Nespojitá Galerkinova metoda pro řešení
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1 Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Václav Kučera Univerzita Karlova v Praze Matematicko-fyzikální Fakulta Katedra numerické matematiky Vytvoření a rozvoj týmu pro náročné technické výpočty na paralelních počítačích na TU v Liberci, 21. června 2010 Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
2 Let Ω R 2 be a bounded domain with boundary Ω = Γ I Γ O Γ W. Find w : Q T = Ω (0, T ) R 4 such that where w t + f s (w) = w = (ρ, ρv 1, ρv 2, e) T R 4, R s (w, w) in Q T, f i (w) = (ρv i, ρv 1 v i + δ 1i p, ρv 2 v i + δ 2i p, (e + p)v i ) T, R i (w, w) = (0, τ i1, τ i2, τ i1 v 1 + τ i2 v 2 + k θ/ x i ) T, τ ij = λδ ij divv + 2µd ij (v), d ij (v) = 1 2 ( vi x j + v j x i ). Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
3 Let Ω R 2 be a bounded domain with boundary Ω = Γ I Γ O Γ W. Find w : Q T = Ω (0, T ) R 4 such that where w t + f s (w) = w = (ρ, ρv 1, ρv 2, e) T R 4, R s (w, w) in Q T, f i (w) = (ρv i, ρv 1 v i + δ 1i p, ρv 2 v i + δ 2i p, (e + p)v i ) T, R i (w, w) = (0, τ i1, τ i2, τ i1 v 1 + τ i2 v 2 + k θ/ x i ) T, τ ij = λδ ij divv + 2µd ij (v), d ij (v) = 1 2 ( vi x j + v j x i ). Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
4 Let Ω R 2 be a bounded domain with boundary Ω = Γ I Γ O Γ W. Find w : Q T = Ω (0, T ) R 4 such that where w t + f s (w) = w = (ρ, ρv 1, ρv 2, e) T R 4, R s (w, w) in Q T, f i (w) = (ρv i, ρv 1 v i + δ 1i p, ρv 2 v i + δ 2i p, (e + p)v i ) T, R i (w, w) = (0, τ i1, τ i2, τ i1 v 1 + τ i2 v 2 + k θ/ x i ) T, τ ij = λδ ij divv + 2µd ij (v), d ij (v) = 1 2 ( vi x j + v j x i ). Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
5 We add the thermodynamical relations p = (γ 1)(e ρ v 2 /2), θ = and the following set of boundary conditions: ( e ρ 1 ) 2 v 2 /c v. Case Γ I : a) ρ ΓI (0,T ) = ρ D, b) v ΓI (0,T ) = v D = (v D1, v D2 ) T, ( ) c) τ ij n i v j + k θ n = 0 on Γ I (0, T ); j=1 i=1 Case Γ W : a) v ΓW (0,T ) = 0, b) θ n = 0 on Γ W (0, T ); Case Γ O : a) τ ij n i = 0, j = 1, 2, b) θ n = 0 on Γ O (0, T ); i=1 Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
6 Let T h be a partition of the closure Ω t into a finite number of closed triangles K T h. By F h we denote the set of all edges of T h. For a given edge Γ F h we define a unit normal n Γ. Γ 8 n Γ8 K 1 Γ 6 n Γ6 Γ 5 n Γ1 K 2 n Γ5 Γ 1 K 5 Γ2 n Γ2 n Γ7 Γ 7 Γ 4 Γ 3 K 3 n Γ4 n Γ3 K 4 Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
7 Let T h be a partition of the closure Ω t into a finite number of closed triangles K T h. By F h we denote the set of all edges of T h. For a given edge Γ F h we define a unit normal n Γ. Γ 8 n Γ8 K 1 Γ 6 n Γ6 Γ 5 n Γ1 K 2 n Γ5 Γ 1 K 5 Γ2 n Γ2 n Γ7 Γ 7 Γ 4 Γ 3 K 3 n Γ4 n Γ3 K 4 Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
8 For each interior face Γ F h there exist two neighbours K (L) Γ, K (R) Γ T h. We use the convention that n Γ is the outer normal to the element K (L) Γ. v (L) = trace of v (L) K Γ v (R) = trace of v (L) K Γ [v] Γ = v (L) v (R), v Γ = 1 2( v (L) + v (R)). on Γ, on Γ, Γ n Γ K (L) Γ K (R) Γ Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
9 For each interior face Γ F h there exist two neighbours K (L) Γ, K (R) Γ T h. We use the convention that n Γ is the outer normal to the element K (L) Γ. v (L) = trace of v (L) K Γ v (R) = trace of v (L) K Γ [v] Γ = v (L) v (R), v Γ = 1 2( v (L) + v (R)). on Γ, on Γ, Γ n Γ K (L) Γ K (R) Γ Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
10 Over T h we define the broken Sobolev space H k (Ω, T h ) = {v; v K H k (K) K T ht } We discretize the continuous problem in the space of discontinuous piecewise polynomial functions S h = {v; v K P p (K) K T ht }, where P p (K) is the space of all polynomials on K of degree p. In order to derive a variational formulation, we multiply the N-S equations by a test function ϕ H 2 (Ω, T h ), apply Green s theorem on individual elements and sum over all elements. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
