DGFEM. Theory and Applications to CFD

DGFEM. Theory and Applications to CFD Miloslav Feistauer Charles University Prague in cooperation with V. Dolejší, V. Kučera, V. Sobotíková and J. Prokopov á Presented at Humboldt-Universität, July 2009

DGFEM for a model scalar nonlinear nonstationary convection-diffusion problem

Goal: to work out a sufficiently accurate, robust and theoretically based method for the numerical solution of compressible flow with a wide range of Mach numbers and Reynolds numbers Difficulties: nonlinear convection dominating over diffusion = boundary layers, wakes for large Reynolds numbers shock waves, contact discontinuities for large Mach numbers instabilities caused by acoustic effects for low Mach numbers

One of promising, efficient methods for the solution of compressible flow is the discontinuous Galerkin finite element method (DGFEM) using piecewise polynomial approximation of a sought solution without any requirement on the continuity between neighbouring elements. Reed&Hill 1973, LeSaint&Raviart 1974, Johnson&Pitkäranta 1986 Cockburn&Shu 1989, Bassi&Rebay, Baumann&Oden 1997,... Hartmann, Houston,... van der Vegt,... M.F., Dolejší, Kučera, Prokopová, Česenek theory for elliptic or parabolic problems: Arnold, Brezzi, Marini, et al, Schwab, Suli,..., Wheeler, Girault, Riviere,... theory for nonstationary (nonlinear) convection-diffusion problems: M.F., Dolejší, Schwab, Sobotíková, Švadlenka, Hájek, Kučera, Prokopová

Continuous model problem Let us consider the problem to find u : Q T = Ω (0, T) IR such that a) u t + d s=1 f s (u) x s = ε u + g in Q T, (1) b) u ΓD (0,T) = u D, c) ε u n Γ N (0,T) = g N, d) u(x,0) = u 0 (x), x Ω. We assume that Ω IR d, d = 2,3, is a bounded polygonal (if d = 2) or polyhedral (if d = 3) domain with Lipschitzcontinuous boundary Ω = Γ D Γ N, Γ D Γ N = and T > 0. The diffusion coefficient ε > 0 is a given constant, g : Q T IR, u D : Γ D (0, T) IR, g N : Γ N (0, T) IR, and u 0 : Ω IR are given functions, f s C 1 (IR), s = 1,..., d, are prescribed fluxes.

DG space semidiscretization Let T h (h > 0) be a partition of the closure Ω of the domain Ω into a finite number of closed triangles (d = 2) or tetrahedra (d = 3) K with mutually disjoint interiors such that Ω = K T h K. (2) We call T h a triangulation of Ω and do not require the standard conforming properties from the finite element method. h K = diam(k), h = max K T h, ρ K = largest ball inscribed into K Let K, K T h. We say that K and K are neighbours, if the set K K has positive (d 1)-dimensional measure. We say that Γ K is a face of K, if it is a maximal connected open subset either of K K, where K is a neighbour of K, or of K Ω.

F h = the system of all faces of all elements K T h, the set of all innner faces: F I h = {Γ F h; Γ Ω}, (3) the set of all Dirichlet boundary faces: F D h = {Γ F h; Γ Ω D }, (4) the set of all Neumann boundary faces: Obviously, F h = Fh I FD h FN h put F N h = {Γ F h, Γ Ω N }. (5). For a shorter notation we F ID h = FI h FD h, FDN h = F D h FN h. (6) For each Γ F h we define a unit normal vector n Γ. We assume that for Γ Fh DN the normal n Γ has the same orientation as the outer normal to Ω. For each face Γ Fh I the orientation of n Γ is arbitrary but fixed. See Figure.

Γ 6 n Γ6 n Γ5 K 1 Γ 5 n Γ1 Γ 1 K 2 K 5 Γ2 n Γ2 n Γ7 Γ 7 Γ 4 Γ 3 K 3 n Γ4 n Γ3 K 4 Elements with hanging nodes d(γ) = diameter of Γ F h.

Broken Sobolev spaces Over a triangulation T h we define the so-called broken Sobolev space with the norm and the seminorm H k (Ω, T h ) = {v; v K H k (K) K T h } (7) v H k (Ω,T h ) = v H k (Ω,T h ) = 1/2 v 2 H k (K) K T h 1/2 v 2 H k (K) K T h (8). (9)

Γ n Γ K (L) Γ K (R) Γ Neighbouring elements For each Γ Fh I K (L) Γ, K(R) Γ there exist two neighbouring elements T h such that Γ K (L) Γ K (R) Γ. We use a convention that K (R) Γ lies in the direction of n Γ and K (L) Γ lies in the opposite direction to n Γ, see Figure. (K (L) are neighbours.) Γ, K(R) Γ

For v H 1 (Ω, T h ) and Γ Fh I, we introduce the following notation: v (L) Γ v (R) Γ = the trace of v K (L) Γ = the trace of v (R) K Γ ( v (L) ) Γ + v (R) Γ, v Γ = 1 2 [v] Γ = v (L) Γ v (R) Γ. on Γ, (10) on Γ, The value [v] Γ depends on the orientation of n Γ, but the value [v] Γ n Γ is independent of this orientation. For Γ F DN h K (L) Γ there exists element K (L) Γ Ω. For v H1 (Ω, T h ), we set v (L) Γ = the trace of v K (L) Γ T h such that Γ on Γ, (11) For Γ Fh DN by v (R) Γ we formally denote the exterior trace of v on Γ given either by a Dirichlet boundary condition or by an extrapolation from the interior of Ω.

The approximate solution sought in the space of discontinuous piecewise polynomial functions S h = S p, 1 h = {v; v K P p (K) K T h }, p > 0 integer, P p (K) the space of all polynomials on K of degree at most p. Derivation of the discrete problem Assume that u sufficiently regular exact solution multiply equation (1), a) by any ϕ H 2 (Ω, T h ) integrate over K T h apply Green s theorem sum over all K T h

After some manipulation we obtain the identity = Ω u t + K T h ϕdx (12) Γ F h Γ K K T K h + K T K h Γ Fh I Γ F D h d Γ s=1 d s=1 ε u ϕdx f s (u)(n K ) s ϕ Γ ds f s (u) ϕ x s dx Γ ε u n Γ[ϕ]dS Ω g ϕdx + Γ ε u n Γ ϕds Γ F N h Γ ε u n Γ ϕds.

To the left-hand side of (12) we add now the terms θ Γ F I h Γ ε ϕ n Γ[u]dS (= 0). (13) Further, to the left-hand side and the right-hand side of (12) we add the terms and θ Γ F D h θ Γ F D h Γ ε ϕ n Γ uds (14) Γ ε ϕ n Γ u D ds, respectively, which are identical due to the Dirichlet condition We consider tho following possibilities: θ = 1 nonsymmetric discretization of diffusion terms(15) (NIPG)

θ = 1 symmetric discretization of diffusion terms (SIPG) θ = 0 incomplete discretization of diffusion terms (IIPG) In view of the Neumann condition, we replace the second term on the right-hand side of (12) by Γ F h N Γ g N ϕds. (16) Because of the stabilization of the scheme we introduce the interior penalty ε Γ Fh I Γ and the boundary penalty ε Γ F h D σ[u][ϕ]ds (= 0) (17) Γ σ u ϕds = ε where σ is a suitable weight. Γ F D h Γ σu DϕdS, (18)

On the basis of above considerations we introduce the following forms defined for u, ϕ H 2 (Ω, T h ): (, ) L 2 (Ω)-scalar product, a h (u, ϕ) = θ θ K T K h Γ F h I Γ F h I Γ Fh D Γ F D h ε u ϕdx (19) Γ ε u n Γ[ϕ]dS Γ ε ϕ n Γ[u]dS Γ ε u n Γ ϕds Γ ε ϕ n Γ uds diffusion form θ = 1 nonsymmetric discretization of diffusion terms (NIPG) θ = 1 symmetric discretization of diffusion terms (SIPG) θ = 0 incomplete discretization of diffusion terms (IIPG)

