OpenAccess Proceedings of the 6th International Conference on Numerical Analysis, pp 198-03 Contents lists available at AMCL s Digital Library. NumAn014 Conference Proceedings Digital Library Triton : http://lib.amcl.tuc.gr A multigrid accelerated high-order pressure correction compact scheme for incompressible Navier-Stokes solvers V. G. Mandikas a, E. N. Mathioudakis a, G. V. Kozyrakis b,c, J. A. Ekaterinaris d and N.A. Kampanis b a Applied Mathematics and Computers Laboratory, Technical University of Crete, University Campus, 7313 Chania, Hellas b Institute of Applied and Computational Mathematics, Foundation for Research and Technology - Hellas, 70013 Heraklion, Hellas c Department of Marine Sciences, University of the Aegean, University Hill, Mytilene 81100, Hellas d Department of Aerospace Engineering, Daytona Beach College of Engineering, Embry-Riddle Aeronautical University, 600 S. Clyde Morris Blvd., Daytona Beach FL 3114, USA bmandikas@science.tuc.gr, manolis@amcl.tuc.gr, gkoz@iacm.forth.gr, ekaterin@iacm.forth.gr, kampanis@iacm.forth.gr Abstract. A high-order accurate compact finite-difference numerical scheme, based on multigrid techniques, is constructed on staggered grids in order to develop an efficient incompressible Navier-Stokes solver. The enforcement of the incompressibility condition by solving a Poisson-type equation at each time step is commonly accepted to be the most computationally demanding part of the global pressure correction procedure of a numerical method. Since the efficiency of the overall algorithm depends on the Poisson solver, a multigrid acceleration technique coupled with compact high-order descretization scheme is implemented to accelerate the iterative procedure of the pressure updates and enhance computational efficiency. The employment of geometric multigrid techniques on staggered grids has an intrinsic difficulty, since the coarse grids do not constitute part of the finer grids. Appropriate boundary closure formulas are developed for the cell-centered pressure approximations of the boundary conditions. Performance investigations demonstrate that the proposed multigrid algorithm can significantly accelerate the numerical solution process, while retaining the high order of accuracy of the numerical method even for high Reynolds number flows. Key words: Global pressure correction, Poisson type equation, Incompressible Navier-Stokes equations, High-order compact schemes, staggered grids, Geometric MultiGrid techniques. 1 Introduction Modern applications in Computational Fluid Dynamics, require high order accurate methods for the numerical solution of the incompressible Navier-Stokes equations. In ISBN: 978-960-8475--9 AMCL/TUC http://lib.amcl.tuc.gr/handle/triton/69
Mandikas V.G. et al. 199 the current work, the N S equations are solved using finite differences with high order accurate compact scheme discretizations. The incompressibility constraint can be enforced by a global pressure correction method based on the (elliptic in nature) Poisson equation and it is obtained by taking the divergence of the momentum equations and invoking continuity. A geometric Multigrid technique [1], [], [8] is employed for the acceleration of the solution of the elliptic BVP part. Recent studies focus on the development of methods based on high-order discretizations of the numerical solution, [5], that avoid the disadvantages of lower order methods and high-order-upwind techniques. The Poisson-type equation is solved with fourthorder accurate, compact, finite difference, energy conserving schemes (and applied to enforce the incompressibility condition to the Navier-Stokes solution) in [5, 7]. The solution of a Poisson-type equation is highly demanding in terms of computational cost, with a large and sparse resulting linear system, that requires an iterative solver to compute. Affordable computational