11 Over T h we define the broken Sobolev space H k (Ω, T h ) = {v; v K H k (K) K T ht } We discretize the continuous problem in the space of discontinuous piecewise polynomial functions S h = {v; v K P p (K) K T ht }, where P p (K) is the space of all polynomials on K of degree p. In order to derive a variational formulation, we multiply the N-S equations by a test function ϕ H 2 (Ω, T h ), apply Green s theorem on individual elements and sum over all elements. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
12 Over T h we define the broken Sobolev space H k (Ω, T h ) = {v; v K H k (K) K T ht } We discretize the continuous problem in the space of discontinuous piecewise polynomial functions S h = {v; v K P p (K) K T ht }, where P p (K) is the space of all polynomials on K of degree p. In order to derive a variational formulation, we multiply the N-S equations by a test function ϕ H 2 (Ω, T h ), apply Green s theorem on individual elements and sum over all elements. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
13 Convective terms w t + f s (w) = R s (w, w) We multiply the convective term by a test function ϕ H 2 (Ω, T h ), integrate over K, apply Green s theorem: K We sum over all K T h : K T h K f s (w) ϕ dx + K f s (w) ϕ dx + F h f s (w)n s ϕ ds, f s (w)n s [ϕ] ds, Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
14 Convective terms w t + f s (w) = R s (w, w) We multiply the convective term by a test function ϕ H 2 (Ω, T h ), integrate over K, apply Green s theorem: K We sum over all K T h : K T h K f s (w) ϕ dx + K f s (w) ϕ dx + F h f s (w)n s ϕ ds, f s (w)n s [ϕ] ds, Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
15 Convective terms w t + f s (w) = R s (w, w) In the second term, incorporate a numerical flux H: F h f s (w)n s [ϕ] ds H(w (L), w (R), n) [ϕ] ds, F h e.g. Lax-Friedrichs H(w (L), w (R), n) =. 1( fs (w (L) )n s +f s (w (R) ) ( )n s +α w (L) w R) 2 Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
16 Convective terms w t + f s (w) = R s (w, w) In the second term, incorporate a numerical flux H: F h f s (w)n s [ϕ] ds H(w (L), w (R), n) [ϕ] ds, F h e.g. Lax-Friedrichs H(w (L), w (R), n) =. 1( fs (w (L) )n s +f s (w (R) ) ( )n s +α w (L) w R) 2 Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
17 Convective terms w t + f s (w) = Finally, we define the convective form: b h (w, ϕ) = K T h K R s (w, w) f s (w) ϕ dx+ H(w (L), w (R), n) [ϕ] ds, F h If Γ Ω, then w (R) is not defined. By providing w (R), we impose boundary conditions in some weak sense. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
18 Convective terms w t + f s (w) = Finally, we define the convective form: b h (w, ϕ) = K T h K R s (w, w) f s (w) ϕ dx+ H(w (L), w (R), n) [ϕ] ds, F h If Γ Ω, then w (R) is not defined. By providing w (R), we impose boundary conditions in some weak sense. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
19 Diffusion terms w t + f s (w) = R s (w, w) Question How does one discretize second order terms using spaces of discontinuous functions? Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
20 Diffusion terms w t + f s (w) = R s (w, w) Question How does one discretize second order terms using spaces of discontinuous functions? Answer Treat the second order terms as a first order system and apply the discretization from the previous slide. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
21 Model problem R s (w, w) = g. Due to properties of R s (w, w) we can write ( K sk (w) w ) = g. x k k=1 We introduce an auxiliary variable σ k and write ( ) K sk (w)σ k = g, k=1 σ k = w x k, k = 1, 2. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
22 Model problem R s (w, w) = g. Due to properties of R s (w, w) we can write ( K sk (w) w ) = g. x k k=1 We introduce an auxiliary variable σ k and write ( ) K sk (w)σ k = g, k=1 σ k = w x k, k = 1, 2. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
23 Model problem R s (w, w) = g. Due to properties of R s (w, w) we can write ( K sk (w) w ) = g. x k k=1 We introduce an auxiliary variable σ k and write ( ) K sk (w)σ k = g, k=1 σ k = w x k, k = 1, 2. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
