Fully Discrete Energy Stable High Order Finite Difference Methods for Hyperbolic Problems in Deforming Domains

Transcription

1 Department of Mathematics Full Discrete Energ Stable High Order Finite Difference Methods for Hperbolic Problems in Deforming Domains Samira Nikkar and Jan Nordström LiTH-MAT-R--/--SE

2 Department of Mathematics Linköping Universit S-8 8 Linköping, Sweden.

3 Full Discrete Energ Stable High Order Finite Difference Methods for Hperbolic Problems in Deforming Domains Samira Nikkar a, Jan Nordström b a Department of Mathematics, Computational Mathematics, Linköping Universit, SE-8 8 Linköping, Sweden (samira.nikkar@liu.se). b Department of Mathematics, Computational Mathematics, Linköping Universit, SE-8 8 Linköping, Sweden (jan.nordstrom@liu.se). Abstract A time-dependent coordinate transformation of a constant coefficient hperbolic sstem of equations which results in a variable coefficient sstem of equations is considered. B appling the energ method, well-posed boundar conditions for the continuous problem are derived. Summation-b-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundar and initial conditions using Simultaneousl Approimation Terms (SATs) lead to a provable full-discrete energ-stable conservative finite difference scheme. We show how to construct a time-dependent SAT formulation that automaticall imposes boundar conditions, when and where the are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automaticall b using SBP-SAT in time and space. The developed technique is illustrated b considering an application using the linearized Euler equations: the sound generated b moving boundaries. Numerical calculations corroborate the stabilit and accurac of the new full discrete approimations. Kewords: deforming domain, initial boundar value problems, high order accurac, well-posed boundar conditions, summation-b-parts operators, stabilit, convergence, conservation, numerical geometric conservation law, Euler equation, sound propagation Preprint submitted to Journal of Computational Phsics Januar 6,

4 . Introduction High order SBP operators together with weak implementation of boundar conditions b SATs, can efficientl and reliabl handle large problems on structured grids for reasonabl smooth geometries [,,,,, 6, 7]. The main reason to use weak boundar procedures together with SBP operators and the energ method is the fact that with this combination, provable stable schemes can be constructed. For comprehensive reviews of the SBP-SAT schemes, see [8, 9]. The developments described above have so far dealt mostl with stead problems while computing flow-fields around moving and deforming objects involves time-dependent meshes [,, ]. We have previousl treated the problems with stead coordinate transformations [,, 6]. In this paper we take the net step, which is the treatment of time-dependent transformations in combination with SBP-SAT schemes. To guarantee stabilit of the full discrete approimation we emplo the recentl developed SBP-SAT technique in time [, ]. The hperbolic constant coefficient sstem that we consider, represents wave propagation problems governed b for eample the elastic wave equation [, 6], Mawell s equations [6, 7, ] and the linearized Euler equations [8, 9, ]. The rest of this paper proceeds as follows. In section, we analze the continuous problem which undergoes a transformation from a deforming domain into a fied domain, and derive characteristic boundar conditions which lead to a strongl well-posed problem. Section deals with the discrete problem where we guarantee stabilit, conservation and the validit of the NGCL. In section, numerical eamples which corroborates the previous theoretical development and confirms the accurac and stabilit of the scheme are considered. An application where sound is generated and propagated b a moving boundar is also studied. Finall we draw conclusions in section.. The continuous problem Consider the following constant coefficient sstem, V t + (ÂV ) + ( ˆBV ) =, (, ) Φ(t), t [, T ], ()