Jh σ (u, ϕ) = Γ F I h interior and boundary penalty l h (ϕ)(t) = θ right-hand side form Ω Γ F D h + ε Γ σ[u][ϕ]ds+ g(t) ϕdx + Γ F D h Γ F N h Γ F D h Γ Γ ε ϕ n Γ u D (t)ds Γ σ u D(t) ϕds σ u ϕds (20) Γ g N(t) ϕds (21)

Finally, the convective terms are approximated with the aid of a numerical flux H = H(u, v, n) by the form d b h (u, ϕ) = f s (u) ϕ dx (22) K T K h s=1 x s + ( u (L) ) Γ, u (R) Γ, n Γ [ϕ] Γ ds Γ Fh I + Γ F DN h Γ H Γ H ( u (L) ) Γ, u (R) Γ, n Γ ϕ (L) Γ ds convective form H numerical flux Definition of the boundary state u (R) Γ (extrapolation) u (L) Γ for Γ Ω : u (R) Γ :=

Assumptions (H): 1. H(u, v,n) is defined in IR 2 B 1, where B 1 = {n IR d ; n = 1}, and Lipschitz-continuous with respect to u, v: H(u, v,n) H(u, v, n) C L ( u u + v v ), 2. H(u, v, n) is consistent: H(u, u,n) = d s=1 3. H(u, v, n) is conservative: u, v, u, v IR, n B 1. f s (u) n s, u IR, n = (n 1,..., n d ) B 1. H(u, v, n) = H(v, u, n), u, v IR, n B 1.

The exact sufficiently regular solution u satisfies the identity ( ) u(t), ϕ h + b h (u(t), ϕ h ) + a h (u(t), ϕ h ) + εjh σ t (u(t), ϕ h) =l h (ϕ h )(t) for all ϕ h S h and for a.a. t (0, T). Discrete problem We say that u h is a DGFE approximate solution of the convection-diffusion problem (1), if a) u h C 1 ([0, T]; S h ), (23) b) ( u h(t), ϕ h ) + a h (u h (t), ϕ h ) + b h (u h (t), ϕ h ) + Jh σ t (u h(t), ϕ h ) =l h (ϕ h )(t) ϕ h S h, t (0, T), c) u h (0) = u 0 h = S h approximation of u 0.

The discrete problem is equivalent to a large system of nonlinear ordinary differential equations. In practical computations: suitable time discretization is applied, e.g. Euler forward or backward scheme, Runge Kutta methods, discontinuous Galerkin time discretization The forward Euler and Runge-Kutta schemes are conditionally stable time step is strongly restricted by the CFLstability condition. Suitable: semi-implicit scheme - leads to a linear algebraic system on each time level Integrals are evaluated with the aid of numerical integration.

Error analysis Assumptions Assumptions (H) The weak solution u of problem (1) is regular, namely Then u t L2 (0, T; H p+1 (Ω)). (24) d dt (u(t), ϕ h) + a h (u(t), ϕ h ) + εj σ h (u(t), ϕ h) (25) +b h (u(t), ϕ h ) = l h (ϕ h )(t), ϕ h S h, for a.e. t (0, T). {T h } h (0,h0 ), h 0 > 0, - regular system of triangulations of the domain Ω: there exists C T > 0 such that h K ρ K C T K T h h (0, h 0 ). (26)

Some auxiliary results Multiplicative trace inequality: There exists a constant C M > 0 independent of v, h and K such that v 2 L 2 ( K) (27) C M ( v L 2 (K) v H 1 (K) + h 1 K v 2 L 2 (K) K T h, v H 1 (K), h (0, h 0 ). ), Inverse inequality: There exists a constant C I > 0 independent of v, h, and K such that v H 1 (K) C Ih 1 K v L 2 (K), v P p (K), K T h, h (0, h 0 ). (28)

S h -interpolation: For v L 2 (Ω) we denote by Π h v the L 2 (Ω)-projection of v on S h : Π h v S h, (Π h v v, ϕ h ) = 0 ϕ h S h. (29) Properties of the operator Π h : There exists a constant C A > 0 independent of h, K, v such that Π h v v L 2 (K) C Ah k+1 K v H k+1 (K), (30) Π h v v H 1 (K) C Ah k K v H k+1 (K), Π h v v H 2 (K) C Ah k 1 K v H k+1 (K), for all v H k+1 (K), K T h and h (0, h 0 ), where k [1, p] is an integer. If u and u h denote the exact and approximate solutions, then we set η(t) = Π h u(t) u(t), ξ(t) = u h (t) Π h u(t)( S h ) for a.e. t (0, T).

Truncation error in the convection form: If Ω D = Ω, Ω N =, then b h (u, ξ) b h (u h, ξ) (31) C ( ξ 2 H 1 (Ω,T h ) + Jσ h (ξ, ξ)) 1/2 ( h p+1 u H p+1 (Ω) + ξ L 2 (Ω)). If Ω N, then b h (u, ξ) b h (u h, ξ) (32) C ( ξ 2 H 1 (Ω,T h ) + Jσ h (ξ, ξ)) 1/2 ( h p+1/2 u H p+1 (Ω) + ξ L 2 (Ω)). Coercivity: An important step in the analysis of error estimates is the coercivity of the form which reads A h (u, v) = a h (u, v) + εjh σ (u, v), (33) A h (ϕ h, ϕ h ) ε 2 ( ϕh 2 H 1 (Ω,T h ) + Jσ h (ϕ h, ϕ h ) ), (34) ϕ h S h, h (0, h 0 ).

We shall discuss the validity of estimate (34) in various situations. (I) Conforming mesh T h Let the mesh T h have the standard properties from the finite element method: if K, K T h, K K, then K K = or K K is a common vertex or K K is a common edge (or K K is a common face in the case d = 3) of K and K. In this case we set σ Γ = C W d(γ), Γ F h. (35) Then the coercivity inequality (34) holds under the following choice of the constant C W : C W > 0 (e. g. C W = 1) for NIPG version, (36) C W 2C M (1 + C I ) for SIPG version, (37) C W C M (1 + C I ) for IIPG version, (38)

where C M and C I are constants from (27) and (28), respectively. (II) Nonconforming mesh T h In this case T h is formed by closed triangles with mutually disjoint interiors with hanging nodes in general. Then the coercivity inequality (34) is guaranteed under conditions (36) (38). However, in this case it is necessary to assume that h K C D d(γ), Γ F h,γ K, (39) in order to prove the estimate J σ h (η, η) Chp u H p+1 (Ω). (40)

(III) Nonconforming mesh T h without assumption (39) It is obvious that condition (39) is rather restrictive in some cases. In order to avoid it, we change the definition of the weight σ: σ Γ = h K (L) Γ σ Γ = C W h K (L) Γ 2C W + h K (R) Γ, Γ F D h., Γ F I h, (41)

Due to theoretical analysis, it is necessary to introduce the assumption of a local quasiuniformity of the mesh: h K (L) Γ C Q h (R), Γ F K h I. (42) Γ (Hence, C Q 1.) Then the coercivity inequality (34) holds under the following choice of C W : C W > 0 (e. g. C W = 1) for NIPG version, (43) C W 2C M (1 + C I )(1 + C Q ) for SIPG version, (44) C W C M (1 + C I )(1 + C Q ) for IIPG version. (45) Proof of the coercivity inequality (34) in the case (III) for SIPG version: Using the definition of the forms a h and Jh σ and the Cauchy and Young s inequalities, we find that for any δ > 0 we have a h (ϕ h, ϕ h ) ε ϕ h 2 H 1 (Ω,T h ) εω ε δ C W J σ h (ϕ h, ϕ h ),

where ω = 1 δ Γ F I h Γ h K (L) Γ + h K (R) Γ 2 ϕ h 2 ds + Γ F D h Γ h K (L) Γ ϕ h 2 ds. In view of (42), ω 1 δ 1 + C Q 2 K T h h K K ϕ h 2 ds. Now, the application of (27) and (28) yields the estimate ω 1 2δ C M(1 + C I )(1 + C Q ) ϕ h 2 H 1 (Ω,T h ). If we set δ := C M (1+C I )(1+C Q ) and use assumption (44), we immediately arrive at (34). In the IIPG case we can proceed similarly.