24 Model problem ( ) K sk (w)σ k = g, k=1 σ k = w x k, k = 1, 2. This first order system for unknowns w, σ 1, σ 2 can be discretized using the discontinuous Galerkin method. Different choices of the numerical flux for this system give different numerical schemes. If the numerical flux is appropriately chosen, it is possible to eliminate σ from the resulting numerical scheme. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
25 Model problem ( ) K sk (w)σ k = g, k=1 σ k = w x k, k = 1, 2. This first order system for unknowns w, σ 1, σ 2 can be discretized using the discontinuous Galerkin method. Different choices of the numerical flux for this system give different numerical schemes. If the numerical flux is appropriately chosen, it is possible to eliminate σ from the resulting numerical scheme. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
26 Nonsymmetric variant of the diffusion form w t + a N h (w, ϕ) = K T h Fh I + K f s (w) = R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds Fh I R s (w, ϕ) n s [w] ds + Here Rk (w, ϕ) := F D h F D h R s (w, w) R s (w, w)n s ϕ ds R s (w, ϕ)n s w ds, K T sk(w) ϕ and R s (w, w) = k=1 K sk (w) w. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
27 Nonsymmetric variant of the diffusion form w t + a N h (w, ϕ) = K T h Fh I + K f s (w) = R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds Fh I R s (w, ϕ) n s [w] ds + Here Rk (w, ϕ) := F D h F D h R s (w, w) R s (w, w)n s ϕ ds R s (w, ϕ)n s w ds, K T sk(w) ϕ and R s (w, w) = k=1 K sk (w) w. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
28 Nonsymmetric variant of the diffusion form w t + a N h (w, ϕ) = K T h Fh I + K f s (w) = R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds Fh I R s (w, ϕ) n s [w] ds + F D h F D h R s (w, w) R s (w, w)n s ϕ ds R s (w, ϕ)n s w ds, Nonsymmetric, coercive and suboptimal convergence rate in L 2 -norm for even p. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
29 Symmetric variant of the diffusion form w t + f s (w) = R s (w, w) a N h (w, ϕ) = K T h Fh I K R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds Fh I R s (w, ϕ) n s [w] ds F D h F D h R s (w, w)n s ϕ ds R s (w, ϕ)n s w ds, Symmetric, not coercive and optimal convergence rate in L 2 -norm. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
30 Symmetric variant of the diffusion form w t + f s (w) = R s (w, w) a N h (w, ϕ) = K T h Fh I K R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds Fh I R s (w, ϕ) n s [w] ds F D h F D h R s (w, w)n s ϕ ds R s (w, ϕ)n s w ds, Symmetric, not coercive and optimal convergence rate in L 2 -norm. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
31 Symmetric variant of the diffusion form w t + f s (w) = R s (w, w) a N h (w, ϕ) = K T h Fh I K R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds Fh I R s (w, ϕ) n s [w] ds F D h F D h R s (w, w)n s ϕ ds R s (w, ϕ)n s w ds, Red terms are a result of applying a numerical flux to the first order system. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
32 Incomplete variant of the diffusion form w t + f s (w) = R s (w, w) a N h (w, ϕ) = K T h Fh I K R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds F D h R s (w, w)n s ϕ ds Not symmetric, not coercive and suboptimal convergence rate in L 2 -norm for even p. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
33 Incomplete variant of the diffusion form w t + f s (w) = R s (w, w) a N h (w, ϕ) = K T h Fh I K R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds F D h R s (w, w)n s ϕ ds Not symmetric, not coercive and suboptimal convergence rate in L 2 -norm for even p. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
34 Incomplete variant of the diffusion form w t + f s (w) = R s (w, w) a N h (w, ϕ) = K T h Fh I K R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds F D h R s (w, w)n s ϕ ds Simplest DG discretization of second order terms. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
