A comparative study of efficient iterative solvers for generalized Stokes equations

2008 ◽  
Vol 15 (1) ◽  
pp. 13-34 ◽  
Author(s):  
Maxim Larin ◽  
Arnold Reusken
Keyword(s):  
Comparative Study ◽  
Stokes Equations ◽  
Iterative Solvers ◽  
Generalized Stokes Equations

An Implicit Multigrid Scheme for the Compressible Navier-Stokes Equations With Low-Reynolds-Number Turbulence Closure

1998 ◽  
Vol 120 (2) ◽  
pp. 257-262 ◽  
Author(s):  
Peter Gerlinger ◽  
Dieter Bru¨ggemann
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Low Reynolds Number ◽  
Convergence Acceleration ◽  
Iterative Solvers ◽  
Turbulence Closure ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Low Reynolds ◽  
Turbulence Transport

A multigrid method for convergence acceleration is used for solving coupled fluid and turbulence transport equations. For turbulence closure a low-Reynolds-number q-ω turbulence model is employed, which requires very fine grids in the near wall regions. Due to the use of fine grids, convergence of most iterative solvers slows down, making the use of multigrid techniques especially attractive. However, special care has to be taken on the strong nonlinear turbulent source terms during restriction from fine to coarse grids. Due to the hyperbolic character of the governing equations in supersonic flows and the occurrence of shock waves, modifications to standard multigrid techniques are necessary. A simple and effective method is presented that enables the multigrid scheme to converge. A strong reduction in the required number of multigrid cycles and work units is achieved for different test cases, including a Mack 2 flow over a backward facing step.

scholarly journals A Comparative Study of Rosenbrock-Type and Implicit Runge-Kutta Time Integration for Discontinuous Galerkin Method for Unsteady 3D Compressible Navier-Stokes equations

2016 ◽  
Vol 20 (4) ◽  
pp. 1016-1044 ◽  
Author(s):  
Xiaodong Liu ◽  
Yidong Xia ◽  
Hong Luo ◽  
Lijun Xuan
Keyword(s):  
Comparative Study ◽  
Discontinuous Galerkin ◽  
Stokes Equations ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Navier Stokes ◽  
Third Order ◽  
Navier Stokes Equations ◽  
Index 2 ◽  
Runge Kutta

AbstractA comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flows to DNS of turbulent flows, are presented to assess the performance of these schemes. Numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.

scholarly journals New Splitting Methods for Convection-Dominated Diffusion Problems and Navier-Stokes Equations

2014 ◽  
Vol 16 (5) ◽  
pp. 1239-1262 ◽  
Author(s):  
Feng Shi ◽  
Guoping Liang ◽  
Yubo Zhao ◽  
Jun Zou
Keyword(s):  
Stokes Equations ◽  
Stokes Problem ◽  
Splitting Method ◽  
Iterative Solvers ◽  
Diffusion System ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Diffusion Problems

AbstractWe present a new splitting method for time-dependent convention-dominated diffusion problems. The original convention diffusion system is split into two sub-systems: a pure convection system and a diffusion system. At each time step, a convection problem and a diffusion problem are solved successively. A few important features of the scheme lie in the facts that the convection subproblem is solved explicitly and multistep techniques can be used to essentially enlarge the stability region so that the resulting scheme behaves like an unconditionally stable scheme; while the diffusion subproblem is always self-adjoint and coercive so that they can be solved efficiently using many existing optimal preconditioned iterative solvers. The scheme can be extended for solving the Navier-Stokes equations, where the nonlinearity is resolved by a linear explicit multistep scheme at the convection step, while only a generalized Stokes problem is needed to solve at the diffusion step and the major stiffness matrix stays invariant in the time marching process. Numerical simulations are presented to demonstrate the stability, convergence and performance of the single-step and multistep variants of the new scheme.

COMPARATIVE STUDY OF THERMAL FLOWS WITH DIFFERENT FINITE VOLUME AND LATTICE BOLTZMANN SCHEMES

2004 ◽  
Vol 15 (02) ◽  
pp. 307-319 ◽  
Author(s):  
AHMAD AL-ZOUBI ◽  
GUNTHER BRENNER
Keyword(s):  
Heat Transfer ◽  
Comparative Study ◽  
Finite Volume ◽  
Lattice Boltzmann ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Hybrid Approach ◽  
Navier Stokes ◽  
Steady Flows ◽  
Parallel Plates

In the present paper, a comparative study of numerical solutions for steady flows with heat transfer based on the finite volume method (FVM) and the relatively new lattice Boltzmann method (LBM) is presented. In the last years, the LB methods have challenged the classical FV methods to solve the Navier–Stokes equations and have proven to be superior in accuracy and efficiency for certain applications. Most of these studies were related to the transport of mass and momentum. In the meantime, significant effort has been invested in the application of the LBM to simulate flows including heat transfer. The studies in the present paper are the analysis of performance and accuracy aspects of LBM applied to the prediction of these flows. For a fully developed laminar flow between parallel plates, analytical solutions for the heat transfer in fully developed thermal boundary layers are available and may be compared with the respective numerical results. Finally, a hybrid approach is proposed to circumvent numerical problems of the thermal LB methods.

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