times can be achieved by incorporating geometric multigrid techniques coupled with the iterative solver. Boundary closure formulas, for Dirichlet, Neumann, Robin or mixed boundary conditions can be accommodated on the physical boundary. For this purpose, appropriate intergrid transfer operators with special treatments for boundary closures are constructed for the multigrid technique. The incompressible Navier-Stokes equations in two dimensional Cartesian (x, y) coordinates are expressed as: @u @x + @v =0, (1) @y @u @t + @F @x + @G @y = rp + 1 @Fv Re @x + @G v, () @y where u =[u, v, ] T and p are the velocity vector and pressure, respectively, and Re is the non-dimensional Reynolds number. F and G are the inviscid flux vectors, while F v and G v are the viscous fluxes given as: F =[u,uv] T, G =[vu, v ] T, F v = h @u @x, @v i T h @u, Gv = @x @y, @v i T. @y The current solution uses 4th Order accurate compact schemes, formulated over a staggered arrangement of variables (Fig. 1). Incompressibility is enforced using a globally defined pressure correction, computed by a Poisson-type equation and temporal discretisation is carried out by the explicit fourth-order Runge-Kutta method [3]. Multigrid acceleration techniques for the numerical solution of the Poisson-type equation are carried out by high order finite difference methods and a fourth order compact scheme is also used for the discretization of the Poisson-type equation. Proceedings of NumAn014 Conference
00 Multigrid accelerated high-order compact scheme for Navier-Stokes solvers Fig. 1: Staggered arrangement of variables on the computational grid. Numerical solution for the Poisson equation for pressure Following the incompressibility constraint, imposed on the N-S equations, the computed velocity field at each time step should be iteratively corrected to satisfy (1). The numerical solver proposed in [5], solves the following Poisson-type equation at each time step, with the pressure correction term p defined globally on, and valid on cell centers M ij (i =1,...,N x, j =1,...N y ). @ ( p) @x with f ij = f(m ij )= 1 a`,` 1 t (r un,` old ) i,j. ij @ ( p) + @y ij = f ij, (3) The Neumann type BC can be applied by taking the normal projection for the momentum equations () on the wall, as shown in [4]. In their work it is suggested that, for the continuous expression of the Poisson equation the Neumann-type BC provides a unique solution for t 0, whereas a Dirichlet type is valid for t>0. For Dirichlet boundary conditions, p is constant on the walls of the domain and therefore p =0. With Neumann conditions, @p/@n =0, which leads to @( p)/@n =0, with n being the outward normal on the boundary. For the development of a fourth order finite difference compact discretization scheme of the pressure correction equation (3), the two one-dimensional fourth order compact finite difference operators P and Q are applied in the form, P y Q x p ij + P x Q y p ij = P y P x f ij + O( 4 ), (4) with P x, Q x, P y and Q y being the corresponding operators for each partial derivative s direction. The above relation is valid for all grid points (x i,y j ), where O( 4 ) denotes the truncated terms of the order of O( x 4 + y 4 ). Proceedings of NumAn014 Conference
Mandikas V.G. et al. 01 3 Multigrid acceleration technique The Multigrid method consists of an iterative solver called smoother, which is a relaxation scheme for the error linear system and two grid-transfer operators, the restriction for mapping residual vectors from the fine h to coarse H grid and the prolongation (interpolation) for returning the corrected error vectors back to the fine grid, [], [10]. 3.1 Prolongation/Interpolation operator The bilinear interpolation operator shown here as I h H, takes the coarse-grid vectors wh i,j and constructs the fine-grid vectors w h i,j according to Ih H wh = w h, w h i,j = 1 16 (9wH i,j +3w H i+1,j +3w H i,j+1 + w H i+1,j+1), (5) for i =1,..., Nx 1, j=1,..., Ny 1. The components of vectors wi+1,j h, wh i,j+1 and wi+1,j+1 h are similarly evaluated. 