By P Wesseling
Multigrid equipment have built swiftly and are used as a strong device for the effective answer of elliptic and hyperbolic equations. this article offers an advent to multigrid tools for partial differential equations, with purposes to functional circulation difficulties.
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Furthermore, relying on the monotonicity property of the kinetic function, it is not hard to see that, to any point ul > 0, we can associate a point ρ (ul ) > ul such that the speed of the classical shock connecting ul to ρ (ul ) be identical with the speed of the nonclassical shock connecting ρ (ul ) to ϕ ◦ ρ (ul ). This latter corresponds to a transition in the Riemann solver described now. In addition, we denote by ϕ− the inverse of the kinetic function. ) ul ϕ (ul ) 0 C or C + N R+N R C ur Figure II-7 : The four wave patterns for the Riemann solution.
2) holds for negative left-hand side and that we have (ϕ0 ◦ ϕ0 )(u) = u, we conclude that ∂u+ E ϕ0 (u), u > 0 and therefore dϕ0 /du < 0. 3. 2. For the choice U (u) = u2 /2 the function ϕ0 is given geometrically by an analogue of Maxwell’s equal area rule. 1) in the form E(u− , u+ ) = − u+ u− f (v) − f (u− ) − f (u+ ) − f (u− ) (v − u− ) dv, u+ − u− we see that the line connecting (u− , f (u− )) to ϕ0 (u− ), f (ϕ0 (u− )) cut the graph of f in two regions with equal areas. 5). a(u−, u+) a(u−, u+) f (u+) f (u−) f (u+) f (u−) Figure II-3 : Compressive and undercompressive shock waves.
1. 5) u(x, t) = u+ , x > λ t, 30 CHAPTER II. 6) are satisﬁed. 6) imply Lax shock inequalities f (u− ) ≥ λ ≥ f (u+ ). 8) a(u− + s(u+ − u− )) ds, where a(u) = f (u), u ∈ IR. ). 7) shows that the characteristic lines impinge on the discontinuity from both sides. The shock wave is said to be compressive and will be referred to as a classical shock. 2. 9) are equivalent. 2. 8) was used to cancel the boundary terms in the integration by parts formula. 6) are equivalent. 6) is also equivalent to f (v) − f (u+ ) f (u+ ) − f (u− ) ≤ v − u+ u+ − u− for all v between u− and u+ .
An Introduction to MultiGrid Methods by P Wesseling