35 Interior and boundary penalty In theory and in practice we need to add the interior and boundary penalty jump terms: 1 J h (w, ϕ) = C W Γ [w][ϕ] ds + C 1 W wϕ ds. Γ F I h This term ensures coercivity, when the constant C W is chosen sufficiently large. The boundary term is balanced on the right-hand side by 1 C W Γ w Bϕ ds, thus enforcing Dirichlet boundary conditions. F D h F D h Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
36 Interior and boundary penalty In theory and in practice we need to add the interior and boundary penalty jump terms: 1 J h (w, ϕ) = C W Γ [w][ϕ] ds + C 1 W wϕ ds. Γ F I h This term ensures coercivity, when the constant C W is chosen sufficiently large. The boundary term is balanced on the right-hand side by 1 C W Γ w Bϕ ds, thus enforcing Dirichlet boundary conditions. F D h F D h Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
37 Discrete Problem Definition We say that w h is a DGFE solution of the compressible Navier-Stokes equations if a) w h C 1 ([0, T ]; S h ), b) d dt (w h(t), ϕ h ) + b h (w h (t), ϕ h ) + J h (w h (t), ϕ h ) + a h (w h (t), ϕ h ) = l h (w h, ϕ h )(t), ϕ h S h, t (0, T ), c) w h (0) = w 0 h, where w 0 h is an S h approximation of the initial condition w 0. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
38 Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) A fully implicit scheme requires the solution of a nonlinear system. In the semi-implicit scheme we linearize the nonlinear terms using their specific properties. We solve only one linear system per time level. The scheme has good stability properties. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
39 Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) A fully implicit scheme requires the solution of a nonlinear system. In the semi-implicit scheme we linearize the nonlinear terms using their specific properties. We solve only one linear system per time level. The scheme has good stability properties. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
40 Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) We introduce a partition 0 = t 0 < t 1 < < t N = T and define τ n = t n+1 t n. Time derivative: d ( wh (t n+1 ), ϕ ) wn+1 h wh n dt τ n Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
41 Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) We introduce a partition 0 = t 0 < t 1 < < t N = T and define τ n = t n+1 t n. Time derivative: d ( wh (t n+1 ), ϕ ) wn+1 h wh n dt τ n Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
42 Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) Convective terms: K T h K f s (w n+1 ) ϕ dx+ H(w (L),n+1, w (R),n+1, n) ϕ ds F h Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
43 Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) Convective terms: K T h K It holds that f s (w n+1 ) ϕ dx+ H(w (L),n+1, w (R),n+1, n) ϕ ds F h f s (w) = A s (w)w, We therefore linearize where A s (w) = Df s(w) Dw. f s (w n+1 ) A s (w n )w n+1. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
44 Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) Convective terms: K T h K f s (w n+1 ) ϕ dx+ H(w (L),n+1, w (R),n+1, n) ϕ ds F h We choose the Vijayasundaram numerical flux for f H(w (L), w (R), n) = P + ( w, n) w (L) + P ( w, n) w (L) and linearize H(w n+1 (L) Γ, wn+1 (R) Γ, n Γ) P + ( w n, n Γ ) w n+1 (L) Γ +P ( w n, n Γ ) w n+1 (R) Γ. P ± = TD ± T 1, where A s n s = TDT 1. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
45 Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) Interior and boundary penalty jump terms are linear J h (w n+1, ϕ) = 1 C W Γ [wn+1 ][ϕ] ds + C W F I h F D h 1 Γ wn+1 ϕ ds. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
46 Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) Diffusion terms (for instance incomplete variant): a I h(w, ϕ) = K T h Fh I K R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds It holds that R s (w, w) = We can linearize k=1 R s (w n+1, w n+1 ) F D h R s (w, w)n s ϕ ds. K sk (w) w. k=1 K sk (w n ) wn+1. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
