L2 Error in Navier Stokes code

mesh
1

Hi,

I am trying to examine the L2 error variation for a refined mesh applied to your Navier-Stokes equation with Poiseuille flow. I observed that the error increases as the mesh is refined, whereas it is expected to decrease.

Specifically, I am varying the values of nx anf ny in the command line:
mesh = create_unit_square(MPI.COMM_WORLD, 10, 10)

within the file ns_code1.ipynb using Binder.
Am I missing something here?

I am attaching three screenshots for the following mesh configurations:
(i) nx = 10, ny = 10

(ii) nx = 50, ny = 50

(iii) nx = 100, ny = 100

100by100
100by100464×482 12.1 KB

Due to media restriction I am just uploading the error screenshot for 100 by 100.
Waiting for responses and suggestions.

2

Let us first eliminate all solver errors by using direct solvers:

# Solver for step 1

solver1 = PETSc.KSP().create(mesh.comm)
solver1.setOperators(A1)
solver1.setType(PETSc.KSP.Type.PREONLY)
pc1 = solver1.getPC()
pc1.setType(PETSc.PC.Type.LU)
pc1.setFactorSolverType("mumps")

# Solver for step 2
solver2 = PETSc.KSP().create(mesh.comm)
solver2.setOperators(A2)
solver2.setType(PETSc.KSP.Type.PREONLY)
pc2 = solver2.getPC()
pc2.setType(PETSc.PC.Type.LU)
pc2.setFactorSolverType("mumps")

# Solver for step 3
solver3 = PETSc.KSP().create(mesh.comm)
solver3.setOperators(A3)
solver3.setType(PETSc.KSP.Type.PREONLY)
pc3 = solver3.getPC()
pc3.setType(PETSc.PC.Type.LU)
pc3.setFactorSolverType("mumps")

With num_steps=500 I get:

nx=ny=10
Time 10.00, L2-error 5.79e-08, Max error 3.17e-07
nx=ny=20
Time 10.00, L2-error 2.56e-06, Max error 2.42e-05
nx=ny=40
Time 10.00, L2-error 9.17e-06, Max error 1.17e-04
nx=ny=80
Time 10.00, L2-error 4.49e-06, Max error 4.39e-05

However, if we increase num_steps to 1000, we get

nx=ny=10
Time 10.00, L2-error 1.28e-13, Max error 6.10e-13
nx=ny=20
Time 10.00, L2-error 8.42e-09, Max error 7.02e-08
nx=ny=40
Time 10.00, L2-error 6.21e-07, Max error 9.66e-06
nx=ny=80
Time 10.00, L2-error 8.52e-07, Max error 1.41e-05

and with 2000 steps I get

Nx=ny=10
Time 10.00, L2-error 1.44e-14, Max error 2.29e-14
nx=ny=20
Time 10.00, L2-error 3.04e-14, Max error 1.43e-13
nx=ny=40
Time 10.00, L2-error 1.31e-09, Max error 1.61e-08
nx=ny=80
Time 10.00, L2-error 7.52e-08, Max error 1.76e-06

I.e. you observe that the error is drastically reduced when dt is decreased as well. The ratio between spatial and temporal discretization links to the CFL condition, which is required to be bounded for stability: Courant–Friedrichs–Lewy condition - Wikipedia

Several other things to note, is that:

  • Splitting schemes are not great for pressure conditions, as imposed in this tutorial.
  • Splitting schemes are not good for steady flow.