5 where the spatial domain Φ is time-dependent. We assume for simplicit that the constant matrices  and ˆB are smmetric and of size l. If the original problem is not smmetric, we smmetrize it b the procedure in [8]. A time-dependent transformation from the Cartesian coordinates into curvilinear coordinates, which results in a fied spatial domain, is = (τ, ξ, η), = (τ, ξ, η), t = τ, ξ = ξ(t,, ), η = η(t,, ), τ = t. The chain-rule is emploed to interpret the sstem () in terms of the curvilinear coordinates as V τ + (ξ t I + ξ  + ξ ˆB)Vξ + (η t I + η  + η ˆB)Vη =, () where ξ, η, τ T. The Jacobian matri of the transformation is ξ ξ [J] = η η, () τ τ where (V ξ, V η, V τ ) T = [J](V, V, V t ) T. The relation between [J], and its inverse, which transforms the derivatives back to the Cartesian coordinates leads to the metric relations Jξ t = η τ τ η, Jξ = η, Jξ = η Jη t = ξ τ ξ τ, Jη = ξ, Jη = ξ, in which J = ξ η η ξ > is the determinant of [J]. B multipling () with J and using (), we replace the coefficients in terms of derivatives of the curvilinear coordinates. Equation () can be rewritten as (JV ) τ + [(Jξ t I + Jξ  + Jξ ˆB)V ]ξ + [(Jη t I + Jη  + Jη ˆB)V ]η = [J τ + (Jξ t ) ξ + (Jη t ) η ]V + [(Jξ ) ξ + (Jη ) η ]ÂV + [(Jξ ) ξ + (Jη ) η ] ˆBV, where I denotes the identit matri of size l. All non-singular coordinate transformations fulfill the Geometric Conservation Law (GCL) [,, ], which is summarized as J τ + (Jξ t ) ξ + (Jη t ) η =, (Jξ ) ξ + (Jη ) η =, (Jξ ) ξ + (Jη ) η =. () () (6) (7)

6 The right hand side of (6) is identicall zero, due to (7), which results in the conservative form of the sstem. The final problem in the presence of initial and boundar conditions that we will consider in this paper is (JV ) τ + (AV ) ξ + (BV ) η =, (ξ, η) Ω, τ [, T ], LV = g(τ, ξ, η), (ξ, η) δω, τ [, T ], V = f(ξ, η), (ξ, η) Ω, τ =, (8) where A = Jξ t I + Jξ Â + Jξ ˆB, B = Jηt I + Jη Â + Jη ˆB, (9) and Ω = [, ] [, ]. In (8), L is the boundar operator, g is the boundar data and f is the initial data... Well-posedness The energ method (multipl with the transpose of the solution and integrate over the domain Ω and time-interval [, T ]) applied to (8) leads to T Ω [V T (JV ) τ + V T (AV ) ξ + V T (BV ) η ] dξ dη dτ =. () B adding and subtracting Vτ T JV +Vξ T T AV +Vη BV to the integral argument in (), we get T Ω T Ω [(V T JV ) τ + (V T AV ) ξ + (V T BV ) η ] dξ dη dτ = [(V T τ JV ) + (V T ξ AV ) + (V T η BV )] dξ dη dτ. () However, the right hand side of () is zero, since the matrices J, A and B are smmetric, and V solves equation JV τ + AV ξ + BV η =. The latter can be seen b multipling () with J and using (9). Integration of (), and the use of Green-Gauss theorem, ields V (T, ξ, η) J = f(ξ, η) J T δω V T [(A, B) n] V ds dτ, ()

7 where the norm is defined b V J = Ω V T J V dξ dη. In (), n = (n, n ) is the unit normal vector pointing outward from Ω, (A, B) n = n A + n B and ds is an infinitesimal element along the boundar of Ω. In order to bound the energ of the solution, boundar conditions must be applied when the matri C = (A, B) n is negative definite. We decompose C = XΛ C X T = XΛ + C XT + XΛ C XT = C + + C where Λ + C and Λ C are diagonal matrices with positive and negative eigenvalues of C, respectivel, on the main diagonal. The energ of the solution is now bounded b data if we impose the characteristic boundar conditions (X T V ) i =(X T V ) i, (Λ C ) ii <, () where the vector V is the solution at the boundar δω. The continuous energ, using () becomes T V (T, ξ, η) J = f(ξ, η) J δω V T C V ds dτ T δω V T C + V ds dτ. () The estimate () guarantees uniqueness of the solution and eistence is given b the fact that we use the correct number of boundar conditions. Hence we can summarize the results obtained so far in the following proposition. Proposition. The continuous problem (8) with the boundar condition in () is strongl well-posed and has the bound (). Remark. The problem (6) is called strongl well-posed since we have an estimate of the solution also for non-zero boundar data. For more details on well-posedness see []. As an eample, assume that we onl need boundar conditions at the south boundar, see Figure, indicated b subscript s, then C s = (A, B) s (, )= B s = X s Λ Bs X T s, and () becomes V (T, ξ, η) J = f(ξ, η) J + T V T B s + V dξ dτ + T V T s B s V s dξ dτ. ()