47 Boundary Conditions - Inlet, outlet Semi-implicit BCs at Γ I, Γ O are imposed by choosing the outside boundary state w (R) in the numerical flux. Appropriate coordinate system, neglecting the tangential derivatives and linearization give: q t + f 1(q) = 0 q x 1 t + A 1(q i ) q = 0, x 1 We seek q j such that the linearized problem has sense. Eigenvectors of A 1 (q i ) form a basis and eigenvalues are real. q i = 4 α s r s, q j = 4 β s r s. Substitution into the system reduces it to four independent equations that have an analytical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
48 Boundary Conditions - Inlet, outlet Semi-implicit BCs at Γ I, Γ O are imposed by choosing the outside boundary state w (R) in the numerical flux. Appropriate coordinate system, neglecting the tangential derivatives and linearization give: q t + f 1(q) = 0 q x 1 t + A 1(q i ) q = 0, x 1 We seek q j such that the linearized problem has sense. Eigenvectors of A 1 (q i ) form a basis and eigenvalues are real. q i = 4 α s r s, q j = 4 β s r s. Substitution into the system reduces it to four independent equations that have an analytical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
49 Boundary Conditions - Inlet, outlet Semi-implicit BCs at Γ I, Γ O are imposed by choosing the outside boundary state w (R) in the numerical flux. Appropriate coordinate system, neglecting the tangential derivatives and linearization give: q t + f 1(q) = 0 q x 1 t + A 1(q i ) q = 0, x 1 We seek q j such that the linearized problem has sense. Eigenvectors of A 1 (q i ) form a basis and eigenvalues are real. q i = 4 α s r s, q j = 4 β s r s. Substitution into the system reduces it to four independent equations that have an analytical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
50 Boundary Conditions - Inlet, outlet Semi-implicit BCs at Γ I, Γ O are imposed by choosing the outside boundary state w (R) in the numerical flux. Appropriate coordinate system, neglecting the tangential derivatives and linearization give: q t + f 1(q) = 0 q x 1 t + A 1(q i ) q = 0, x 1 We seek q j such that the linearized problem has sense. Eigenvectors of A 1 (q i ) form a basis and eigenvalues are real. q i = 4 α s r s, q j = 4 β s r s. Substitution into the system reduces it to four independent equations that have an analytical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
51 Boundary Conditions - Inlet, outlet Semi-implicit BCs at Γ I, Γ O are imposed by choosing the outside boundary state w (R) in the numerical flux. Appropriate coordinate system, neglecting the tangential derivatives and linearization give: q t + f 1(q) = 0 q x 1 t + A 1(q i ) q = 0, x 1 We seek q j such that the linearized problem has sense. Eigenvectors of A 1 (q i ) form a basis and eigenvalues are real. q i = 4 α s r s, q j = 4 β s r s. Substitution into the system reduces it to four independent equations that have an analytical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
52 Boundary Conditions - Inlet, outlet Semi-implicit Conclusion: depending on the sign of eigenvalues of A 1 (q i ) we either prescribe or extrapolate α s, β s When prescribing β s, we evaluate from an appropriate state (e.g. far-field). Finally q j := q i = Tα α = T 1 q i, q 0 j = Tβ β = T 1 q 0 j. 4 γ s r s = Tγ, where γ s = { α s, λ s 0, β s, λ s < 0. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
53 Boundary Conditions - Inlet, outlet Semi-implicit Conclusion: depending on the sign of eigenvalues of A 1 (q i ) we either prescribe or extrapolate α s, β s When prescribing β s, we evaluate from an appropriate state (e.g. far-field). Finally q j := q i = Tα α = T 1 q i, q 0 j = Tβ β = T 1 q 0 j. 4 γ s r s = Tγ, where γ s = { α s, λ s 0, β s, λ s < 0. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
