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Some examples illustrating how to use the relevant namelists to set the numerical methods for solving the equations

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Context : Heat driven incompressible flow

  • Numerical scheme for solving the governing equations of velocity and temperature
    • Time discretization : semi-implicit formulation with the 2nd order Backward Differentiation formula (BDF2)
    • Viscous and conduction fluxes (2nd order centered scheme selected by default with BDF2)
    • convective flux for momentum equation : 2nd order centered scheme, conservative form
    • advective flux for temperature equation : 2nd order centered scheme, conservative form
    • Solving Poisson's equation : see the following examples

Example 1 : Partial diagonalization method

Only one namelist is required : “Numerical_Methods”. You first find the old version and then the new one. The both versions are strictly equivalent.
The old version : 

    &Numerical_Methods  Numerical_Scheme= 1,
                        Convective_Flux_Discretization_Type            = 1 , 
                        Temperature_Advective_Flux_Discretization_Type = 1 ,  
                        Numerical_Method_Poisson_Equation   = 3    /

The corresponding new version : 

  &Numerical_Methods  NS_NumericalMethod= "BDF2-SchemeO2",
                      MomentumConvection_Scheme="Centered-O2-Conservative" , 
                      TemperatureAdvection_Scheme="Centered-O2-Conservative" ,  
                      Poisson_NumericalMethod="Home-PartialDiagonalization"  /

Example 2 : An iterative method coupled with a multigrid procedure using a "in-house" development

Suitable setting :

  • A SOR solver with a relaxation coefficient of 1.7, using a red-black alogorithm in a MPI-parallel context.
  • The nV-cycle multigrid procedure is composed of 5 grid levels, with a maximum number of cycles n= 10.
  • The number of SOR iterations is :
    • 5 on the restriction step (going from finnest to the coarsest grid)
    • 20 on the coarsest grid
    • 15 on the prolongation step (going from coarsest to the finnest grid)
  • The stopping criterion based on the residu of the computation is 1E-08

As the fluid is incompressible, the matrix coefficients of the Poisson's equation are constant.
As a “homemade” method is used, two ways are possible :

Using the namelist "Numerical_Methods" only (old version)

               
                        
    &Numerical_Methods  Numerical_Scheme= 1,
                        Convective_Flux_Discretization_Type            = 1 , 
                        Temperature_Advective_Flux_Discretization_Type = 1 , 
                         
                        Numerical_Method_Poisson_Equation   = 1   
                        Iterative_Method_Selection          = 3   ,
                        Number_max_Grid = 5                         ,
                        Number_max_Cycle= 10                      ,
                        Number_Iteration= 15,
                        Number_Iteration_FineToCoarseGrid= 5,
                        Number_Iteration_CoarsestGrid    = 15,
                        Number_Iteration_CoarseToFineGrid= 10,
                        Relaxation_Coefficient           = 1.70 , 
                        Convergence_Criterion            = 1.D-08 /

Using the namelists "Numerical_Methods" and "HomeData_PoissonSolver" (new version)

    &Numerical_Methods  NS_NumericalMethod= "BDF2-SchemeO2",
                        MomentumConvection_Scheme="Centered-O2-Conservative" , 
                        TemperatureAdvection_Scheme="Centered-O2-Conservative" ,  
                        Poisson_NumericalMethod="Home-SORMultigrid-ConstantMatrixCoef"/
                        
    &HomeData_PoissonSolver   SolverName="SOR",
                              Number_max_Grid = 5                         ,
                              Number_max_Cycle= 10                      ,
                              Number_Iteration= 15,
                              Number_Iteration_FineToCoarseGrid= 5,
                              Number_Iteration_CoarsestGrid    = 15,
                              Number_Iteration_CoarseToFineGrid= 10,
                              Relaxation_Coefficient           = 1.70 , 
                              Convergence_Criterion            = 1.D-08 /

Example 3 : An iterative method coupled with a multigrid procedure using the HYPRE library

Suitable setting :

  • Selection of the PFMG method using a SOR relaxation method for non symmetrical matrix (even though the Poisson's operator could be symmetric in this context)
  • The number of iterations is :
    • The maximum iteration number is 20
    • 5 relaxation sweeps before coarse-grid correction
    • 10 relaxation sweeps after coarse-grid correction
  • The tolerance convergence is 1E-08

Using the namelists "Numerical_Methods" and "HypreData_PoissonSolver" (new version only)

               
         
    &Numerical_Methods  NS_NumericalMethod= "BDF2-SchemeO2",
                        MomentumConvection_Scheme="Centered-O2-Conservative" , 
                        TemperatureAdvection_Scheme="Centered-O2-Conservative" ,  
                        Poisson_NumericalMethod="Home-Multigrid-ConstantMatrixCoef"/
                        
    &HypreData_PoissonSolver  SolverName="SOR-Redblack-Nonsym",
                              Number_Iteration                 = 20 ,
                              Number_Iteration_FineToCoarseGrid= 5, 
                              Number_Iteration_CoarseToFineGrid= 10, 
                              Convergence_Criterion            = 1.D-08 /

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