Fully Discrete Energy Stable High Order Finite Difference Methods for Hyperbolic Problems in Deforming Domains: An Initial Investigation

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1 Full Discrete Energ Stable High Order Finite Difference Methods for Hperbolic Problems in Deforming Domains: An Initial Investigation Samira Nikkar and Jan Nordström Abstract A time-dependent coordinate transformation of a constant coefficient hperbolic sstem of equations is considered. We use the energ method to derive well-posed boundar conditions for the continuous problem. Summation-b-Parts (SBP) operators together with a weak imposition of the boundar and initial conditions using Simultaneousl Approimation Terms (SATs) guarantee energ-stabilit of the full discrete scheme. We construct a time-dependent SAT formulation that automaticall imposes the boundar conditions, and show that the numerical Geometric Conservation Law (GCL) holds. Numerical calculations corroborate the stabilit and accurac of the approimations. As an application we stud the sound propagation in a deforming domain using the linearized Euler equations. Introduction High order SBP-SAT schemes, can efficientl and reliabl handle large problems on structured grids for reasonabl smooth geometries [7, ]. The developements within this framework, have so far dealt mostl with stead meshes while computing flow-fields around moving and deforming objects involves time-dependent meshes [3, 3]. In this paper (and the full paper [5]) we treat the time-dependent transformations in a SBP-SAT framework. To guarantee stabilit of the full discrete approimation we emplo the recentl developed SBP-SAT technique in time [8, 4]. Samira Nikkar Linköping Universit, SE Linköping, samira.nikkar@liu.se Jan Nordström Linköping Universit, SE Linköping, jan.nordstrom@liu.se

2 Samira Nikkar and Jan Nordström The continuous problem The following hperbolic smmetrized constant coefficient sstem, V t + (ÂV ) + ( ˆBV ) =, (,) Φ(t), t [, T ], () can, with the use of the GCL [3], be rewritten as (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 =, () through a time-dependent transformation from the Cartesian coordinates into curvilinear coordinates as (τ,ξ,η) ξ (t,,), (τ,ξ,η) η(t,,), t = τ. (3) In (), A = Jξ t I + Jξ Â + Jξ ˆB, B = Jη t I + Jη Â + Jη ˆB, also Ω = [, ] [, ]. Moreover, L is the boundar operator, g is the boundar data, f is the initial data. and J = ξ η η ξ > is the determinant of the Jacobian of the transformation.. Well-posedness The energ method (multipl () with the transpose of the solution and integrate over the domain Ω and time-interval [, T ]) is applied to (), and the term Vτ T JV + Vξ T AV + V η T BV = is added to the integral argument. Then, integration together with the use of Green-Gauss theorem gives V (T,ξ,η) J = f (ξ,η) J T δω V T [(A,B) n] V ds dτ, (4) where the norm is defined b V J = Ω V T J V dξ dη. In (4), n is the unit normal pointing outward from the Ω, and ds is an infinitesimal element along the boundar, δω. 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 + X T + XΛC X T = C + +C where Λ C + and Λ C are diagonal matrices with positive and negative eigenvalues of C, respectivel. We choose the characteristic boundar conditions, in order to bound the energ of the solution as where V is the solution at δω. The continuous energ, using (5) is estimated as (X T V ) i =(X T V ) i, (Λ C ) ii <, (5)

3 Energ Stable High Order FDM for Hperbolic Problems in Deforming Domains 3 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. (6) The estimate (6) 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 () with the boundar condition in (5) is strongl well-posed and has the bound (6). 3 The discrete problem The spatial domain, Ω, is a square in ξ, η coordinates, and discretized using N and M nodes in ξ and η directions respectivel. In time we use L time levels from to T. The first derivative u ξ is approimated b D ξ u, where D ξ is a so-called SBP operator, see []. A multi-dimensional finite difference approimation (including the time discretization [8, 4]), on SBP-SAT form, is constructed b etending the one-dimensional SBP operators in a tensor product fashion as D τ =P τ Q τ I ξ I η I, D ξ =I τ P Q ξ I η I, D η =I τ I ξ Pη Q η I (7) ξ where represents the Kronecker product []. In (7), I denotes the identit matri with a size consistent with its position in the Kronecker product. In [5] it is shown that the operators in (7) commute. The SBP-SAT approimation of () including the penalt terms for the weak boundar conditions (we onl consider the boundar along which η =, namel the south boundar, denoted b subscript s), 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 (8) s [V V ], in which the bold face of the variables corresponds to the approimated values. Σ i and Σ s are the penalt matrices for 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). Also, the vectors V and f contain the boundar data at η = and initial data at τ = respectivel. Note that in (8), the splitting technique described in [6] is used prior to the discretizations, in order to get similar energ estimate as the one in the continuous case.