54 Boundary Conditions - Inlet, outlet Semi-implicit Conclusion: depending on the sign of eigenvalues of A 1 (q i ) we either prescribe or extrapolate α s, β s When prescribing β s, we evaluate from an appropriate state (e.g. far-field). Finally q j := q i = Tα α = T 1 q i, q 0 j = Tβ β = T 1 q 0 j. 4 γ s r s = Tγ, where γ s = { α s, λ s 0, β s, λ s < 0. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
55 Shock Capturing Semi-implicit In transonic and supersonic flows it is common that solutions develop discontinuities. In these cases spurious under and overshoots occur on elements near the discontinuity. Especially in the semi-implicit case, it is desirable to avoid such phenomena. We therefore locally add artificial diffusion to suppress these effects. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
56 Shock Capturing Semi-implicit To the scheme we add two artificial viscosity forms. Internal diffusion: Φ 1 h (wn h, wn+1 h, ϕ) = ν 1 h K G n (K) w n+1 h ϕ dx K T K h with ν 1 = O(1) a given constant. Here G(K) is a discontinuity indicator which measures interelement jumps of the solution: { G k 1 if interelement jumps of wh n (K) = are large near K i, 0 otherwise. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
57 Shock Capturing Semi-implicit Interelement diffusion: Φ 2 h (wn h, wn+1 h, ϕ) = ν 2 G n [w n+1 Fh I h ] [ϕ] ds, with ν 2 = O(1) a given constant. This term allows to strengthen the influence of neighbouring elements and improves the behavior of the method in the case, when strongly unstructured and/or anisotropic meshes are used. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
58 Flow around cylinder, M = 10 4 Semi-implicit Figure: Velocity isolines of exact and numerical solution, respectively. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
59 Flow around cylinder, M = 10 4 Semi-implicit Figure: Velocity distribution on cylinder surface. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
60 Semi-implicit Corner eddies around cylinder, M = 10 4 L.E. Fraenkel: On Corner Eddies in Plane Inviscid Shear Flow, 1961 Figure: Streamlines, exact solution. Figure: Streamlines, numerical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
61 Semi-implicit Corner eddies around cylinder, M = 10 4 L.E. Fraenkel: On Corner Eddies in Plane Inviscid Shear Flow, Figure: Velocity distribution on cylinder surface: exact solution of incompressible flow, numerical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
62 Semi-implicit Flow around Žukovsky profile, M = 10 4 Figure: Velocity isolines: exact solution of incompressible flow, numerical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
63 Semi-implicit Flow around Žukovsky profile, M = Figure: Velocity distribution on profile surface: exact solution of incompressible flow, numerical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
64 Semi-implicit Flow around Žukovsky profile, M = Figure: Pressure distribution on profile surface: exact solution of incompressible flow, numerical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
65 Semi-implicit Supersonic flow around Žukovsky profile, M = 2.0 Figure: M = 0.8, Mach number isolines. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
66 Semi-implicit Supersonic flow around Žukovsky profile, M = 2.0 Figure: M = 2.0, Mach number isolines. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
67 Flow in GAMM channel, M = 0.67 Semi-implicit Figure: Mach number isolines. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
68 Flow in GAMM channel, M = 0.67 Semi-implicit Figure: Entropy isolines. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
69 Flow in GAMM channel, M = 0.67 Semi-implicit Figure: Entropy. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
70 Flow in GAMM channel, M = 0.67 Semi-implicit Figure: Entropy. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
71 NACA 0012 viscous flow Semi-implicit Figure: M = 0.5, Re = 5000, α = 2, Mach isolines. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
72 NACA 0012 viscous flow Semi-implicit Figure: M = 0.5, Re = 5000, α = 25, Mach isolines. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec / 39
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