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Jian-Guo Liu 1 and Chi-Wang Shu 2 We Are Interested in Solving the Following 2D Time-Dependent Incompressible Euler Equations In

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Jian-Guo Liu 1 and Chi-Wang Shu 2 We Are Interested in Solving the Following 2D Time-Dependent Incompressible Euler Equations In Journal of Computational Physics 160, 577–596 (2000) doi:10.1006/jcph.2000.6475, available online at http://www.idealibrary.com on Jian-Guo Liu 1 and Chi-Wang Shu 2 Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland 20742; and Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912 E-mail: ¡ [email protected], [email protected] Received June 4, 1999; revised January 26, 2000 In this paper we introduce a high-order discontinuous Galerkin method for two- dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The stream function is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite el- ement spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no nu- merical dissipation and total enstrophy stability. The method is efficient for inviscid or high Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments. c 2000 Academic Press Key Words: incompressible flow; discontinuous Galerkin; high-order accuracy. 1. INTRODUCTION AND THE SETUP OF THE SCHEME We are interested in solving the following 2D time-dependent incompressible Euler equations in vorticity stream-function formulation; t u 0 u (1.1) u n given on 1 Research supported by NSF Grant DMS-9805621. 2 Research supported by ARO Grant DAAG55-97-1-0318, NSF Grants DMS-9804985, ECS-9906606, and INT-9601084, NASA Langley Grant NAG-1-2070 and Contract NAS1-97046 while this author was in residence at ICASE, NASA Langley Research Center, Hampton, VA 23681-2199, and AFOSR Grant F49620-99-1-0077. 577 0021-9991/00 $35.00 Copyright c 2000 by Academic Press All rights of reproduction in any form reserved. 578 LIU AND SHU where y x . Notice that the boundary condition, plus the fact that u n , recovers on the boundary (up to a constant) in a simple connected domain b (1.2) We are also interested in solving the Navier–Stokes equations with high Reynolds numbers Re 1: 1 u t Re u (1.3) u given on The boundary condition is now (1.2) plus the non-slip type boundary condition: u (1.4) n b For simplicity, we only consider the no-flow, no-slip boundary conditions b 0 ub 0 and periodic boundary conditions. We first emphasize that, for Euler equations (1.1) and high Reynolds number (Re 1) Navier–Stokes equations (1.3), it is advantageous to treat both the convective terms and the viscous terms explicitly. The methods discussed in this paper are stable under standard CFL conditions. Since the momentum equation (the first equation in (1.1) and (1.3)) is treated explicitly in the discontinuous Galerkin framework, there is no global mass matrix to invert, unlike conventional finite element methods. This makes the method highly efficient for parallel implementation, see for example [3]. As any finite element method, our approach has the flexibility for complicated geometry and boundary conditions. The method is adapted from the Runge–Kutta discontinuous Galerkin methods discussed by Cockburn et al. in a series of papers [7–13, 20]. The main difficulties in solving incompressible flows are the incompressibility condition and boundary conditions. The incompressibility condition is global and is thus solved by the standard Poisson solver for the stream function using continuous finite elements. One advantage of our approach is that there is no matching conditions needed for the two finite element spaces for the vorticity and for the stream function . The incompressibility con- dition, represented by the stream function , is exactly satisfied pointwise and is naturally matched with the convective terms in the momentum equation. The normal velocity u n is automatically continuous along any element boundary, allowing for correct upwinding for the convective terms and still maintaining a total energy conservation and total enstrophy stability. There is an easy proof for stability, both in the total enstrophy and in the total energy, which does not depend on the regularity of the exact solutions. For smooth solutions error estimates can be obtained. We use the vorticity stream-function formulation of the Navier–Stokes equations. This formulation with the local vorticity boundary condition has been revitalized by the recent work of E and Liu [14, 15, 23]. The main idea is to use convectively stable time-stepping procedure to overcome the cell Reynolds number constraint, explicit