8 η c d b d' c' Ω n a a' b' ξ ab bc cd da a'b' : South (s) b'c' : East (e) c'd' : North (n) d'a' : West (w) Figure : A schematic of the moving and fied domains and boundar definitions.. The discrete problem The spatial computational domain Ω is a square in ξ, η coordinates, see Figure, and discretized using N and M nodes in the direction of ξ and η respectivel. In time we use L time levels from to T. The full-discrete numerical solution is a column vector of size llmn organized as follows V = V. [V k ]. V L ; [V k ]= V. [V i ]. V N ; [V i ] k = k V. [V j ]. V M ; [V j ] ki = ki v v. v l = V kij, (6) kij where V kij = [v, v,, v l ] T kij approimates V (τ k, ξ i, η j ). The first derivative u ξ is approimated b D ξ u, where D ξ is a so-called 6

9 SBP operator of the form D ξ = P ξ Q ξ, (7) and u = [u, u,, u N ] T is the solution evaluated in each grid point. P ξ is a smmetric positive definite matri, and Q is an almost skew-smmetric matri that satisfies Q ξ + Q T ξ = E E =B = diag(,,...,, ). (8) In (8), E =diag(,,..., ) and E =diag(,...,, ). The η and τ directions are discretized in the same wa. A first derivative SBP operator is a s-order accurate central difference operator which is modified close to the boundaries such that it becomes one-sided. Together with a diagonal norm P, the boundar closure is s- order accurate, making a stable first order approimation s+ order accurate globall [, ]. For more non-standard SBP operators see [6, 7, 8, 9]. A finite difference approimation including the time discretization [], on SBP-SAT form, is constructed b etending the one-dimensional SBP operators in a tensor product fashion as D τ = Pτ Q τ I ξ I η I, D ξ = I τ P ξ Q ξ I η I, D η = I τ I ξ Pη Q η I (9) where represents the Kronecker product []. All matrices in the first position are of size L L, the second position N N, the third position M M and the fourth position l l. I denotes the identit matri with a size consistent with its position in the Kronecker product. The Kronecker product is bilinear and associative. For square matrices the following rules eist (A B)(C D)=(AC BD), (A B) =A B, (A B) T =A T B T. () For later reference we need Lemma. The difference operators in (9) commute. Proof. The properties () of the Kronecker product lead to 7

10 D τ D ξ = (Pτ Q τ I ξ I η I)(I τ P ξ Q ξ I η I) = Pτ Q τ P ξ Q ξ I η I = (I τ P ξ Q ξ I η I)(Pτ Q τ I ξ I η I) = D ξ D τ. The proof is analogous for the other coordinate combinations. To obtain an energ estimate similar to the continuous one, we use the splitting technique described in []. We split the equation in (8) as [(JV ) τ +JV τ +J τ V ]+ [(AV ) ξ +AV ξ +A ξ V ]+ [(BV ) η +BV η +B η V ]=. () The SBP-SAT approimation of () including the penalt terms for the boundar procedure (we onl consider the south boundar), and a weak initial condition, is constructed as [D τ(jv) + JD τ V + J τ V] + [D ξ(av) + AD ξ V + A ξ V]+ [D η(bv) + BD η V + B η V] = P i Σ i (V f) + P s Σ s X T s [V V ]. () In (), J and J τ are diagonal matrices approimating J and J τ values pointwise. Moreover, A, B, A ξ and B η are block-diagonal matrices approimating A, B, A ξ and B η pointwise respectivel, i.e. (A ξ )... A ξ = (A ξ ) k... (A ξ )... (A ξ ) k = (A ξ ) i... (A ξ )... (A ξ ) ik = (A ξ ) j... ; (A ξ ) L ; (A ξ ) N k ; (A ξ ) M ki () 8