Error estimates Assumptions: (H), regularity of u, regularity of the mesh, u 0 h = Π hu 0, σ, d(γ), h K and C W satisfy assumptions from the cases (I) or (II) or (III). Then the error e h = u u h satisfies the estimate max t [0,T] e h (t) 2 L 2 (Ω) t (46) + ε 2 ( e h(ϑ) 2 0 H 1 (Ω,T h ) + Jσ h (e h(ϑ), e h (ϑ))) dϑ C h 2p, h (0, h 0 ), (47) with a constant C > 0 independent of h.

Sketch of the proof Let us subtract the relations valid for the exact and approximate solutions, set ϕ h = ξ and use the coercivity inequality: 1 d 2 dt ξ(t) 2 L 2 (Ω) + ε 2 ξ(t) 2 H 1 (Ω,T h ) + ε 2 Jσ h (ξ(t), ξ(t)) (48) ( ) η(t) b h (u(t), ξ(t)) b h (u h (t), ξ(t)), ξ(t) t a h (η(t), ξ(t)) εjh σ (η(t), ξ(t)) for a.a. (0, T). Now we estimate individual terms in (48): d dt ξ 2 L 2 (Ω) + ε ξ 2 H 1 (Ω,T h ) + εjσ h (ξ, ξ) (49) C {( Jh σ (ξ, ) ( ξ)1/2 + ξ H 1 (Ω,T h ) ξ L 2 (Ω) + hp+1/2 u H (Ω)) p+1 +h p+1 u/ t H p+1 (Ω) ξ L 2 (Ω) +ε h p ( u H p+1 (Ω) J σ h (ξ, ξ) 1/2 + ξ H 1 (Ω,T h ))} Now we apply Young s inequality:

d dt ξ 2 L 2 (Ω) + ε ξ 2 H 1 (Ω,T h ) + εjσ h (ξ, ξ) (50) ε ( {( J σ 2 h (ξ, ξ) + ξ 2 H 1 (Ω,T h )) + C 1 + 1 ) ξ 2 ε L 2 (Ω) + 1 ( (ε 2 h 2p + h 2p+1 ) u 2 ) ε H p+1 (Ω) + h 2p+2 u/ t 2 H (Ω)} p+1 a. e. in (0, T). The integration of (50) from 0 to t [0, T] and the relation ξ(0) = u 0 h Π hu 0 = 0 yield t ξ(t) 2 L 2 (Ω) + ε ( ξ(ϑ) 2 2 0 H 1 (Ω,T h ) + Jσ h (ξ(ϑ), ξ(ϑ))) dϑ (51) {( C 1 + 1 ) t ε 0 ξ(ϑ) 2 L 2 (Ω) dϑ + 1 t ( ε h2p (ε 2 + h) u(ϑ) 2 ) 0 H p+1 (Ω) dϑ + h 2p+2 t } 0 u(ϑ)/ t 2 H p+1 (Ω) dϑ, t [0, T]. Using Gronwall s lemma, we get

ξ(t) 2 L 2 (Ω) + ε 2 t 0 ( ξ(ϑ) 2 H 1 (Ω,T h ) + Jσ h (ξ(ϑ), ξ(ϑ))) dϑ (52) C ( (ε + h/ε) u 2 L 2 (0,T;H p+1 (Ω)) + h2 u/ t 2 L 2 (0,T;H p+1 (Ω)) h 2p exp ( C 1 + ε t ε ), t [0, T], (C and C are constants independent of t, h, ε, u). ) Now, since e h = ξ + η and thus, e h 2 L 2 (Ω) 2 ( ξ 2 L 2 (Ω) + η 2 L 2 (Ω)), (53) e h 2 H 1 2 ( ξ 2 (Ω,T h ) H 1 (Ω,T h ) + η 2 H 1 (Ω,T h )), Jh σ (e h, e h ) 2 ( Jh σ (ξ, ξ) + Jσ h (η, η)), we use the above result, estimate the terms with η and obtain the sought error estimate.

Optimal error estimates The error estimate (46) is optimal in the L 2 (H 1 )-norm, but suboptimal in the L (L 2 )-norm. We carried out the analysis of the L (L 2 )-optimal error estimate under the following assumptions. Assumptions (B): the discrete diffusion form a h is symmetric (i.e. we consider the SIPG version), the polygonal domain Ω is convex, the exact solution u satisfies the regularity condition, conditions (H) are satisfied, u 0 h = Π hu 0, Γ D = Ω and Γ N =. The application of the Aubin-Nitsche technique based on the use of the elliptic dual problem considered for each z L 2 (Ω): ψ = z in Ω, ψ Ω = 0. (54)

Then the weak solution ψ H 2 (Ω) and there exists a constant C > 0, independent of z, such that ψ H 2 (Ω) C z L 2 (Ω). (55) For each h (0, h 0 ) and t [0, T] we define the function u h (t) as the A h-projection of u(t) on S h, i. e. a function satifying the conditions u h (t) S h, A h (u h (t), ϕ h) = A h (u(t), ϕ h ) ϕ h S h, (56) and set χ = u u h. Using the elliptic dual problem (54), we proved the existence of a constant C > 0 such that χ L 2 (Ω) Chp+1 u H p+1 (Ω), (57) χ t L 2 (Ω) Chp+1 u t H p+1 (Ω), h (0, h 0). (58) This, the estimate of the truncation error in the form b h (31), multiple application of Young s inequality and Gronwall s lemma represent important tools for obtaining the L (L 2 )-error estimate:

Theorem. Let assumptions (B) be fulfilled. Then the error e h = u u h satisfies the estimate e h L (0,T;L 2 (Ω)) Chp+1, (59) with a constant C > 0 independent of h. Remark The constant C in the error estimates is of the order O(exp( CT/ε), which blows up for ε 0+. = a consequence of the application of necessary tools for overcoming the nonlinear convective terms, namely Young s inequality and Gronwall s lemma.

Space-time DG method for nonstationary convection-diffusion problems

Goal: to develop a sufficiently accurate and robust method for the numerical simulation of compressible flow Promising space discretization: discontinuous Galerkin method with interior and boundary penalty Time discretization???: first-order: forward or backward Euler, higher-order: explicit Runge-Kutta, Crank-Nicolson, BDF, DG in time

Analysis of the space-time DG for convectiondiffusion problems Nonlinear problem u t + d f s (u) = ε u + g in Q T = Ω (0, T), s=1 x s u Ω (0,T) = u D, (60) u(x,0) = u 0 (x), x Ω f s C 1 (IR), f s C, ε > 0, d = 2, 3 Ω IR d bounded, polygonal (polyhedral) domain, T > 0

Space semidiscretation by the DGFEM Mesh: let d = 2 T h = partition of the closure Ω of the domain Ω into a finite number of closed triangles K with mutually disjoint interiors such that Ω = K T h K. F h = the system of all faces of all elements K T h, the set of all interior faces: Fh I = {Γ F h; Γ Ω}, the set of all boundary faces: Fh B = {Γ F h; Γ Ω}

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 Elements with hanging nodes Γ F h unit normal vector n Γ. For Γ Fh B the normal n Γ has the same orientation as the outer normal to Ω. d(γ) = diameter of Γ F h.