4 4 Samira Nikkar and Jan Nordström 3. Stabilit The energ method (multipling from the left with V T (P τ P ξ P η I)) is applied to (8) 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(J τ + 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, (9) where P = (P τ P ξ P η I), B τ = [(Q + Q T ) τ P ξ P η I], B ξ = [P τ (Q + Q T ) ξ P η I], and B η = [P τ P ξ (Q+Q T ) η I]. The following Lemma is proved in []. Lemma. The numerical GCL holds: J τ + A ξ + B η =. In (9), b using Lemma we get V T J(E L P ξ η I)V=V T (E P ξ η I)(J + Σ i )V f T (E P ξ η I)Σ i V V T (E P ξ η I)Σ i f + V T (P τ,ξ E I)(B s + Σ s X T s + X s Σ T s )V V T (P τξ E I)Σ s X T s (V ) s (V ) T s X s Σ T bs (P τξ E I)V, () where P ξ η = P ξ P η, P τξ = P τ P ξ, B s = (I τ I ξ E I η I)B, and E, E L are zero matrices ecept at the one entr corresponding to the initial and final time, respectivel. Proposition. The problem (8) is stable if 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. 4 Numerical eperiments We consider the two-dimensional linearized smmetrized Euler equations in a deforming domain described b (), where V =[ cρ/( γ ρ), u, v, T /( c γ(γ ))] T, and ρ,u,v,t and γ are respectivel the densit, the velocit components in and directions, the temperature and the ratio of specific heats [, 4]. An equation of state in form of γ p = ρt + ρ T closes the sstem (), in which the bar 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 The deforming domain 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

5 Energ Stable High Order FDM for Hperbolic Problems in Deforming Domains 5 corresponding polar coordinate sstem. We transform the deforming domain from Cartesian coordinates,,, into polar coordinates, r, φ, and scale the polar coordinates such that Ω = [, ] [, ], see Figure, as ξ (,,t) = r(,,t) r (t) r (t) r (t), η(,,t) = φ(,,t) φ (t) φ (t) φ (t). () "! )!! &!!!! )*! $! )+! '! $%! &%! (! "! #! Fig. A schematic of the Cartesian-polar transformation, and illustrations of r, r, φ and φ ; Also boundar definitions as west: ad a d, east: bc b c, south: ab a b, north: dc d c (%!! '%!!! 4. Order of accurac We move the boundaries b the transformation r (t) =. π sin(πt), φ (t) =.5 π sin(πt) r (t) = +. π sin(πt), φ (t) = π +.5 π sin(πt), (3) and construct the matrices  and ˆB for a state where ū =, v =, ρ =, γ =.4 and c =. To verif the order of accurac of our method, we use the method of manufactured solution [9], and impose the characteristic boundar conditions as derived in (5). The numerical solution for a scheme with SBP63 in space and SBP84 in time, converges to the eact solution at T= with the convergence rate presented in table. Moreover, the scheme is tested with SBP and SBP4 and the convergence rates are quantified as and 3 respectivel [5]. 4. The sound propagation application We consider a deforming domain where the west boundar is moving, see Figures and 3. Note that these schematics are for illustration purposes onl, the numerical eperiments are carried out on finer meshes. The movements are defined b

