Towards Higher-Order Schemes for Compressible Flow

Transcription

1 WDS'6 Proceedings of Contributed Papers, Part I, 5 4, 6. ISBN MATFYZPRESS Towards Higher-Order Schemes for Compressible Flow K. Findejs Charles Universit, Facult of Mathematics and Phsics, Prague, Czech Republic. Abstract. Two different approaches to the construction of higher-order schemes for the stationar compressible Euler equations are presented. The former is a generalization of first order Finite Volume Method (FVM) based on a higher-order polnomial reconstruction. The latter one is the so-called Residual Distribution Method (RDM) which is basicall second order, but some generalizations to higher-order are alread known (e.g. [Abgrall, 5], [Caraeni, 5]). We made a comparative stud and present the numerical results for the Ringleb flow problem. Introduction The Finite Volume Method is formall of first order of accurac. This causes that discontinuities in a solution are often smeared and some details are not resolved sufficientl accuratel. To avoid this deficienc, one tries to generalize the FVM to higher-order. One wa is to use a higher-order polnomial reconstruction of piecewise constant approimation used in the FVM. This works onl if sufficientl large stencils are used and the special attention has to be paid to the reconstruction in the vicinit of discontinuities. Another ver popular approach is the Discontinuous Galerkin Method. Similarl to the FVM it uses discontinuous piecewise polnomial data representation and it turns out to be ver accurate and robust method. Its onl drawback is the fact, that the computational and memor costs are ver high. The Residual Distribution Method, which is considered in this paper, uses continuous piecewise polnomial data representation. Therefore, the number of degrees of freedom is lower than in the Discontinuous Galerkin Method. The results of man authors show that this is ver powerful method even for problems with discontinuous solution. Formulation of the problem Let us consider the flow of an inviscid gas in a bounded domain Ω IR N where N = or for D or D flow, respectivel. We suppose that Ω is polgonal in D or polhedral in D, respectivel. Further, we suppose that the flow is adiabatic and we neglect the outer volume force. Our goal is to find stationar solution of the Euler equations equipped with the initial condition w t + N s= with a given vector function w and boundar conditions f s (w) s = in Ω (, + ) () w(, ) = w (), Ω, () B(w(, t)) = for(, t) Ω (, + ). () Here B is suitable boundar operator. The specifications of the boundar conditions and their approimation can be found in [Feistauer et al., ]. The state vector w = (ρ, ρv,...,ρv N, E) T IR m, m = N+ (i.e. m = 4 or 5 for D or D flow, respectivel). Here ρ, v,..., v N and E are phsical quantities denoting the densit, the velocit components and the total energ, respectivel. The flues f s = (ρv s, ρv v s + δ s p,..., ρv N v s + δ Ns p, (E + p)v s ) are m-dimensional mappings of w. Here p denotes the pressure and δ is Kronecker s smbol. Further we denote b A s (w) = Df s (w)/dw the Jacobi matrices of the flues f s, which pla ver important role in the design of presented numerical schemes. Their forms and properties can be found e.g. in [Feistauer et al., ]. 5