Γ n Γ K (L) Γ K (R) Γ Neighbouring elements there exist two neighbouring ele- For each Γ Fh I ments K (L) Γ, K(R) Γ T h such that Γ K (R) Γ We use a convention that K (R) Γ lies in the direction of n Γ and K (L) Γ to n Γ, see Figure. K(L) Γ. lies in the opposite direction

h K - diameter of K T h, ρ K - radius of the largest ball inscribed in K, h = max K Th h K. Broken Sobolev space H k (Ω, T h ) = {ϕ; ϕ K H k (K) K T h } with seminorm ϕ H k (Ω,T h ) = K T h ϕ 2 H k (K) 1/2 ϕ H 1 (Ω, T h ) - in general discontinuous on interfaces Γ Fh I ϕ (L) Γ and ϕ (R) Γ - the values of ϕ on Γ considered from the interior and the exterior of K (L) ϕ Γ := (ϕ (L) Γ + ϕ(r) Γ )/2, [ϕ] Γ := ϕ (L) Γ Γ, ϕ(r) Γ.

Approximate solution sought in the space S p h = {ϕ L2 (Ω); ϕ K P p (K) K T h } p 1 is an integer. Derivation of a DG space semidiscretization - multiply the PDE by any ϕ h S p h, - integrate over each K T h, - apply Green s theorem, - sum over all elements - in convective terms use numerical flux - add suitable terms - either vanishing or canceling on the basis of the Dirichlet BC.

= Space semi-discrete problem: Find u h C 1 ([0, T]; S p h ) such that ( ) uh t, ϕ h + A h (u h (t), ϕ h ) = l h (ϕ h )(t) ϕ h S p h t (0, T), (61) (u h (0), ϕ h ) = (u 0, ϕ h ) ϕ h S p h. (u, ϕ) = Ω uϕ dx, A h (u, ϕ) = a h (u, ϕ) + b h (u, ϕ) + εj h (u, ϕ),

Definitions of the forms a h, b h,... Let C W > 0 be a fixed constant. We introduce the notation h(γ) = h(γ) = h K (L) Γ + h K (R) Γ 2C W for Γ F I h, h K (L) Γ C W for Γ F B h. a h (u, ϕ) = ε ε Γ F I h ε Γ F B h K T K h u ϕ dx Γ ( u n Γ[ϕ] + θ ϕ n Γ [u]) ds Γ (( u n Γ)ϕ + θ( ϕ n)u) ds,

J h (u, ϕ) = h(γ) 1 Γ [u][ϕ] ds + Γ F I h h(γ) 1 Γ uϕ ds, Γ F B h l h (ϕ)(t) = θε Γ F B h +ε Γ F B h Ω g(t)ϕ dx Γ h(γ) 1 u D (t)ϕ ds Γ u D(t)( ϕ n) ds θ = 1, 0, 1 for NIPG, IIPG, SIPG

b h (u, ϕ) = d f s (u) ϕ dx K T K h s=1 x s + ( Γ H u (L) ) Γ, u(r) Γ, n Γ [ϕ] Γ ds Γ F I h + Γ F B h ( Γ H u (L) ) Γ, u(l) Γ, n Γ ϕ (L) Γ ds H numerical flux: Lipschitz continuous, consistent: H(v, v, n) = d s=1 f s (v) n s, conservative: H(v, w, n) = H(w, v, n)

Time discretization by the discontinuous Galerkin method: (C. Johnson, V. Thomée, C. Schwab, D. Schötzau,... for ODE s or for parabolic problems in combination with conforming FEM in space) Partition 0 = t 0 < t 1 <... < t M = T of [0, T] I m = (t m 1, t m ), τ m = t m t m 1, m = 1,..., M Notation: ϕ ± m = ϕ(t m±) = lim t t m ± ϕ(t) {ϕ} m = ϕ + m ϕ m.

For each time interval I m, m = 1,..., M, we can consider, in general, a different triangulation T h,m of the domain Ω. Therefore, for different intervals I m we have different S p h,m, a h,m, b h,m, J h,m, l h,m, A h,m, etc. - L 2 (Ω)-norm, DG-norm in H 1 (Ω, T h,m ): ϕ DG,m = K T h,m K ϕ 2 dx + J h,m (ϕ, ϕ) 1/2 h m = max K T h,m h K, h = max m=1,...,m h m, τ = max m=1,...,m τ m. Let p, q 1 be integers. (q = 0 - backward Euler - see M.F., V. Dolejší, J. Hozman, CMAME 2007)

Approximate solution: U(x, t) S p,q h,τ (62) = satisfying ϕ L2 (Q T ); ϕ Im = q i=0 t i ϕ i, with ϕ i S p h,m, ( (U, ϕ) + Ah,m (U, ϕ) ) dt (63) I m +({U} m 1, ϕ + m 1 ) = I m l h,m (ϕ) dt, m = 1,..., M, ϕ S p,q h,τ, (U 0, ϕ) = (u0, ϕ) ϕ S p h,1.

Space-time interpolation of the exact solution: for m = 1,..., M πu S p,q h,τ, (64) (πu u, ϕ ) dt = 0 ϕ S p,q 1 I m h,τ, πu(t m) = Π m u(t m), Π m is L 2 -projection on S p h,m in space: if v L 2 (Ω), then Π m v S p h,m and (Π mv v, ϕ) = 0 for all ϕ S p h,m. Main goal: the derivation of the estimation of the error e = U u = ξ + η, ξ = U πu S p,q h,τ, η = πu u

= = I m I m ( (ξ, ϕ) + A h,m (ξ, ϕ) ) dt + ( {ξ m 1 }, ϕ + ) m 1 ( bh,m (u, ϕ) b h,m (U, ϕ) ) dt ( (η, ϕ) + A h,m (η, ϕ) ) ( dt {η} m 1, ϕ + ) m 1 I m ϕ S p,q h,τ.

Derivation of error estimates Some assumptions We consider a system of triangulations T h,m, m = 1,..., M, h (0, h 0 ) which is shape regular, quasiuniform: h K ρ K c R, K T h,m, m = 1,..., M, h (0, h 0 ), h K (L) Γ C Q h K (R) Γ Γ F I h,m.

Auxiliary results Multiplicative trace inequality: v 2 ( ) L 2 ( K) C M v L 2 (K) v H 1 (K) + h 1 K v 2 L 2, (K) v H 1 (K), K T h,m, h (0, h 0 ). Inverse inequality: v H 1 (K) C Ih 1 K v L 2 (K), v P p (K), K T h,m, h (0, h 0 ). Approximation properties of Π m : µ = min(r, p) Π m v v L 2 (K) C hµ+1 K v H r+1 (K), Π m v v H 1 (K) C hµ K v H r+1 (K), Π m v v H 2 (K) C hµ 1 K v H r+1 (K), v H r+1 (Ω), K T h,m, h (0, h 0 ),

Coercivity of the form A h,m : A h,m (ξ, ξ) ε 2 ξ 2 DG,m provided C W > 0 for NIPG, C W C M (1 + C I )(1 + C Q ) for IIPG, C W 2C M (1 + C I )(1 + C Q ) for SIPG. Consistency of b h,m : For any ϕ S p,q b h,m (u, ϕ) b h,m (U, ϕ) h,τ and k > 0, ε k ϕ 2 DG,m + C ε where σ m 2 (η) = ( ξ 2 + σ 2 m (η) ), ( η 2 ) L 2 (K) + h2 K η 2 H 1 (K). K T h,m

I) Derivation of estimates for ξ Let us substitute ϕ := ξ and analyze individual terms. Integration by parts, Young s inequality and above estimates = C ε ξ m 2 ξ m 1 2 + 1 2 {ξ} m 1 2 + ε I m ξ 2 dt + 2 η m 1 2 + C where k > 0 and R m (η) = εσ 2 m (η) + 1 ε σ2 m (η), ( 1 2 ) ξ 2 k I DG,m dt m I m R m (η)dt, σm 2 (η) = η 2 DG,m + h 2 K η 2 H 2 (K) K T h,m