6 t 6 Samira Nikkar and Jan Nordström N, M ρ u v p Table Convergence rates at T=, for a sequence of mesh refinements, SBP63 in space, SBP84 in time (L=) r (t) = + sin(4πt)/(4π), φ (t) = π/4, r (t) = 5, φ (t) = 3π/4. (4) τ Fig. A schematic of the deforming mesh at different times, sound propagation η.5.5 Fig. 3 A schematic of the fied mesh at different times, sound propagation. ξ.5.5 We choose γ =.4, c =, ρ = and manufacture ū and v such that the mean flow satisfies the solid wall no-penetration condition at the moving boundar b ū = τ /ep(ξ ), v = τ /ep(ξ ). (5) Consider the eigenvalue matri, C = XΛX T at the west boundar, in which Λ = R diag ( ˆω, ˆω, ˆω c, ˆω + c), where ˆω = (Jξ t + Jξ ū b + Jξ v b )/R and R = (Jξ ) + (Jξ ). The no-penetration condition for the mean flow at the moving boundar results in ˆω =, which takes (6) to T V (T, ξ, η) J = f (ξ, η) c(ṽ 4 ṽ 3) dη + BT. (6) In (6), Ṽ = X T V = [ṽ, ṽ, ṽ 3, ṽ 4 ] T, and BT is the contribution at the other boundaries. An boundar condition of the form ṽ 3 = ±ṽ 4 is well-posed. We choose ṽ 3 +ṽ 4 =, 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 (.5, 3.5). We have used N = M = 5, L = and SBP4 in space and time. The velocit field at two different time levels, with non-penetrating flow close to the solid wall, are presented in Figures 4-7.

7 Energ Stable High Order FDM for Hperbolic Problems in Deforming Domains 7 6 The velocit field at t =.44 The reference domain The deforming domain The velocit field at t =.44 The reference domain The deforming domain Fig. 4 The global velocit field Fig. 5 A blow-up of the velocit field. 6 The velocit field at t =.66 The reference domain The deforming domain.6 The velocit field at t =.66 The reference domain The deforming domain Fig. 6 The global velocit field Fig. 7 A blow-up of the velocit field. The reference domain in Figures 4-7 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. 5 Summar and conclusions We have considered a constant coefficient hperbolic sstem of equations in timedependent curvilinear coordinates. The sstem is transformed into a fied coordinate frame, resulting in variable coefficient sstem. We show that the energ method applied to the transformed sstems together with time-dependent appropriate boundar conditions leads to strongl well-posed problem. 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 and high order accurate numerical

8 8 Samira Nikkar and Jan Nordström 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 shown to hold numericall. We have tested the scheme for high order accurate SBP operators in space and time using the method of manufactured solution. Numerical calculations corroborate the stabilit and accurac of the full-discrete approimations. Finall, as an application, sound propagation b the linearized Euler equations in a deforming domain is illustrated. References. Abrabanel, S., Gottleib, D.: Optimal time splitting for two- and three-dimensional Navier- Stokes equations with mied derivatives, Journal of Computational Phsics, 4, -43, (98).. Charles F. Van Loan: The ubiquitous Kronecker product, Journal of Computational and Applied Mathematics, 3, 85-, (). 3. Farhat, C., Geuzaine, P., Grandmont, C.: The discrete geometric conservation law and the nonlinear stabilit of ALE schemes for the solution of flow problems on moving grids, Journal of Computational Phsics, 74, (). 4. Lundquist. T., Nordström, J.: The SBP-SAT Technique for Initial Value Problems, Journal of Computational Phsics, 7, 86-4, (4). 5. Nikkar, S., Nordström, J., Full Discrete Energ Stable High Order Finite Difference Methods for Hperbolic Problems in Deforming Domains, LiTH- MAT-R, 4:5, 4, Department of Mathematics, Linköping Universit. 6. Nordström, J.: Conservative Finite Difference Formulations, Variable Coefficients, Energ Estimates and Artificial Dissipation, Journal of Scientific Computing, 9, (6). 7. Nordström, J., Carpenter, H.: High-order finite difference methods, multidimensional linear problems and curvilinear coordinates, Journal of Computational Phsics, 73, ( ). 8. Nordström, J., Lundquist. T.: Summation-B-Parts in Time, Journal of Computational Phsics, 5, , (3). 9. Salari, K.: Code Verification b the Method of Manufactured Solutions, doi:.7/ Strand B.: Summation b Parts for Finite Difference Approimations of d/d, Journal of Computational Phsics,, (994).. Abe, Y., Izuka, N., Nonomura, T., Fuji, K.: Smmetric-conservative metric evaluations for higher-order finite difference scheme with the GCL identities on the three-dimensional moving and deforming mesh, ICCFD7 ().. Svärd, M., Nordström, J.: A stable high-order finite difference scheme for the compressible Navier Stokes equations: no-slip wall boundar conditions, Journal of Computational Phsics, 7 (), (8). 3. Thomas, P.D., Lombard, C.K.: Geometric Conservation Law and Its Application to Flow Computations on Moving Grids, AIAA Journal, 7, (979). 4. Turkel, E.: Smmetrization of the fluid dnamics matrices with applications, Math. Comp., 7, , (973).