Residual distribution schemes for advection and advection-diffusion problems on quadrilateral and hybrid meshes

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1.2 Classical and weak solutions 9 1.2 Classical and weak solutions Definition 1.2. A classical solution to the Cauchy problem is a C 1 function Q which satis- fies (1.2) and (1.4) point-wisely. Unfortunately for a system of conservation laws there does not exist, in general, a classical solution beyond some finite time interval, even for initial conditions given by a smooth function Q0. Indeed discontinuities may arise. Here we introduce weak solutions, a wider class which embraces both continuous and discontinuous solutions. Let C 10 (Rd× [0,∞[) denote the space of C 1 functions with compact support in Rd× [0,∞[ , with the vector function ϕ = (ϕ1, . . . , ϕp) made of p scalar functions ϕm ∈ C 10 (Rd× [0,∞[) for m = 1, . . . , p. Let us start from equation (1.2), multiplying by ϕ and integrating on Rd× [0,∞[ ; then by Green’s theorem (or integrating by parts) we get ∫ +∞ 0 ∫ Rd [ ∂Q ∂t +∇·F ] ·ϕ dxdt = − ∫ +∞ 0 ∫ Rd [ Q · ∂ϕ ∂t + F ·∇ϕ ] dxdt + ∫ +∞ 0 ∫ Rd [ ∂(Q · ϕ) ∂t + ∇· (F · ϕ) ] dxdt. The second term at the right-hand-side can be further simplified noticing that ∫ +∞ 0 ∫ Rd ∂ ∂t (Q · ϕ) dxdt = ∫ Rd ∣ ∣ ∣Q · ϕ ∣ ∣ ∣ +∞ 0 dx = − ∫ Rd Q(x, 0) ·ϕ(x, 0) dx , where, since ϕ has compact support in Rp it turns out that lim t→∞ϕ(x, t) = 0 ∀x ∈ R d . As a consequence, the term for t→ ∞ vanishes. Moreover, the other term related to the flux vector vanishes since: ∫ +∞ 0 ∫ Rd ∇· (F · ϕ) dxdt = ∫ +∞ 0 [∮ ∂D→∂Rd ϕ · (F · nˆ) dS ] dt = 0 , exploiting, once more, the compact support of ϕ in Rp. Finally the following equivalence is obtained: − ∫ +∞ 0 ∫ Rd [ ∂Q ∂t + ∇·F ] ·ϕ dxdt = = ∫ +∞ 0 ∫ Rd [ Q · ∂ϕ ∂t + F ·∇ϕ ] dxdt + ∫ Rd Q(x, 0) ·ϕ(x, 0) dx , and it is a simple matter to check that a classical solution of the Cauchy problem satisfies both

Anteprima della Tesi di Dante Tommaso Rubino

Anteprima della tesi: Residual distribution schemes for advection and advection-diffusion problems on quadrilateral and hybrid meshes, Pagina 9

Tesi di Dottorato

Dipartimento: Macchine ed Energetica

Autore: Dante Tommaso Rubino Contatta »

Composta da 150 pagine.

 

Questa tesi ha raggiunto 381 click dal 15/01/2007.

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