3. Restriction operator The reverse intergrid operator of prolongation is called restriction and transports the residual vectors from the fine grid wi,j h to the coarse grid wh i,j. It is shown here as, Ih Hwh = w H. Choosing the full-weighted operator (FW) IH h and satisfying the relation all interior coarse grid values can be written as: I h H = 1 4 IH h, (6) w H i,j = 1 64 (wh i,j +3w h i 1,j +3w h i,j + w h i+1,j +3w h i,j 1 +9w h i 1,j 1 +9w h i,j 1 +3w h i+1,j 1 +3w h i,j +9w h i 1,j +9w h i,j +3w h i+1,j + w h i,j+1 +3w h i 1,j+1 +3w h i,j+1 + w h i+1,j+1), for i =,..., Nx 1, j =,..., Ny 1. (7) 4 Numerical Results and Discussion The results presented hereafter are computed for a steady-state and a time-dependent flow problem. For the steady Kovasznay flow [6], exists an analytical solution of the form, u =1 e x cos( y) v = e x sin( y) p = 1 (1 e x ), (8) Proceedings of NumAn014 Conference
0 Multigrid accelerated high-order compact scheme for Navier-Stokes solvers where the following convergence estimates in Table 1. = Re q Re 4 +4. Comparison with the numerical solution to (8), yields Table 1 Computational error and convergence estimates for the Kovasznay flow grid size u u L p p L Error Order Error Order 8 4.51e- - 5.13e- - 16 3.79e-3 3.57 6.40e-3 3.01 3.3e-4 4.09 5.04e-4 3.67 64 1.41e-5 3.98 3.50e-5 3.85 18 8.87e-7 3.99.46e-6 3.9 The unsteady flow problem describes the decay of an ideal vortex (Oseen vortex) with uniform pressure and initial velocity distribution, v (r, t = 0) = r, (9) where is the strength of the vortex and r is the distance from origin. This vortex decays under viscous dissipation with the velocity distribution at time t described by an exact solution [9], v (r, t) = r (1 e r Re 4t ). (10) Fig. compares the results for the velocity distribution in (10) over equidistanced, Cartesian grids (64x64, 18x18) with the exact solution, at time T =4. Fig. : Analytical and approximated velocity distribution over 64x64 and 18x18 grid points at T =4for the Oseen vortex decay. Proceedings of NumAn014 Conference
Mandikas V.G. et al. 03 Evidently, the approximate solution in both flow cases agrees very well with the analytical expressions and as seen in Table 1 and Fig. 3, the solution is fourth-orderaccurate both in space and time. Fig. 3: decay. L norms of the spatial (left) and temporal (right) error at T =1for the Oseen vortex References 1. A. Brandt, Multi-level adaptive solutions to boundary value problems, Mathematics of Computation 31 (1997) 333 390.. W. L. Briggs, V. E. Henson, S. McCormick, A Multigrid Tutorial, SIAM, Philadelphia, 000. 3. J. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta Methods and General Linear Methods, J. Wiley and Sons, Chichester, 1987. 4. P. M. Gresho, R. L. Sani, On pressure boundary conditions for the incompressible Navier- Stokes equations, International Journal for Numerical Methods in Fluids 7 (10) (1987) 1111 1145. 5. N. Kampanis, J. Ekaterinaris, A staggered grid, high-order accurate method for the incompressible Navier-Stokes equations, J. Comp. Physics 15 (006) 589 613. 6. L. I. G. Kovasznay, Laminar flow behind a two-dimensional grid, Proc. Camb. Phil. Soc. 44 (1948) 58 6. 7. V. Mandikas, E. Mathioudakis, E. Papadopoulou, N. Kampanis, in: Proc. of the World Congress on Engineering 013 (WCE013, London, U.K., July 3-, 013), vol. 1, pp. 74 79, award Certificate of Merit for The 013 International Conference of Applied and Engineering Mathematics. 8. S. McCormick, Multigrid Methods, SIAM, Philadelphia, 1987. 9. R. L. Panton, Incompressible Flow, J. Wiley and Sons, Chichester, 1984. 10. P. Wesseling, An Introduction to Multigrid Methods, J. Wiley and Sons, Chichester, U.K., 199. Proceedings of NumAn014 Conference