Discontinuous Fluctuation Distribution for Time-Dependent Problems

Transcription

1 Discontinos Flctation Distribtion for Time-Dependent Problems Matthew Hbbard School of Compting, University of Leeds, Leeds, LS2 9JT, UK Introdction For some years now, the flctation distribtion approach to approimating mltidimensional systems of conservation laws has been able to prodce accrate simlations of comple steady state flid flow phenomena sing nstrctred meshes [DSA]. More recent research has illstrated their potential for providing a similar level of accracy in the simlation of time-dependent problems (see the notes in [VKI5] for a recent overview of sch methods). Even so, comptational simlation of compressible flid flow problems is still dominated by the finite volme approach. This is changing, with the emergence of the discontinos Galerkin (DG) approach, which can be treated as a natral generalisation of the finite volme techniqe which acconts directly for variation of the soltion within each mesh cell rather than dealing with cell-averaged vales. Flctation distribtion schemes do, however, have inherent advantages over finite volme and discontinos Galerkin schemes, both of which se nmerical fles across cell bondaries in the pdate of the dependent variables. They instead consider how the variation within each cell (loosely speaking, a generalised fl difference) shold affect the local evoltion of the dependent variable. Flctation distribtion schemes are also typically designed to incorporate the most important nderlying physical processes: making se of the flctation/fl difference instead of the fl provides an environment in which it is simpler to accrately model, not only geninely mltidimensional flow physics, bt also sorce terms when these represent processes which have a natral balance with the fles. Since they are essentially alternative formlations of continos finite element methods, eisting flctation distribtion schemes are similarly restricted by the continity imposed on the nmerical soltion. This can make it difficlt to apply h- and p-adaptivity or constrct high order schemes which are free of nmerically indced oscillations. Recent research, presented in

2 2 Matthew Hbbard [Hb7, Hb8], has led to the proposal of a discontinos flctation distribtion scheme, designed to overcome sch problems. It provides an alternative discontinos model to DG which avoids the constrction of nmerical fles and has been sccessflly sed to design a second order accrate, positive algorithm (a generalisation of the PSI scheme) which can be applied to the Eler eqations of gas dynamics. This paper will discss the etension of these schemes to time-dependent problems, and present preliminary reslts for scalar conservation laws in one space dimension. 2 Discontinos Flctation Distribtion Consider the scalar conservation law governing the evoltion of an nknown qantity (, t) and given by t + f = or t + λ = () on a domain Ω, with bondary conditions imposed on the inflow part of Ω and appropriate initial conditions. Here λ = f/ defines the advection velocity associated with the conservation law (). These eqations will be approimated by discretising an integrated form of the conservation law, assming that the representation of is piecewise polynomial with discontinities allowed at the interfaces between the cells of the comptational mesh. Integrating the spatial derivative terms over the whole domain gives Ω f dω = N c j= C j f dω + N f lim ǫ k= F k ǫ f dω, (2) in which N c, N f are the nmbers of cells and faces in the mesh, respectively, and the final term represents the integrals over the interfaces, which are being treated as limiting cases of degenerate cells whose widths (ǫ) perpendiclar to the adjacent cell faces tend to zero. Now assme that, in d space dimensions, the comptational mesh cells are d-dimensional simplices, that varies linearly with these cells, and that an appropriate (conservative) linearisation eists for the system [VKI5]. The cell spatial flctations can now be written φ j = f dω = f n dγ = i λ ni, (3) C j C j 2 i C j where the symbol indicates an appropriately linearised qantity. The inde i loops over the vertices of the mesh cell and n i is the inward nit normal to the i th face (opposite the i th verte) mltiplied by the length of that edge. The interface spatial flctations can also be evalated eactly, giving

3 Discontinos Flctation Distribtion 3 ψ k = lim f dω, = [f n] dγ = ǫ F [ i ] ǫ k F k 2 n, i F k (4) where λ represents a second (different) set of conservatively averaged vales, and [ ] represents the jmp in a qantity across an interface (where i is considered to be dal-valed), the sign of the difference being dictated by the direction chosen for n. This term is simply the integral over the interface of the fl difference across it. Each mesh node corresponds to many cell vertices and mltiple vales of. When all of the cell- and interface-based flctations are distribted, each j i (the vale associated with verte i of cell j) can receive contribtions from precisely one cell and d interfaces (sbject to the application of bondary conditions). When this is combined with a simple forward Eler discretisation of the time derivative it leads to an iterative pdate of the form ( j i )n+ = ( j i )n + (d + ) t S j ( α j i φ j + ) d α k i ψ k, (5) in which t is the time-step, S j is the volme of cell j, α j/k i are the distribtion coefficients which indicate the appropriate proportions of the flctations to be sent from cell j/interface k to verte i of cell j. Conservation is assred as long as i C j α j i = i F k α k i =, j, k, i.e. the whole of each flctation is distribted to the cell vertices. The precise properties of the scheme depends on the choice of the distribtion coefficients. In particlar, the discontinos PSI scheme described in [Hb7, Hb8] is conservative, positive for an appropriate limit on t, given by k= t S j/(d + ) l C j (k j l )+ cells j, (6) linearity preserving (and hence second order accrate for piecewise linear ), compact, pwind and continos. Note that k l = 2 λ n l are the standard inflow parameters which govern the pwinding. 2. Time-Dependent Problems The development of time-dependent flctation distribtion schemes in which is continos has tended to treat the time derivative in a slightly different manner to the spatial derivatives (see, for eample, [AM3, RCD5]). However, for the prposes of this discssion the time dimension will be treated precisely as an additional spatial dimension, in which the soltion is being advected with speed λ t =. Eqation () can now be written as t f t = or λ t t =, (7) in which t, f t and λ t have all been agmented appropriately. The discontinos flctation distribtion schemes otlined earlier in Section 2 can now be applied to these eqations, albeit on a d + -dimensional space-time mesh.