II) Estimation of ξ 2 dt I m Approach proposed by Ch. Makridakis based on the Radau quadrature formula: 0 < ϑ 1 < < ϑ q+1 = 1, w i > 0, i = 1,..., q + 1 weights = 1 q+1 0 ϕ(t)dt w i ϕ(ϑ i ) i=1 - exact for polynomials of degree 2q Transformed to the interval I m : I m ϕ(t) dt = tm t m 1 ϕ(t) dt τ m q+1 i=1 w i ϕ(t m,i ): t m,i = t m 1 + ϑ i τ m, i = 1,..., q + 1

Set ϕ := ξ = interpolation of the function τ m ξ(t)/(t t m 1 ) at the points t m,i, i = 1,..., q + 1 Notation: v m := q+1 τ m i=1 w i ϑ 1 i v(t m,i ) 2 1/2

Auxiliary estimates: a) b) (ξ, ξ)dt + (ξ + ) I m m 1, ξ m 1 1 ( ξm 1 2 2 + 1 ξ 2 m τ m ξ + m 1 2 c 1 ξ 2 m τ m ) c) I m A h,m (ξ, ξ)dt ε I m ξ 2 DG,m dt d) A I h,m (η, ξ)dt m ε ξ k Im 2 DG,m dt + C ε σ 2 I m m(η)dt.

e) I m (η, ξ)dt + ( {η} m 1, ξ m 1 + ) η m 1 ξ + m 1 f) I m (b h,m (u, ξ) b h,m (U, ξ))dt ε k ξ 2 I DG,m dt + C ξ 2 dt + C m ε I m ε I m σ 2 m (η)dt. R m (η) = εσ 2 m (η) + 1 ε σ2 m (η), σm(η) 2 = η 2 DG,m + h 2 K η 2 H 2 (K) K T h,m σ m(η) 2 = ( η 2 L 2 (K) + h2 K η 2 H 1 (K) K T h,m )

= under the assumption that τ m = O(ε) we have I m ξ 2 dt c τ m ( ξ m 1 2 + η m 1 2 + I m R m (η)dt). The above estimates imply that ξ m 2 + ε 2 ( 1 + c ε τ m + C I m ( ) I m ξ 2 DG,m dt ξ m 1 2 + C 1 + τ m ε ) ( 1 + τ m ε ) η m 1 2 R m (η)dt, m = 1,..., M, with constants c, C > 0.

Gronwall s lemma, ξ 0 Abstract error estimate: e m 2 + ε 2 Ce c ε t m m j=1 m j=1 +2 η m 2 + ε m = 0, e = ξ + η = I m e 2 DG,j dt η j 2 + m j=1 j=1 I j I j η 2 DG,j ( 1 + τ ) j ε R j (η)dt dt, m = 1,..., M.

III) Estimates of expressions with η in terms of h, τ Lemma Let u H = H q+1 (0, T; H 1 (Ω)) C([0, T]; H p+1 (Ω)), assume the regularity of meshes T h,m and let Ĉ S h 2 K τ m not necessary, if the meshes at all time levels are identical. Then I m η 2 H 1 (Ω,T h,m ) dt Ch 2p u 2 L 2 (I m ;H p+1 (Ω)) + Cτ2q+2 m u 2 H q+1 (I m ;H 1 (Ω)),

I m η 2 L 2 (Ω) dt Ch 2p+2 u 2 L 2 (I m ;H p+1 (Ω)) + Cτ2q+2 m u 2 H q+1 (I m ;L 2 (Ω)), h 2 I m η 2 H 2 (Ω,T h,m ) dt Ch2p u 2 L 2 (I m ;H p+1 (Ω)) +Cτ 2q+2 m u 2 H q+1 (I m ;H 1 (Ω)), M 1 m=1 η m 2 L 2 (Ω) Ch2p u 2 C([0,T];H p+1 (Ω)). I m J h,m (η, η) dt Ch 2p u 2 L 2 (I m ;H p+1 (Ω)) +Cτ 2q m u 2 H q+1 (I m ;L 2 (Ω)) loss of order of accuracy in time!!!

IV) Finally, we finish the proof of the error estimates: Assumptions: a) regularity of u: u H = H q+1 (0, T; H 1 (Ω)) C([0, T]; H p+1 (Ω)), b) shape regularity of the space-time meshes = error estimate max m=1,...,m e m 2 + M ε e 2 m=1 2 I DG,m dt m ( C h 2p u 2 C([0,T];H p+1 (Ω)) + τ2q u 2 H q+1 (0,T;H 1 (Ω)) ).

Improvement of error estimates in time We expect the error in time in the previous estimate of order O(τ 2q+2 ) The loss of accuracy caused by the estimate of the penalty term J h,m (η, η) It is necessary to estimate I m J h,m (π(π m u) Π m u, π(π m u) Π m u)dt. Thus, for Γ F I h,m we have to estimate I m (h(γ) 1 Γ [π(π mu) Π m u] 2 ds)dt

Using the notation D q+1 = q+1 tq+1, relations D q+1 [Π m u(x, )] = [D q+1 Π m u(x, )], [D q+1 u] = 0, D q+1 (Π m u u) = Π m (D q+1 u) D q+1 u and approximation properties of π, we get (h(γ) 1 I m Γ [π(π mu) Π m u] 2 ds)dt C τm 2q+2 (h(γ) 1 I m Γ [Π m(d q+1 u) D q+1 u] 2 ds)dt. Multiplicative trace inequality v L 2 ( K) C M( v L 2 (K) v H 1 (K) + h 1 K v L 2 (K) ) and approximation properties of Π m =

I m ( Γ F I h,m h(γ) 1 Γ [π(π mu) Π m u] 2 ds)dt C τ 2q+2 m u 2 H q+1 (I m,h 1 (Ω)). If Γ Fh,m B, the situation is more complicated. Necessary to assume that u D (x, t) = with ψ j H p+1/2 ( Ω). q j=0 ψ j (x) t j

Then a similar process and above results lead to the expected estimate: J h,m (η, η) C (h 2p u 2 L 2 (I m,h p+1 (Ω)) +τ2q+2 u 2 H q+1 (0,T;H 1 (Ω)) ) If u D is not polynomial in t of degree q, then it is necessary to slightly modify the scheme.

Remark: The use of Young s inequality and Gronwall s lemma = the constant in the error estimate C exp(c/ε). Uniform in ε 0 error estimate obtained for a linear convection-diffusion-reaction equation (in this case Young s inequality, Gronwall s lemma and assumption τ m = O(ε) NOT USED)

Examples Q T = (0,1) 2 (0,1), u + v u ε u + cu = g t v = (v 1, v 2 ), v 1 = v 2 = 1, c = 0.5, ε = 0.005 (parabolic case) and ε = 0 (hyperbolic case). The right-hand side g, boundary and initial conditions such that they conform to the exact solution u ex (x 1, x 2, t) = (1 e t ) (2x + 2y xy + 2(1 e v ) 1(x 1 1)/ν )(1 e v 2 (x 2 1)/ν ),

ν = 0.05 determines the steepness of the boundary layer in the exact solution. Coarse and fine meshes h τ e τh L2 (L 2 ) e τh εl 2 (H 1 ) EOC 0 space EOC 0 time EOC 1 space EOC 1 time 0.2838 0.2500 4.5853E-02 1.9970E-01 - - - - 0.2172 0.2000 3.5474E-02 1.5372E-01 0.96 1.15 0.98 1.17 0.1540 0.1667 2.2387E-02 1.2782E-01 1.34 2.52 0.54 1.01 0.1035 0.1000 1.2945E-02 8.9991E-02 1.38 1.07 0.88 0.69 0.0768 0.0769 5.3557E-03 5.7493E-02 2.95 3.36 1.50 1.71 0.0532 0.0526 2.3742E-03 3.7567E-02 2.22 2.14 1.16 1.12 0.0398 0.0400 1.3345E-03 2.6438E-02 1.98 2.10 1.21 1.28 0.0270 0.0270 5.2577E-04 1.6779E-02 2.40 2.38 1.17 1.16 Global order of convergence 2.07 2.11 1.07 1.11 ε = 0.005, p = 1, q = 1 (parabolic case)