2 Finite volume method The sstem () () can be discretized b an eplicit finite volume scheme w k+ i = w k i τ k H(w k i, w k D i j,n ij ) Γ ij, D i D h, t k [, T). (4) j S(i) This scheme is based on the construction of a finite volume mesh D h = {D i } i J, where J Z + = {,,...} is an inde set and h >. Here D i, i J, are closed polgons or polhedrons, if N = or, respectivel, with mutuall disjoint interiors such that Ω = i J D i. If two elements D i, D j D h, i j, contain a nonempt common open face, we call them neighbours. For two neighbours D i, D j D h we set Γ ij = D i D j = Γ ji. (5) We denote b D i or Γ ij the N-dimensional measure of D i or the (N )-dimensional measure of Γ ij, respectivel and b S(i) J the inde set of all neighbours of volume D i. We construct a partition = t < t <... of the time interval [, T] and denote b τ k = t k+ t k the time step between t k and t k+. Scheme (4) represents the algorithm for finding the approimations w k i of the integral averages D i w(, t k )d/ D i of the quantit w over the finite volume D i at time instant t k. Further, we approimate the flu N s= f s(w)(n ij ) s of the quantit w through the face Γ ij in the direction n ij with the aid of a numerical flu H(w k i, wk j, n ij), depending on the value of the approimate solution w k i on the finite volume D i, the value w k j on D j, and on the normal n ij : tk+ t k Γ ij s= n s f s (w) ds dt τ k H(w k i, wk j, n ij) Γ ij. (6) The strateg for finding the stationar solution will be following. We will iterate using the scheme (4) unless the following stopping criterion is satisfied: ρ k i ρk+ i τ k ρ k < TOL, (7) i i J where ρ k i is densit corresponding to the state wk i and TOL is the prescribed tolerance. Etension to higher-order at stead state To increase the order of the FVM, we use a higher-order polnomial reconstruction. For the scheme of order p we define vectors of (p )-degree polnomials ŵ k i P p satisfing ŵ k i D i () d = wk i, (8) D i w(, t k ) Di = ŵ k i () + O(h p ), if w C p (Ω). (9) In the higher-order approach the space integral in (6) is approimated using the quadrature rule with integration points µ and weights α µ tk+ τ k Γ ij t k Γ ij K n s f s (w) ds dt Ĥ(wk i, wk j, n µ ij) = α µ H(ŵ k i ( µ), ŵ k j ( µ), n ij ). () s= µ= The numerical flu H in scheme (4) is then replaced b higher-order flu scheme in space. Ĥ to obtain higher-order Residual distribution method In this section we present a different approach. We define the approimate solution w h of problem () () as a continuous piecewise linear mapping. We denote b P i the nodes of N-simplicial mesh and b w k i the approimate solution at the node P i at the time instant t k. First the cell residual Φ D on the element D will be computed as Φ D = n s f s (w h ) ds D s= s= w h A s ( w) D s d = D s= A s ( w) w h s, () D 6

3 where n s are components of outer unit normal to the boundar D and w is average state obtained using Roe s parameter vector z = ρ(, v...v N, H) T, where H = (E + p)/ρ is the enthalp. It is well known that both state vectors w = w(z) and flues f s (w) = f s ( w(z)) = f s (z) are quadratic functions P i D z i, P i are the vertices of N-simple D of the components of z. Then w = w( z), where z = N+ and z i are computed from data values w i. According to the scalar case, the cell residual of each component of w h D is distributed to the downstream nodes of each element D using the matrices B i which have to satisf the consistenc condition P i D BD i = I: Φ D i = B D i ΦD () The use of Euler eplicit time integration leads to the following scheme which is formall scheme of the tpe (4), but the difference is that w k i are now data values in nodes P i: w k+ i = w k i τ k D i D j,p i D j Φ Dj i. () Here Di = N+ D j,p i D j D j is the area of the median dual cell around the node P i and τ k = t k+ t k is the time step. Further, we will be concerned with the definition of the distribution matrices B D i. In what follows the superscript D will be omitted and the nodes of the element D will be indeed locall b numbers,..., N +. The construction of the matrices B i is direct generalization of the scalar case. For details see e.g. [Deconinck et al., 5]. First we define the matrices K i = N (n i ) s A s ( w), (4) s= where n i is the inward scaled normal to the side of N-simple D opposite to the node P i and its magnitude equals the length of the side. A ver useful propert is that these matrices are diagonalizable: K i = RΛ\R. (5) Using this, we can split the matrices K i into the positive and negative part: K + i = RΛ\ + R, K i = RΛ\ R, (6) where the matrices Λ\ +, Λ\ contain positive or negative eigenvalues, respectivel. Several was are known to define the suitable distribution matrices B i and the properties of the resulting scheme strongl depend on them. One possibilit is to construct the matrices B i in order to get desired second order. This leads to so-called LDA scheme (low diffusion advective), which uses following definition of nodal residuals: Φ LDA i = B LDA i Φ D, where B LDA i = K + i ( N+ i= K + i ). (7) On the other hand it is known from Godunov s theorem (see e.g. [Deconinck et al., 5]) that a linear scheme cannot be both positive and more than first order. This means that LDA scheme cannot be positive which causes a bad behaviour of the solution near discontinuities. Another useful choice is the so-called N scheme - the linear monotone scheme satisfing positivit and therefore (in view of Godunov s theorem) not reaching the second order. The distribution matrices B i are now dependent also on the solution w k i. The are not eplicitl defined, but we can easil epress the nodal residuals directl as Φ N i = K + i N+ j= K + j N+ K j (w i w j ). (8) Our goal is to construct a positive scheme which is satisfactor accurate. The idea is to combine previous two linear schemes into a non-linear LDA-N blended scheme whose nodal residuals are epressed as j= Φ i = LΦ N i + (I L)Φ LDA i, (9) 7