11 Note that in (), (A ξ ) kij A ξ (τ k, ξ i, η j ). () (A ξ ) kij = [(Jξ t ) ξ + (Jξ ) ξ Â + (Jξ ) ξ ˆB]kij (B η ) kij = [(Jη t ) η + (Jη ) η Â + (Jη ) η ˆB]kij, () where (Jξ t ) ξ, (Jξ ) ξ, (Jξ ) ξ, (Jη t ) η, (Jη ) η and (Jη ) η approimate (Jξ t ) ξ, (Jξ ) ξ, (Jξ ) ξ, (Jη t ) η, (Jη ) η and (Jη ) η pointwise respectivel. In (), the variables with a bold face correspond to the ones with regular face in the continuous problem. This notation is emploed to be able to use similar names for the variables that are inherentl (not eactl) the same in the continuous and the discrete problem, regardless of the structure of the variables. Moreover, Σ i and Σ s are the penalt matrices corresponding to the weak initial condition and the south boundar procedure. Furthermore P i = Pτ E I ξ I η I, P s = I τ I ξ Pη E I, and X s = (I τ I ξ E X). All the numerical matrices defined so far are of size llmn llmn. V is a zero vector of the same size as V ecept at the position η = where the zeros are replaced with the boundar data. Moreover f is a zero vector, of same size as V, ecept at the position τ = where the initial data (compatible with the reference solution at the boundaries) is imposed... Stabilit The energ method (multipling from the left with V T (P τ P ξ P η I)) is applied to (), the properties () are emploed and the equation is added to its transpose. The result is V T ( B τ J+ B ξ A+ B η B)V + V T P (Jτ +A ξ +B η )V = V T (E P ξ P η I)Σ i (V f) + (V f) T Σ T i (E P ξ P η I)V + V T (P τ P ξ E I)Σ s X T s [V V ] + [V V ] T X s Σ T s (P τ P ξ E I)V. (6) where P = (P τ P ξ P η I), Bτ = [(Q + Q T ) τ P ξ P η I], Bξ = [P τ (Q + Q T ) ξ P η I], and B η = [P τ P ξ (Q + Q T ) η I]. We have 9

12 used that the diagonal matrices B τ, B ξ and B η commute with the smmetric matrices J, A and B respectivel. We will need Lemma. The NGCL: J τ + A ξ + B η =, holds. Proof. Consider the following definitions, J τ = diag[d τ (D η M () D ξ M () )] (Jξ t ) ξ = diag[d ξ (D τ M () D η M () )] (Jη t ) η = diag[d η (D ξ M () D τ M () )] (Jξ ) ξ = diag[d ξ (D η )] (Jξ ) ξ = diag[ D ξ (D η )] (Jη ) η = diag[ D η (D ξ )] (Jη ) η = diag[d η (D ξ )] (7) in which and are the discrete Cartesian coordinates in Φ. Also M () = diag()(d ξ ), M () = diag()(d η ) and M () = diag()(d τ ). We evaluate the term J τ + A ξ + B η, and substitute A ξ and B η with the definitions () and (), which results in [ J τ +A ξ +B η = J τ + (Jξ t ) ξ + (Jξ ) ξ + (Jη ) η ] [ (8) + (Jη t ) η + (Jξ ) ξ + (Jη ) η ]ˆB, where  = I τ I ξ I η Â, ˆB = I τ I ξ I η ˆB. Now we insert (7) into (8) and obtain J τ +A ξ +B η = diag[d τ (D η M () D ξ M () )] + diag[d ξ (D τ M () D η M () )] + diag[d ξ (D η ) D η (D ξ )] + diag[d η (D ξ M () D τ M () )] + diag[d η (D ξ ) D ξ (D η )]ˆB. (9) B Lemma we find that the right hand side of (9) is zero. Remark. The NGCL as a consequence of commuting operators is previousl reported in []. On the left hand side of the equalit in (6), we keep the terms corresponding to the initial time, final time and the south boundar and ignore