h τ e τhl2 (L 2 ) e τh εl2 (H 1 ) EOC 0 space EOC 0 time EOC 1 space EOC 1 time 0.2838 0.2500 2.0470E-02 8.7193E-02 - - - - 0.2172 0.2000 1.0103E-02 5.5539E-02 2.64 3.16 1.69 2.02 0.1540 0.1667 4.3992E-03 3.4110E-02 2.42 4.56 1.42 2.67 0.1035 0.1000 1.6821E-03 1.7835E-02 2.42 1.88 1.63 1.27 0.0768 0.0769 4.9668E-04 7.7827E-03 4.08 4.65 2.78 3.16 0.0532 0.0526 1.6550E-04 3.3350E-03 3.00 2.90 2.31 2.23 0.0398 0.0400 7.7630E-05 1.8029E-03 2.61 2.76 2.12 2.24 0.0270 0.0270 2.7654E-05 7.0749E-04 2.66 2.63 2.41 2.39 Global order of convergence 2.89 2.78 2.05 2.41 ε = 0.005, p = 2, q = 2 (parabolic case) h τ e τh L2 (L 2 ) EOC 0 space EOC 0 time 0.2838 0.2500 4.9212E-02 - - 0.2172 0.2000 3.8843E-02 0.89 1.06 0.1540 0.1667 2.5997E-02 1.17 2.20 0.1035 0.1000 1.5581E-02 1.29 1.00 0.0768 0.0769 6.9089E-03 2.72 3.10 0.0532 0.0526 3.2904E-03 2.02 1.95 0.0398 0.0400 1.8620E-03 1.96 2.07 0.0270 0.0270 7.5458E-04 2.32 2.30 Global order of convergence 1.95 1.99 ε = 0, p = 1, q = 1 (hyperbolic case)

h τ e τh L2 (L 2 ) EOC 0 space EOC 0 time 0.2838 0.2500 2.3451E-02 - - 0.2172 0.2000 1.2484E-02 2.36 2.83 0.1540 0.1667 6.1746E-03 2.05 3.86 0.1035 0.1000 2.6342E-03 2.14 1.67 0.0768 0.0769 8.0848E-04 3.95 4.50 0.0532 0.0526 2.6400E-04 3.05 2.95 0.0398 0.0400 1.0761E-04 3.09 3.27 0.0270 0.0270 2.7962E-05 3.47 3.44 Global order of convergence 2.87 2.98 ε = 0, p = 2, q = 2 (hyperbolic case)

DGFEM for the solution of compressible flow

Importance of the simulation of compressible flow: design of airplanes (investigation of wings and tails vibrations) design of steam turbomachines (vibrations of blades) car industry (in order to avoid noise) civil engineering (interaction of a strong wind with structures - TV towers, cooling towers, bridges etc.) medicine (creation of voice)

In all these examples: flow of gases, i.e. compressible flow for low Mach numbers often incompressible model used sometimes the compressibility plays an important role often, the flow in time-dependent domains has to be studied

Continuous problem Consider compressible flow in a bounded domain Ω t IR 2 depending on time t [0, T]. Let the boundary of Ω t consist of three different parts: Ω t = Γ I Γ O Γ Wt Γ I - inlet Γ O - outlet Γ Wt - impermeable walls that may move in dependence on time.

Basic equations: continuity eq., Navier-Stokes eq s, energy equation Notation w t + N s=1 f s (w) x s = N s=1 R s (w, w) x s (65) w = (ρ, ρv 1,..., ρv N, E) T IR m, m = N + 2, (66) w = w(x, t), x Ω t, t (0, T), f i (w) = (f i1,..., f im ) T = (ρv i, ρv 1 v i + δ 1i p,..., ρv N v i + δ Ni p,(e + p)v i ) T R i (w, w) = (R i1,..., R im ) T = (0, τ i1,..., τ in, τ i1 v 1 + + τ in v N κ θ/ x i ) T, τ ij = λdivvδ ij + 2µ d ij (v), d ij (v) = 1 2 v i + v j x j x i.

ρ density, p pressure, E total energy, v = (v 1,..., v N ) velocity, θ absolute temperature Thermodynamical relations: p = (γ 1)(E ρ v 2 /2), θ = E ρ 1 2 v 2 /c v. (67) γ > 1 Poisson adiabatic constant, c v > 0 specific heat, µ > 0, λ = 2µ/3 viscosity coefficients, κ heat conduction Initial condition: w(x,0) = w 0 (x), x Ω 0, (68)

Boundary conditions For each t (0, T) we prescribe the following conditions: Inlet: a) ρ ΓI = ρ D, b) v ΓI = v D = (v D1,..., v DN ) T, c) θ ΓI = θ D ; Wall - with a moving part: a) v ΓWt = z D = velocity of a moving wall, b) θ n Γ Wt = 0;

Outlet: a) N i=1 τ ij n i = 0, j = 1,..., N, b) θ n = 0 on Γ O Obstacles: hyperbolic-parabolic character of the system nonlinear singularly perturbed system shock waves, contact discontinuities, boundary layers, wakes and their interaction moving boundary lack of theoretical results

ALE formulation Let N = 2 = m = 4. The dependence of the domain on time is taken into account with the aid of the arbitrary Lagrangian- Eulerian (ALE) method (proposed by T. Hughes et al.), based on a regular one-to-one ALE mapping of the reference domain Ω 0 onto the current configuration Ω t : A t : Ω 0 Ω t, i.e. A t : X Ω 0 x = x(x, t) Ω t. The ALE mapping A t.

Domain velocity: z(x, t) = t A t(x), t [0, T], X Ω 0, (69) z(x, t) = z(a 1 t (x), t), t [0, T], x Ω t (z ΓWt = z D ) ALE derivative of a function f = f(x, t) defined for x Ω t, t [0, T]: where D A Dt f(x, t) = f t (X, t) X=A 1 t (x), (70) f(x, t) = f(a t (X), t), X Ω 0.

It is possible to show that D A f Dt = f t + z grad f = f t + div(zf) f divz. (71) = ALE formulation of the Navier-Stokes equations: D A w Dt + 2 s=1 g s (w) x s + wdivz = 2 s=1 R s (w, w) x s, g s, s = 1,2, - ALE modified inviscid fluxes: g s (w) := f s (w) z s w. (72)

Space semidiscretation by the DGFEM Mesh: Ω ht = polygonal approximation of Ω t T ht = partition of the closure Ω ht of the domain Ω ht into a finite number of closed triangles K with mutually disjoint interiors such that Ω ht = K T ht K.

F ht = the system of all faces of all elements K T ht, the set of all interior faces: F I ht = {Γ F ht; Γ Ω}, the set of all boundary faces: F B ht = {Γ F ht; Γ Ω ht }, the set of all Dirichlet boundary faces: F D ht = { Γ F B ht ; a Dirichlet condition on 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 Elements with hanging nodes Γ F ht unit normal vector n Γ. For Γ Fht B the normal n Γ has the same orientation as the outer normal to Ω ht. d(γ) = diameter of Γ F ht.

Γ n Γ K (L) Γ K (R) Γ Neighbouring elements For each Γ Fht I there exist two neighbouring elements K (L) Γ, K(R) Γ T h such that Γ K (R) Γ. We use a convention that K(R) Γ the direction of n Γ and K (L) Γ K (L) Γ lies in lies in the opposite direction to n Γ, see Figure. (K (L) Γ, K(R) Γ neighbours.) are called

Space of approximate solutions Discontinuous piecewise polynomial functions: S ht = [S ht ] 4, (73) S ht = {v; v K P r (K) K T ht }, r 0 integer and P r (K) denotes the space of all polynomials on K of degree r. ϕ S ht - in general discontinuous on interfaces Γ F I ht ϕ (L) Γ and ϕ (R) Γ - the values of ϕ on Γ considered from the interior and the exterior of K (L) Γ, ϕ Γ = (ϕ (L) Γ + ϕ(r) Γ )/2, [ϕ] Γ = ϕ (L) Γ ϕ(r) Γ.