treatment of the vis- cous terms and the local vorticity boundary condition. This results in a decoupling of the HIGH-ORDER DISCONTINUOUS GALERKIN METHOD 579 computation of stream function and vorticity at every time step. This method is very efficient and accurate for moderate to high Reynolds number flows, as demonstrated in [14, 15, 23]. Our method, as it stands, can only compute 2D flows. In 3D, the normal velocity u n is no longer continuous along an element boundary, hence making the method more complicated to design and to analyze. Similar approaches for the stream-function vorticity formulation, or for the primitive variable formulation, suitable for 3D calculations, are under investigation. We do not advocate our method for modest or low Reynolds number flows. In such regime viscous terms should be treated implicitly for efficiency. This is a much more challenging task in terms of space matching characterized by the Babuska–Brezzi–Ladyzenskaja˘ con- dition, projection type methods, and global vorticity boundary conditions; see, for example [4, 17–19, 25, 26, 28] etc. We remark that the only problem of our method for modest or low Reynolds number flows is the small time step dictated by the stability of the explicit time discretization. Of course, if the objective is to resolve the full viscous effect, hence a small time step is justified for accuracy, then it is still adequate to use our method. For convection-dominated flows, as we investigate in this paper, we mention the work of Bell et al. [1] for second-order Godunov-type upwinding methods; see also Levy and Tadmor [22] and E and Shu [16]. This is still an active field for research. We now describe the setup of the scheme. We start with a triangulation h of the domain , consisting of polygons of maximum size (diameter) h, and the following two approximation spaces k k k k Vh : K P K K h W0 h Vh C0 (1.5) where Pk K is the set of all polynomials of degree at most k on the cell K . k For the Euler equations (1.1), the numerical method is defined as follows: find h Vh k and h W0 h, such that k t h K huh K uh n h e 0 Vh (1.6) e K k h h W0 h (1.7) with the velocity field obtained from the stream function by uh h (1.8) Here is the usual integration over either the whole domain or a subdomain denoted by a subscript; similarly for the L2 norm . Notice that the normal velocity uh n is continuous across any element boundary e,but both the solution h and the test function are discontinuous there. We take the values of the test function from within the element K denoted by . The solution at the edge is taken as a single-valued flux h, which can be either a central or an upwind-biased average. For example, the central flux is defined by 1 h 2 h h (1.9) where h is the value of h on the edge e from outside K , the complete upwind flux is 580 LIU AND SHU defined by h if uh n 0 h (1.10) h if uh n 0 and the Lax–Friedrichs upwind biased flux is defined by 1 uh n h 2 [uh n h h h h ] (1.11) where is the maximum of uh n either locally (local Lax–Friedrichs) or globally (global Lax–Friedrichs). k We remark that, for general boundary conditions (1.2), the space W0 h in (1.5) should be modified to take the boundary value into consideration. Moreover, additional physical vorticity boundary condition for any inlet should be given. Navier–Stokes equations (1.3) can be handled in a similar way, with the additional viscous terms treated by the local discontinuous Galerkin technique in [13], and with a local vorticity boundary condition in [14]. The details are left to Section 3. Section 2 is devoted to the discussion of stability and error estimates for the Euler equations. Accuracy check and numerical examples are given in Section 4. Concluding remarks are given in Section 5. 2. STABILITY AND ERROR ESTIMATES FOR THE EULER EQUATIONS For stability analysis, we take the test function h in (1.6), obtaining d 1 2 1 2u u n 0 dt 2 h K 2 h h K h h h e e K where we have used the exact incompressibility condition satisfied by uh for the second term. Performing an integration by parts for the second term, we obtain d 1 2 u n 1 2 0 dt 2 h K h h h 2 h e e K Now, using the fact that 1 2 2 2 [ ] [] where 1 2 [ ] we obtain d 1 1 2 u n 1 2 u n [ ] 0 dt 2 h K h h h 2 h e 2 h h h h e e K e K Notice that the second term is of opposite sign for adjacent elements sharing a common edge e, hence it becomes zero after summing over all the elements K (using the no-flow HIGH-ORDER DISCONTINUOUS GALERKIN METHOD 581 boundary condition on the physical boundary or periodic boundary conditions).
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