13 the other boundar terms. B also using Lemma we get V T J(E L P ξ P η I)V = V T (E P ξ P η I)(J + Σ i )V f T (E P ξ P η I)Σ i V V T (E P ξ P η I)Σ i f + V T (P τ P ξ E I)(B s + Σ s X T s + X s Σ T s )V V T (P τ P ξ E I)Σ s X T s (V ) s (V ) T s X s Σ T s (P τ P ξ E I)V. () In (), B s = (I τ I ξ E I)B, and E, E L are zero matrices ecept at the one entr corresponding to the initial and final time, respectivel. We can prove Proposition. The discrete problem () is stable if holds. J + Σ i, Σ s X T s + X s Σ T s + B s () Proof. With zero boundar and initial data the solution at the final time is clearl bounded. A particularl nice result is obtained with Σ i = J. Let Σ s = X s Σs, where X s = (I τ I ξ E I)X, X is a block diagonal matri of X and Σ s is diagonal. B inserting B s = X s Λ B X T s, the second condition in () becomes X s ( Σ s + Λ Bs )X T s, and is fulfilled if Σ s is defined such that the following relations hold, ( Σ s ) ii (Λ Bs ) ii if (Λ Bs ) ii > () ( Σ s ) ii = if (Λ Bs ) ii. A time-dependent penalt matri that automaticall adjusts for stabilit according to () is given b The final numerical energ estimate becomes Σ s = (Λ Bs + Λ Bs )/. () V L J(E L P ξ P η I) = f J(E P ξ P η I) + (V ) s B + s (P τ P ξ E I) +V s T (P τ P ξ E I)B s V s V f J(E P ξ P η I) V s (V ) s B + s (P τ P ξ E I). ()

14 In () the definition Z H = ZT HZ is used for the norms, where Z is a vector and H is a positive definite matri. As an eample V L J(E L P ξ P η I) = V T L J(E L P ξ P η I)V L in which J(E L P ξ P η I) is positive definite. Moreover, V L denotes the numerical solution restricted to the last time level. We have proved Proposition. The discrete problem () is strongl stable and satisfies () if Σ i = J, Σ s = X s Σs and Σ s = (Λ s + Λ s )/ holds. Remark. Equation () shows that the numerical energ estimate is similar to that of the continuous one shown in () with etra damping terms coming from the weak impositions of initial and boundar conditions. Remark. The problem () is strongl stable since the estimate () contains the boundar data, see [] for more details on stabilit definitions... Conservation and weak form of the governing equations In a conservation law, the total quantit of a conserved variable in an region changes onl as a result of the flu through the boundaries of the region. To show that the solution is conserved, we integrate (8) in space and time and obtain T (JV ) T dξ dη + (F, G) n dτ =, () Ω in which F = AV and G = BV. In this paper, we will however, rel on a broader definition of conservation motivated b the original proof of the La-Wendroff theorem []. We demand that all moments of the flu against an arbitrar test function telescope across the domain. This additional constraint demands an equivalence between the weak forms of the continuous and discrete operators []. We multipl (8) with an arbitrar test function φ(τ, ξ, η) H that vanishes on the temporal and spatial boundaries followed b integration with respect to time and space as T Ω φ T [(JV ) τ + (AV ) ξ + (BV ) η ] dξ dη dτ =. (6)