Derivation of the discrete problem - multiply system (71) by a test function ϕ h S ht - integrate over K T ht - use Green s theorem - sum over all K T ht - introduce the concept of the numerical flux - introduce suitable terms mutually vanishing for a regular exact solution = K T ht K D A w ϕ Dt h dx +b h (w, ϕ h ) + a h (w, ϕ h ) + J h (w, ϕ h ) +d h (w, ϕ h ) = l h (w, ϕ h )

Convection form b h b h (w, ϕ h ) = + Γ Fht I + Γ F B ht H g - numerical flux K T ht K 2 s=1 g s (w) ϕ h x s dx Γ H g(w (L) Γ, w(r) Γ, n Γ) [ϕ h ] Γ ds, Γ H g(w (L) Γ, w(r) Γ, n Γ) ϕ (L) hγ ds consistent with the fluxes g s : H g (w, w, n) = 2 s=1 g s (w)n s (n = (n 1, n 2 ), n = 1), conservative: H g (u, w, n) = H g (w, u, n) locally Lipschitz-continuous

Viscous form a h (IIPG) a h (w, ϕ) = Γ F I ht Γ F D ht Γ Γ K T ht 2 s=1 2 s=1 K 2 s=1 R s (w, w) ϕ x s dx R s (w, w) (n Γ ) s [ϕ] ds R s (w, w)(n Γ ) s ϕ ds Interior and boundary penalty J h (w, ϕ) = + Γ F D ht Γ Γ F I ht Γ σw ϕ ds σ Γ = C W µ/d(γ) σ[w] [ϕ] ds

The form d h d h (w, ϕ) = K T ht K (w ϕ h )divz dx Right-hand side form l h l h (w, ϕ) = Γ F D ht Γ 2 s=1 σw B ϕ ds. The state w B defined on the basis of the Dirichlet BC s and extrapolation: w B = (ρ D, ρ D v D1, ρ D v D2, c v ρ D θ D + 1 2 ρ D v D 2 ) on Γ I, w B = w (L) Γ on Γ O, w B = (ρ (L) Γ, ρ(l) Γ z 1, ρ (L) Γ z 2, c v ρ (L) Γ θ(l) Γ + 1 2 ρ(l) Γ z 2 ) on Γ Wt

The approximate solution is defined as w h (t) S ht such that K T ht K D A w h (t) Dt ϕ h dx +b h (w h (t), ϕ h ) + a h (w h (t), ϕ h ) +J h (w h (t), ϕ h ) + d h (w h (t), ϕ h ) = l h (w h (t), ϕ h ) holds for all ϕ h S ht, all t (0, T) and w h (0) = w 0 h =approximation of the initial state w0

Time discretization partition 0 = t 0 < t 1 < t 2... of the time interval [0, T] τ k = t k+1 t k - time step z(t n ) z n, w h (t n ) w n h S ht n defined in Ω htn - causes problems in transition t k t k+1 introduce the function ŵ k h = wk h A t k A 1 t k+1 - defined in the domain Ω htk+1

Approximation of the ALE derivative at time t k+1 by the finite difference D A w h (x, t k+1 ) = w h Dt t (X, t k+1) (X) w k h (X) wk+1 h τ k = wk+1 h (x) ŵ k h (x), x = A tk+1 (X) Ω htk+1. τ k Notation: (, ) - scalar product in L 2 (Ω htk+1 )

Possible full discretization: (a) w k+1 h S htk+1, (b) wk+1 h τ k ŵ k h, ϕ h +b h (w k+1 h, ϕ h ) + a h (w k+1 h, ϕ h ) +J h (w k+1 h, ϕ h ) + d h ( w k+1 h, ϕ h ) ϕ h S htk+1, k = 0,1,.... -strongly nonlinear algebraic system!!! = l h (w k+1 h, ϕ h )

Semi-implicit linearized scheme Linearization of the form b h b h (w, ϕ h ) = + Γ Fht I + Γ F B ht K T ht On the basis of the relation K 2 s=1 g s (w) ϕ h x s dx Γ H g(w (L) Γ, w(r) Γ, n Γ) [ϕ h ] Γ ds, Γ H g(w (L) Γ, w(r) Γ, n Γ) ϕ (L) hγ ds g s (w k+1 h ) = (A s (w k+1 h ) zs k+1 I)w k+1 h (A s (ŵ k h ) zk+1 s I)w k+1 h

we linearize the first term of b h (.,.): K T htk+1 K K T htk+1 2 s=1 K g s (w k+1 h ) ϕ h x s 2 s=1 (A s (ŵ k h ) zk+1 s I)w k+1 h ) ϕ h x s dx. The second term of b h (.,.) is linearized with the aid of the Vijayasundaram numerical flux: H g (w k+1(l) hγ, w k+1(r) hγ ), n Γ ) P + g ( ŵ k h Γ, n Γ )w k+1(l) hγ where P + g and P g + P g ( ŵ k h Γ, n Γ )w k+1(r) hγ, are positive and negative parts of the matrix P g (w, n Γ ) = 2 s=1 (A s (w) zs k+1 I)(n Γ ) s

= = b h (ŵ k h, wk+1 h, ϕ h ) K T htk+1 + Γ Fht I k+1 ( +P g + Γ F B ht k+1 +P g 2 K s=1 { P + Γ g ŵ k h Γ ( ŵ k h { (A s (ŵ k (x)) z k+1 Γ, n Γ P + g ( ŵ k ) h, n Γ w k+1(l) Γ hγ ( Γ, n Γ ) w k+1(r) hγ ŵ k h ) Γ, n Γ } ) w k+1(r) } hγ s (x))i)w k+1 (x)) ϕ h (x) dx, x s [ϕ h ] Γ ds w k+1(l) hγ ϕ hγ ds

Linearization of the form a h based on the fact that R s (w, w) is nonlinear in w but linear in w = a h (w k+1, ϕ) â h (ŵ k, w k+1, ϕ) := K T htk+1 K 2 s=1 R s (ŵ k, w k+1 ) ϕ x s dx Γ Fht I k+1 Γ 2 s=1 R s (ŵ k, w k+1 ) (n Γ ) s [ϕ] ds Γ Fht D k+1 Γ 2 s=1 R s (ŵ k, w k+1 )(n Γ ) s ϕ ds

= Semi-implicit discrete problem: (a) w k+1 h S htk+1, (b) wk+1 h τ k ŵ k h, ϕ h +ˆb h (ŵ k h, wk+1 h, ϕ h ) + â h (ŵ k h, wk+1 h, ϕ h ) +J h (w k+1 ( h, ϕ h ) + d h w k+1 ) h, ϕ h = l(w k B, ϕ) ϕ h S htk+1, k = 0,1,.... linear with respect to w k+1 h

Remarks: Time discretization is of order 1. Possibility to construct higher order time discretizations using BDF formula and extrapolation in nonlinear terms Discrete problem is equivalent on each time level to a linear algebraic system - solved by UMF- PACK (direct solver) or GMRES with a block diagonal preconditioning. Scheme for the solution of inviscid flow: µ = λ = κ = 0.

