Examples » KozCG: Nonlinear energy growth

Equations solved

The system of equation solved is

detailed at the KozCG page describing its numerical method, with the exact solution

and the source terms

For more details, see Papers.

Problem setup

The simulation domain is a cube centered around the point {0,0,0}. The initial conditions are sampled from the analytic solution at t=0. We set Dirichlet boundary conditions on the sides of the cube, sampling the analytic solution. The numerical solution is time-dependent. The simulation was run for a single time unit computing the numerical errors during time stepping and the errors at t=1 are used to assess the convergence rate. A time step of 0.001 was used for the coarsest mesh and was successively halved for the finer meshes.

To estimate the order of convergence, the numerical solution is computed using three different meshes, whose properties of are tabulated below, where h is the average edge length.

Code revision to reproduce

To reproduce the results below, use code revision 8c9d1db and the control file below.

Control file

-- vim: filetype=lua:

print "Euler equations computing nonlinear energy growth"

term = 1.0
ttyi = 10

solver = "kozcg"
fct = false

dt = 0.001      -- 750K
--dt = 0.0005     --   6M
--dt = 0.00025      --  48M

part = "rcb"

problem = {
  name = "nonlinear_energy_growth"
  alpha = 0.25,
  beta = { 1.0, 0.75, 0.5 },
  r0 = 2.0,
  ce = -1.0,
  kappa = 0.8
}

mat = { spec_heat_ratio = 5/3 }

bc_dir = {
  { 1, 1, 1, 1, 1, 1 },
  { 2, 1, 1, 1, 1, 1 },
  { 3, 1, 1, 1, 1, 1 },
  { 4, 1, 1, 1, 1, 1 },
  { 5, 1, 1, 1, 1, 1 },
  { 6, 1, 1, 1, 1, 1 }
}

fieldout = {
  iter = 1000
}

diag = {
  iter = 1,
  format = "scientific"
}

Run using the 750K mesh on 32 CPUs

./charmrun +p32 Main/inciter -i unitcube_750K.exo -c nonlinear_energy_growth.q

Visualization

ParaView can be used for interactive visualization of the numerically computed fields as

paraview out.e-s.0.32.0

The figures below show the density, energy, and pressure distributions at four different times on center planes through the origin.

Numerical errors and convergence rate

The table below shows the numerical L1 errors of the flow quantities computed. The L1 error is computed as

where n is the total number of points of the mesh, and are the exact and computed solutions at mesh point v, and denotes the volume associated to mesh point v. The order of convergence is estimated as

where m is a mesh index. The following table summarizes the L1 errors in all flow variables at t=1.

Here r, u, v, w, and e are the fluid density, velocity in the x, y, z directions, and the internal energy, respectively. The measured convergence rate in all variables approaches 2.0, consistent with expectations.

References