., Falcovitz J.
This monograph supplies a scientific presentation of the GRP technique, ranging from the underlying mathematical ideas, via easy scheme research and scheme extensions (such as reacting stream or two-dimensional flows related to relocating or desk bound boundaries). An array of instructive examples illustrates the variety of purposes, extending from (simple) scalar equations to computational fluid dynamics. history fabric from mathematical research and fluid dynamics is equipped, making the booklet obtainable to either researchers and graduate scholars of utilized arithmetic, technological know-how, and engineering.
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Extra resources for Generalized Riemann Problems in Computational Fluid Dynamics
16) is of second-order accuracy ( p = 2). 15). To study its validity we prove the following claim. 42 3. The GRP Method for Scalar Conservation Laws ˜ t) be smooth in x ∈ [x j−1/2 , x j+1/2 ] and t ≥ tn . 18) n+1/2 ˜ j+1/2 , t) evaluated [namely, f j+1/2 = the linear approximation (in t) of f u(x at the midpoint tn+1/2 = tn + k2 ]. Proof This is a direct consequence of Taylor’s theorem and the fact that λ = k/ x = constant. To simplify notation, we introduce the functions ˜ j+1/2 , t) , g j+1/2 (t) = f u(x −∞ < j < ∞.
When it is a centered rarefacthe right, according to the sign of S = f (uuRR)− −u L tion wave, it propagates to the right (resp. to the left) when the initial jump satisﬁes the additional relation 0 < f (u L ) < f (u R ) (resp. f (u L ) < f (u R ) < 0). However, in the case of a rarefaction wave there is a third possibility, which we refer to as the “sonic case,” where f (u L ) ≤ 0 ≤ f (u R ). Here the line x = 0 is contained within the rarefaction fan, coinciding with the characteristic line that moves at zero speed.
U˜ + ) is the value behind (resp. ahead of) the shock, u ± (x(tn ), tn ) = U nj+1/2,± . Thus, σ (t) and the value of t=tn = −f U nj+1/2,+ 2 s nj+1 + f U nj+1/2,− U nj+1/2,+ − U nj+1/2,− 2 s nj , ∂ u˜ (x j+1/2 , tn ) is determined according to whether ±σ ∂t (tn ) > 0. The last technical step in the description of the GRP algorithm is con. In the language common to cerned with a modiﬁcation of the slope s n+1 j numerical schemes, it is a “postprocessing” step applied to the new results , s n+1 . 25), are never modiﬁed.