Further important ingredients - Realization of boundary conditions in the form ˆb h, i.e. the determination of the state w k+1(r) hγ if Γ Ω ht - with the aid of a linearized local initialboundary value Riemann problem - Use of isoparametric elements at a curved boundary - Avoiding the Gibbs phenomenon in high-speed flow = spurious overshoots and undershoots at discontinuities in the numerical solution - with the aid of a discontinuity indicator (M.F., V. Dolejší, C. Schwab, 2003) and local artificial viscosity (M.F., V. Kučera, 2007):

a)define the discontinuity indicator g k (i) proposed by M.F., Dolejší and Schwab: Math. Comput. Simul. (2003): g k (K) = K [ˆρk h ]2 ds/(h K K 3/4 ), K T htk+1. (74) b)define the discrete indicator G k (K) = 0 if g k (K) < 1, G k (K) = 1 if g k (K) 1, K T htk+1. (75) c)to the left-hand side of of the scheme we add the artificial viscosity form β h (ŵ k h, wk+1 h, ϕ) = ν 1 h K G k (K) K T htk+1 K wk+1 h ϕ dx

d)augment the left-hand side of the scheme by adding the form J h (ŵ k h, wk+1 h, ϕ) 1 = ν 2 Γ Fht I 2 (Gk (K (L) Γ ) + Gk (K (R) Γ ) k+1 Γ [wk+1 h ] [ϕ]ds,

Resulting scheme: (a) w k+1 h S htk+1, (b) wk+1 h τ k ŵ k h, ϕ h +ˆb h (ŵ k h, wk+1 h, ϕ h ) + â h (ŵ k h, wk+1 h, ϕ h ) +J h (w k+1 ( h, ϕ h ) + d h w k+1 ) h, ϕ h +β h (ŵ k h, wk+1 h, ϕ h ) + J h (ŵ k h, wk+1 h, ϕ h ) = l(w k B, ϕ) ϕ h S htk+1, k = 0,1,.... This method successfully overcomes problems with the Gibbs phenomenon in the context of the semiimplicit scheme.

Important: G k (i) vanishes in regions where the solution is regular. = The scheme does not produce any nonphysical entropy in these regions. Example Burgers equation u t + u u x 1 + u u x 2 = 0 in Ω (0, T), (76) where Ω = ( 1,1) ( 1,1), equipped with initial

condition u 0 (x 1, x 1 ) = 0.25+0.5sin(π(x 1 +x 2 )), (x 1, x 2 ) Ω, (77) and periodic boundary conditions Exact solution at t = 0.45

1 0.5 0-0.5-1 -1-0.5 0 0.5 1 Triangulation used for the numerical solution

Numerical solution computed by DGFEM, at t = 0.45

Numerical solution computed by DGFEM with limiting, at t = 0.45

Examples: quadratic elements Flow in fixed domains 1) Inviscid flow a) Low Mach number flow at incompressible limit Stationary flow past a Joukowski profile constant far field quantities = the flow is irrotational and homoentropic complex function method: exact solution of incompressible inviscid irrotational flow satisfying the Kutta Joukowski trailing condition, provided

the velocity circulation around the profile, related to the magnitude of the far field velocity, γ ref = 0.7158 Compressible flow: M = 10 4, #T h = 5418 The maximum density variation in compressible flow ρ max ρ min = 1.04 10 8. Computed velocity circulation related to the magnitude of the far field velocity: γ refcomp = 0.7205, = the relative error 0.66%

Compressible low Mach flow past a Joukowski profile, approximate solution, streamlines

1.4 1.2 1 0.8 0.6 0.4 0.2 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Velocity distribution along the profile: exact solution of incompressible flow, approximate solution of compressible low Mach flow

b) Transonic and hypersonic flow with shock waves past the Joukowski profile with far field Mach number M = 0.8 and M = 2.0, respectively The maximum density variation: ρmax ρmin = 0.94 for M = 0.8 and ρmax ρmin = 2.61 for M = 2.0

Mach number isolines of the flow past a Joukowski profile with M = 0.8 (left) and M = 2.0 (right)

Entropy isolines of the flow past a Joukowski profile with M = 0.8 (left) and M = 2.0 (right)

2) Viscous compressible flow a) Stationary viscous flow past NACA0012 profile θ = 0 IIPG far-field Mach number M = 0.5 angle of attack α = 2 Reynolds number Re = 5000 NACA0012 α = 2 viscous flow, Mach number

isolines (left), pressure isolines (right) entropy isolines.

b) Non-stationary viscous flow past NACA0012 profile far-field flow has Mach number M = 0.5 angle of attack α = 25 Reynolds number Re = 5000 possible to observe an unsteady vortex shedding from the airfoil figures illustrate the flow situation at time t = 8.5

NACA0012 α = 25 viscous flow, Mach number isolines (left), streamlines (right) NACA0012 α = 25 viscous flow, entropy isolines

c) Hypersonic viscous flow Flow past NACA0012 profile: Far field Mach number M = 2, α = 10 Reynolds number = 1000-1 0 1 2 3 4

Mesh for viscous flow - constructed by ANGENER - V. Dolejší -1 0 1 2 3 4 Mach number isolines for viscous flow

Distribution of the Mach number for viscous flow

Flow in time-dependent domains 1) Compressible inviscid flow in a channel with the initial rectangular shape Ω 0 = ( 2,2) (0,1), where the lower wall of the channel is moving in the interval X 1 ( 1,1): 0.45sin(0.4t)(cos(πX 1 ) + 1), X 1 ( 1,1). (78) This movement is interpolated to the whole domain resulting in the ALE mapping A t. Inlet Mach number= 0.12

Figure 1: velocity isolines at different time instants during one period The solution contains a vortex formation, when the lower wall starts to descend, convected through the domain. Moreover, we see that a contact discontinuity is developed, when the channel becomes narrow.

2 2 2 2 2 2 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 2 1 0 1 2 0.8 0.8 0.4 0.4 0 0 0.4 0.4 0.8 2 1 0 1 2 0.8 0.8 0.8 0.8 0.4 0.4 0.4 0 0 0 0.4 0.4 0.4 0.8 0.8 0.8 2 1 0 1 2 0.8 0.8 0.4 0.4 0 0 0.4 0.4 0.8 2 1 0 1 2 0.8 0.8 0.8 0.4 0.4 0 0 0.4 0.4 0.8 2 1 0 1 2 0.8 0.8 0.8 0.4 0.4 0 0 0.4 0.4 0.8 2 1 0 1 2 0.8 0.8 0.8 0.4 0.4 0 0

2) Compressible viscous flow in a channel with the initial rectangular shape Ω 0 = ( 5,5) (0,1), where the lower wall of the channel is moving in the interval X 1 ( 1,1): 0.3sin 2 (0.04(t 250.5))(cos(πX 1 )+1), X 1 ( 1,1). Re = 6976.74, inlet Mach number= 0.12. t = 10.08s

3) Compressible flow past a moving airfoil

Conclusion DGFEM = a promising robust method for the solution of compressible flow combination with ALE method allows the solution of flow problems in time dependent domains Further goals: coupling with structural models applications to complex FSI problems theoretical analysis Thank you for your attention

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compressible flow with wide range of Mach numbers. Computing and Visualization in Science, 10 (2007), 17-27. V. Dolejší, M. Feistauer, J. Hozman: Analysis of semi-implicit DGFEM for nonlinear convectiondiffusion problems on nonconforming meshes. Comput. Methods Appl. Mech. Engrg. 196 (2007) 2813-2827 (ISSN 0045 7825). V. Sobotíková, M. Feistauer: On the effect of numerical integration in the DGFEM for nonlinear convection-diffusion problems. Numerical

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M. Feistauer, V. Kučera: A new technique for the numerical solution of the compressible Euler equations with arbitrary Mach numbers. In: Hyperbolic Problems: Theory, Numerics, Applications, Proceedings of the 11th International Conference on Hyperbolic Problems (HYP2006) held in Lyon, S. Benzoni-Gavage, D. Serre, eds., Springer, Berlin, 2008, 523-532 (ISBN 978-3- 540-75711-5). V. Dolejší, M. Feistauer, V. Kučera, V. Sobotíková: An optimal L (L 2 )-error estimate of the discon-

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dent domains by the DG method. In: Numerical Analysis and Applied Mathematics (ICNAAM 2008), T.E. Simos, G. Psihoyios, Ch. Tsitouras, eds., American Institute of Physics, Melville, New York 2008, AIP Conference Proceedings 1048, 851-854, ISBN 978-0-7354-0576-9, ISSN 0094-243X. V. Kučera, M. Feistauer, J. Prokopová: Discontinuous Galerkin solution of compressible flow in time-dependent domains. Mathematics and Computers in Simulations (to appear).