15 Integration b parts and the fact that J, A and B are smmetric lead to T V T [Jφ τ + Aφ ξ + Bφ η ] dξ dη dτ =. (7) Ω B adding and subtracting the term V T (J τ φ + A ξ φ + B η φ) to the integral argument in (7) we obtain T V T [(Jφ) τ + (Aφ) ξ + (Bφ) η ] dξ dη dτ = V T [(J τ + A ξ + B η )φ] dξ dη dτ. Ω Ω }{{} T :=RHS (8) In (8), the GCL leads to RHS =. Finall we split the integral argument on the left hand side and obtain T Ω V T [(Jφ) τ +Jφ τ +J τ φ+(aφ) ξ +Aφ ξ +A ξ φ+(bφ) η +Bφ η +B η φ]dξ dη dτ =. Equations (7) and (9) are different weak forms of (8) including the broader definition of conservation mentioned above. The form (9) is of specific interest, since we use a scheme based on that form of splitting. The discrete conservation is shown b multipling the scheme in () from the left with φ T (P τ P ξ P η I). The Kronecker rules (), together with the SBP properties (8) leads to [φt ( Q T τ P ξ P η I)JV + φ T J( Q T τ P ξ P η I)V+ φ T P Jτ V] + [φt (P τ Q T ξ P η I)AV+φ T A(P τ Q T ξ P η I)V+ φ T P Aξ V]+ [φt (P τ P ξ Q T η I)BV+φ T B(P τ P ξ Q T η I)V+φ T P Bτ V] =, () in which all the penalt terms are ignored. Equation () now becomes (9) [(D τφ) T J P V + (D τ Jφ) T P V + φ T J τ P V]+ [(D ξφ) T A P V + (D ξ Aφ) T P V + φ T A ξ P V]+ () [(D ηφ) T B P V + (D η Bφ) T P V + φ T B η P V] =. B considering (), and the fact that P commutes with J, A, B, J τ, A ξ, B η we obtain VT P [Dτ Jφ+(JD τ φ+j τ φ)+d ξ Aφ+(AD ξ φ+a ξ φ)+d η Bφ+(BD η φ+b η φ)]=. ()

16 The relation () mimics the weak conservative formulation (9) perfectl. We have proved Proposition. The discrete problem () is conservative.. Numerical eperiments We consider the two-dimensional constant coefficient smmetrized Euler equations in a deforming domain described b (), where c V =[ ρ, u, v, γ ρ c γ(γ ) T ]T. () In (), ρ, u, v, T and γ are respectivel the densit, the velocit components in and directions, the temperature and the ratio of specific heats. An equation of state in form of γp= ρt + ρ T closes the sstem (), in which the bar sign denotes the state around which we have linearized. Moreover the matrices in () are ū c/ γ c/ v c/ γ γ ū γ Â= γ c ū, v ˆB = c/ γ v γ γ c, () γ γ c ū γ γ c v where γ =., c = and ρ =. These parameter values are used throughout the section while ū and v are changed. is chosen to be a portion of a ring-shaped geometr where the boundaries are moving while alwas coinciding with a coordinate line in the corresponding polar coordinate sstem, see Figure. We transform the deforming domain from Cartesian coordinates,,, into polar coordinates, r, φ, and scale the polar coordinates such that the fied domain in ξ, η coordinates becomes a square, see Figure, as r(,, t) = (t) + (t), φ(,, t) = tan ( (t) ) (t) ξ(,, t) = r(,,t) r (t), η(,, t) = φ(,,t) φ (t). () r (t) r (t) φ (t) φ (t) Note that the scaling depends on the movements of the boundaries of the deforming domain, and is defined b arbitrar functions of time through r (t), r (t), φ (t) and φ (t).

17 r η c φ φ r d r b d' c' a φ a' b' ξ west: ad a d east: bc b c south: ab a b north: dc d c Figure : A schematic of the Cartesian-polar transformation in the phsical domain, the computational domain, and definition of r, r, φ and φ... Time-dependent automatic SAT formulations To show that the scheme automaticall imposes the correct number of boundar conditions, we move the boundaries b r (t) =, φ (t) = π 8 + π 8 sin(πt π ) r (t) =, φ (t) = π, (6) and construct the matrices  and ˆB b choosing (ū, v) = (, ). The eigenvalue matri Λ C in (), is evaluated at the moving boundar as ω Λ C = ω ω R c, (7) ω + R c in which ω = Jη t + Jη ū + Jη v, R = (Jη ) + (Jη ). As in (), we construct Σ s such that the elements on the main diagonal are non-zero if and onl if the corresponding eigenvalues are negative. Depending on the signs of the eigenvalues we can have zero (super-sonic outflow), one (sub-sonic outflow), three (sub-sonic inflow) or four boundar conditions (super-sonic