4 4 Matthew Hbbard 2.2 One Space Dimension The new scheme is most easily illstrated in one space dimension. Figre shows two possible configrations for the discretisation of a rectanglar space time block of dimensions t, in which it is sbdivided into two triangles. The interfaces between the triangles, at which the discontinities occr, are indicated by the dashed rectangles. The reverse configration also illstrates the behavior of the distribtion when the flow velocity is in the opposite direction. The arrows indicate the pwind directions in which the flctations are distribted and, importantly, show that since λ t =, pwinding always sends the flctations arising from the discontinities at a fied time level forward in time. This allows the soltion to be fond seqentially, stepping forward in time and solving at each time level instead of having to approimate the fll space-time domain at once. time flow space Fig.. Flctation distribtion for a one-dimensional space-time block for flow from left to right (showing both orientations for the diagonal). Solid arrows show vertices to which a proportion of the flctation will always be distribted. Distribtion indicated by the dotted arrows is dependent on the magnitde of the velocity λ. The system can now be approimated at the new time level by iterating the following to convergence: ( ) ( j i )(m+) = ( j i )(m) + 2 τ d α j i t φ j + α k i ψ k. (8) The vales of 5 and 6 (see Figre ) form the soltion at the new time level. Note that this method is positive for any vale of t, thogh the above iteration is only positive at each stage for vales of τ governed by (6). The time derivative does not have to be treated eactly like the space derivatives. Using the approach typical of the continos time-dependent schemes [AM3, RCD5] (which do not sally sbdivide the space-time cells k=

5 Discontinos Flctation Distribtion 5 into simplices) to distribte the cell flctations in the discontinos case leads to a scheme which is only first order accrate. It is not yet clear why this shold be so and other possibilities are being investigated to try to remove the asymmetry which appears in the mesh of simplices. 3 Nmerical Reslts The one-dimensional scalar advection eqation was modelled, with { G() for (, ) = elsewhere, being advected with a constant velocity (λ = ) across the domain [, 2]. Periodic bondary conditions are applied and niform space-time meshes are sed to prodce all of the following reslts. Figre 2 shows the otcomes for two initial profiles, G() = cos 2 ((.5)π) and G() =, obtained on a 5 node niform spatial mesh with a CFL of.5. The effect of increasing the CFL nmber and reversing the advection velocity are illstrated in Figre 3. The scheme has also been applied sccessflly to the one-dimensional inviscid Brgers eqation. (9) period periods periods period periods periods Fig. 2. Nmerical approimation of the scalar advection eqation for a smooth (left) and a discontinos (right) initial profile after, and periods. 4 Smmary A framework has been proposed for the development of flctation distribtion schemes for approimating time-dependent problems when the nderlying representation of the dependent variable is allowed to be discontinos

6 6 Matthew Hbbard cfl =.5 cfl = 2. cfl =. period periods periods Fig. 3. Nmerical approimation of the scalar advection eqation for a smooth initial profile for different CFL nmbers (left) and for λ = (right). across space-time mesh interfaces. It has been sccessflly applied to onedimensional, scalar problems, for which second order accracy in space and time and nconditional L stability have been verified. Frther work is reqired to apply the approach in higher space dimensions and improve the efficiency of the approach, as well as removing the asymmetry inherent in the crrent space-time mesh sed. References [AM3] [AS8] Abgrall, R., Mezine, M.: Constrction of second order accrate monotone and stable residal distribtion schemes for nsteady flow problems. J. Compt. Phys., 88, 6 55 (23) Abgrall, R., Sh, C.-W.: Development of residal distribtion schemes for the Discontinos Galerkin methods: the scalar case with linear elements. to appear in Commn. Compt. Phys. [DSA] Deconinck, H., Sermes, K., Abgrall, R.: Stats of mltidimensional pwind residal distribtion schemes and applications in aeronatics. AIAA paper (2) [Hb7] Hbbard, M.E.: A framework for discontinos flctation splitting. Int. J. Nmer. Meth. Flids, 56, 55 3 (28) [Hb8] Hbbard, M.E.: Discontinos flctation distribtion. sbmitted to J. Compt. Phys. [RCD5] Ricchito, M., Csík, Á., Deconinck, H.: Residal distribtion for general time-dependent conservation laws. J. Compt. Phys., 29, (25) [VKI5] High Order Discretization Methods. 34th Comptational Flid Dynamics corse, von Karman Institte for Flid Dynamics, Lectre Series (25)