4 where L is the diagonal matri with the diagonal values l ii. This matri should be defined in such a wa that the scheme () behaves like N scheme near discontinuities and like LDA scheme elsewhere. Some heuristic was are known, but the optimal definition of such a matri L is still an open task. As a stopping criterion for the iteration process we again use (7) as in Finite Volume Method where J is the inde set of nodes. Numerical eperiment For a comparison of Residual Distribution Schemes and Finite Volume Schemes we have chosen the so-called Ringleb problem. Its eact stationar solution is smooth and can be epressed analticall. It is depicted in Fig.. It is widel used for testing higher-order methods, because the first order methods do not give satisfactor results. Four numerical schemes were compared: first order Finite Volume Scheme, second order Finite Volume Scheme with least squares linear reconstruction (FVM-LS), N scheme and LDA scheme on four different meshes. As numerical flu for FVM the HLLCF lu was used. The least squares reconstruction (LS) was constructed on stencil containing all volumes D j having at least one verte with D i in common. Let us denote Ŝ(i) the set of indees of such volumes. The reconstructed gradient ŵk i was then computed as the least squares solution of the following sstem of linear algebraic equations ŵ k i ( i j ) = w k i wk j, for j Ŝ(i), () where i and j are centers of gravit of the volumes D i and D j, respectivel. The reconstructed linear function is then ŵ k i () = w k i + ŵ k i ( i ). () All computations were done unless the condition (7) with tolerance TOL = 6 for stationar error was satisfied. For all methods were used the same constant time steps restricted with respect to used mesh to satisf the CFL stabilit condition of Finite Volume Method. The errors in L norm were computed using three point quadrature rule of second order Figure. Computational mesh with nodes (left) and densit isolines of the eact solution (right) 8

5 FVM scheme FVM - LS scheme N scheme LDA scheme Figure. Densit isolines in the domain Ω.95.9 FVM scheme FVM-LS scheme N scheme LDA scheme eact line FVM scheme FVM-LS scheme N scheme LDA scheme Figure. Densit distribution along the boundar (left) and histor of convergence to stationar solution (right) 9

6 Table. Comparison of errors in Densit Discussion method number L norm L norm L norm L norm of elements error order error order FVM scheme FVM-LS scheme N scheme LDA scheme The results in Tab. show that using Finite Volume Method with Least Squares reconstruction we obtained a more accurate solution than using second order LDA scheme. Further we see from Fig. and that the Residual Distribution Scheme suffers from numerical boundar conditions. This causes the loss of accurac close to inlet. Farther from the boundar the scheme gives quite satisfactor results. The convergence histor depicted in Fig. on the right-hand side shows, that the reconstruction used in the Finite Volume Scheme causes a slowdown of convergence to stationar solution. The second order LDA scheme converges much faster. Conclusion From these results we can conclude that RDM is probabl more sensitive to the treatment of boundar conditions near curved boundaries. However, in view of convergence to a stationar solution the RDM seems to be more suitable method than FVM with reconstruction. Acknowledgments. The work is a part of the research project MSM 689 financed b MŠMT and was partl supported b the grant No. 4/5/B-MAT/MFF of the Grant Agenc of the Charles Universit. The author acknowledges this support. References Abgrall R., Construction of high order residual distribution schemes for scalar problems preliminar results for sstems, 4th VKI Lecture Series CFD, von Karman Institute, 5. Abgrall R. and Barth T., Residual distribution schemes for conservation laws via adaptive quadrature, SIAM J. Sci. Comput., Vol. 4, No., pp ,. Caraeni D., A Third order residual distribution methods for stead/unstead simulations: formulation and benchmarking including LES, 4th VKI Lecture Series CFD, von Karman Institute, 5. Deconinck H., Abgrall R., Introduction to residual distribution methods, 4th VKI Lecture Series CFD, von Karman Institute, 5. De Palma P. Pascazio G., Rossiello G., Napolitano M., Implicit third-order-acurate residual distribution schemes for unstead hperbolic problems, 4th VKI Lecture Series CFD, von Karman Institute, 5. Feistauer M., Felcman J., Straškraba I. Mathematical and Computational Methods for Compressible Flow, Oford Universit Press, Oford,. Felcman J. and Findejs K., Higher-Order finite volume method for multidimensional compressible flow, Proceedings of ICNAAM 5 (ed. T. E. Simons), WILEY-VCH, Weinheim, 86-89, 5. Nishikawa H., and Roe P. L., Towards high-order fluctuation-splitting schemes for Navier-Stokes equations, Department of Aerospace engineering, The Universit of Michigan, MI