18 .. τ =. τ =.7.. τ =.. τ =. τ =. Figure : A schematic of the deforming domain at five time levels. inflow). Note that in time-dependent domains, the relative normal velocit between boundar and fluid distinguishes between inflow, outflow, sub- and super-sonic states. A schematic of the deforming domain at five time levels is shown in Figure. In Figure, the number of boundar conditions imposed automaticall at the moving boundar is depicted. Initiall we have sub-sonic inflow and three boundar conditions are imposed (state a). Then the boundar encounters sub-sonic outflow which requires one boundar condition (state b). Net, no boundar conditions are imposed for the super-sonic outflow (state c). A sub-sonic outflow following b a sub-sonic inflow occurs afterwards which demand one and three boundar conditions respectivel. Finall, in the case of super-sonic inflow (state d), four boundar conditions are imposed, and this procedure is continued automaticall b the scheme. 6

19 The number of boundar conditions, ω and φ 8 6 d d d a a a a a a a 6 b b b b b c c c The nr. of b.c. s ω φ 8... τ Figure : The number of boundar conditions, the position of the moving boundar in terms of φ, and ω over time... Order of accurac We consider a deforming domain where the boundaries are moving b r (t) =. sin(πt), φ π (t) =. sin(πt) π r (t) = +. sin(πt), φ π (t) = π +. sin(πt). (8) π A schematic of the deforming and fied domains at different time steps is presented in Figure and Figure 6 respectivel (note that these schematics are for illustration purposes onl, the numerical eperiments are carried out on finer meshes). Here we use (ū, v) = (, ) and c = to construct the matrices  and ˆB. To verif the order of accurac of our method, we use the manufactured solution V V = [ sin( t), cos( t), sin( t), cos( t) ] T, (9) which is injected in () through a forcing function. The characteristic boundar conditions () are automaticall imposed b the scheme. 7

20 t τ Figure : A schematic of the deforming mesh at different times, order of accurac.... η. Figure 6: A schematic of the fied mesh at different times, order of accurac.. ξ.. The convergence rate is defined as p = log () V V () J P V () V () J P log (NM)() (NM) (). () where superscripts () and () denote two mesh levels with (N M) () and (N M) () grid points respectivel, also P = E L P τ P ξ P η I. In order to show the convergence rate in space we use the th order accurate SBP8 in time. The numerical solution in our eperiments converges to the eact solution at T= with the convergence rates presented in Tables, and. The convergence rates are in agreement with the theor mentioned above, and we conclude that our scheme works as it should. N, M 8 6 ρ u v p Table : Convergence rates at T=, for a sequence of mesh refinements, SBP in space, SBP8 in time (L=) 8

21 N, M 8 6 ρ u v p Table : Convergence rates at T=, for a sequence of mesh refinements, SBP in space, SBP8 in time (L=8) N, M 6 7 ρ u v p Table : Convergence rates at T=, for a sequence of mesh refinements, SBP6 in space, SBP8 in time (L=).. Free-stream preservation Consider the domain described in (8) with (ū, v) = (, ) when constructing the matrices  and ˆB. In order to show that the scheme in () preserves the state of a uniform flow, we use the manufactured solution V = [,,, ] T. () The characteristic boundar conditions () are automaticall imposed b the scheme. We use the th order accurate SBP8 in time (L=) and rd order accurate SBP in space (N=M=). The uniform flow (eamplified b the densit component) is preserved up to T= as presented in Figure 7. Moreover, the norm of the error (the difference between the numerical and manufactured solution) versus time is on machine precision level, see Figure 8... The sound propagation application We consider sound propagation in a deforming domain where the west boundar is moving and other boundaries are fied. A schematic of the deforming and fied domains at different time steps is presented in Figures 9 9

22 # - The ; component The u component The v component The T component. v (,, ).. Norm of the errors (). () Time Figure 8: The norm of the error for all solution components versus time Figure 7: The ρ component of the solution at T= and (note that these schematics are for illustration purposes onl, the numerical eperiments are carried out on finer meshes). The movements are defined b r (t) = + sin(πt)/(π), φ (t) = π/, () r (t) =, φ (t) = π/ τ t η.... ξ Figure 9: A schematic of the deforming mesh at different times, sound propagation. Figure : A schematic of the fied mesh at different times, sound propagation. We manufacture u and v such that the mean flow satisfies the solid wall no-penetration condition at the moving boundar b ( u, v ) = (τ, τ )/ ep(ξ). () Consider the eigenvalue matri, C = XΛX T evaluated at the west boundar, in which Λ = R diag (ˆ ω, ω ˆ, ω ˆ c, ω ˆ + c ), where ω ˆ = (Jξt + Jξ u b +

23 Jξ v b )/R and R = (Jξ ) + (Jξ ). The no-penetration condition for the mean flow results in ˆω =, which takes () to V (T, ξ, η) J = f(ξ, η) R T c(ṽ ṽ ) dη + BT. () In (), Ṽ = X T V = [ṽ, ṽ, ṽ, ṽ ] T, and BT is the contribution at the other boundaries. An boundar condition in form of ṽ = ±ṽ is well-posed. We choose ṽ + ṽ =, which is the no-penetration boundar condition. Also we impose characteristic boundar conditions with zero data at the other boundaries, and initialize the solution with zero data for densit and velocities, together with an initial pressure pulse centered at (.,.). The velocit field with tangential flow close to the solid wall, the pressure distribution and the rate of dilatation at five different time levels are presented in Figures -. We have used SBP in both space and time (N=M= and L=) to obtain these results. 6 The velocit field at t =. The velocit field at t = Figure : The global velocit field..... Figure : A blow-up of the velocit field. in Figures - illustrate the movements of the south boundar relative to its initial location. As seen in the Figures - the flow stas tangential to the moving solid boundar all the time, as it should for an Euler solution. The pressure pulse and the corresponding rate of dilatation move from left to right, and leave the domain as time passes.. Summar and conclusions We have considered a constant coefficient hperbolic sstem of equations in time-dependent curvilinear coordinates. The sstems is transformed into

24 6 The velocit field at t =. The velocit field at t = Figure : The global velocit field..... Figure : A blow-up of the velocit field. 6 The velocit field at t =.66.6 The velocit field at t = Figure : The global velocit field Figure 6: A blow-up of the velocit field. a fied coordinate frame, resulting in variable coefficient sstems. We show that the energ method applied to the transformed sstems together with time-dependent appropriate boundar conditions leads to strongl well-posed problem. The continuous energ estimate that we obtain provides the target for the numerical approimations. B using a special splitting technique, summation-b-parts operators in space and time, weak imposition of the boundar and initial conditions and the discrete energ method, a full-discrete strongl stable high order accurate and conservative numerical scheme is constructed. The full-discrete energ estimate is similar to the continuous one with small added damping terms. Furthermore, b the use of SBP operators in time, the Geometric Conservation Law is proven to hold numericall.

25 6 The velocit field at t =.88.8 The velocit field at t = Figure 7: The global velocit field..... Figure 8: A blow-up of the velocit field. 6 The velocit field at t =..8 The velocit field at t = Figure 9: The global velocit field..... Figure : A blow-up of the velocit field. We have tested the scheme for high order accurate SBP operators in space and time using the method of manufactured solution. It was shown that the scheme automaticall imposes the correct number of boundar conditions for the time-dependent domain. Numerical calculations corroborate the stabilit and accurac of the full-discrete approimations, and free-stream preservation was shown. Finall, as an application, sound propagation b the linearized Euler equations in a deforming domain was illustrated.

26 6 The pressure distribution at t =. 6 The rate of dilatation at t = Figure : The pressure distribution. Figure : The rate of dilatation. 6 The pressure distribution at t =. 6 The rate of dilatation at t = Figure : The pressure distribution. Figure : The rate of dilatation. 6 The pressure distribution at t =.66 6 The rate of dilatation at t = Figure : The pressure distribution. Figure 6: The rate of dilatation.

27 6 The pressure distribution at t =.88 6 The rate of dilatation at t = Figure 7: The pressure distribution. Figure 8: The rate of dilatation. 6 The pressure distribution at t =. 6 The rate of dilatation at t = Figure 9: The pressure distribution. Figure : The rate of dilatation.

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