Functions
	exact_density (x, t)
	Density of the entropy wave advected with u=1 over [-0.1,0.1].

	solver_name (rk_order)
	Name of the solver: EulerFV for rk_order None, EulerFD4 otherwise.

	patch_file (directory, rk_order)

	build_and_run (rk_order, volume_size, end_time, time_in_between_plots, directory, peano_root)
	Generate, compile and run one setup within directory.

	snap (offset, size)
	Snap a patch's offset and size onto the regular mesh over the domain.

	errors (directory, rk_order)
	Return a list of (time stamp, max error, rms error) of the density per snapshot.

	density_along_x_axis (directory, rk_order, snapshot)
	Return time stamp plus grid cell centres and densities along y=0.

	label (rk_order)

	plot_convergence (runs)
	Max error at t=ConvergenceEndTime over the grid cells per axis.

	plot_profile (runs)
	Density along y=0 after one period on the coarsest mesh.

	plot_instability (runs)
	Max error over time for the explicit Euler time stepping vs the others.

	directory (name)

Variables
	Here = os.path.dirname(os.path.abspath(__file__))

	PeanoRoot = os.path.abspath(os.path.join(Here, "..", "..", ".."))

str	Flux

str	MaxEigenvalue

str	InitPointWisely

float	DomainOffset = -0.1

float	DomainWidth = 0.2

list	VolumeSizes = [(0.0075, 27), (0.0025, 81), (0.00083, 243)]

float	ConvergenceEndTime = 0.05

float	PeriodEndTime = 0.2

dict	RungeKuttaColours = {1: "C3", 2: "C1", 3: "C2", 4: "C0"}

	parser

	default

	help

	nargs

	choices

	args = parser.parse_args()

	work_directory

dict	runs = {}

	d

Function Documentation

◆ build_and_run()

12_higher-order-solvers.build_and_run	(	rk_order,
		volume_size,
		end_time,
		time_in_between_plots,
		directory,
		peano_root
	)

Generate, compile and run one setup within directory.

Parameters

rk_order	int or None None selects the Finite Volumes of the tutorial 10_euler, an integer the fourth-order Finite Differences with the Runge-Kutta scheme of this order.
peano_root	str Configured and built Peano directory the executable links against.

Definition at line 123 of file 12_higher-order-solvers.py.

References solver_name().

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◆ density_along_x_axis()

12_higher-order-solvers.density_along_x_axis	(	directory,
		rk_order,
		snapshot
	)

Return time stamp plus grid cell centres and densities along y=0.

Definition at line 245 of file 12_higher-order-solvers.py.

References tarch.plotter.api.notebook.read_patches(), tarch.plotter.api.notebook.read_snapshots(), and snap().

Referenced by plot_instability(), and plot_profile().

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◆ directory()

12_higher-order-solvers.directory ( name )

Definition at line 443 of file 12_higher-order-solvers.py.

◆ errors()

12_higher-order-solvers.errors	(	directory,
		rk_order
	)

Return a list of (time stamp, max error, rms error) of the density per snapshot.

Definition at line 217 of file 12_higher-order-solvers.py.

References exact_density(), tarch.plotter.api.notebook.read_patches(), tarch.plotter.api.notebook.read_snapshots(), and snap().

Referenced by plot_convergence(), and plot_instability().

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◆ exact_density()

12_higher-order-solvers.exact_density	(	x,
		t
	)

Density of the entropy wave advected with u=1 over [-0.1,0.1].

Definition at line 105 of file 12_higher-order-solvers.py.

Referenced by errors(), plot_instability(), and plot_profile().

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◆ label()

12_higher-order-solvers.label ( rk_order )

Definition at line 263 of file 12_higher-order-solvers.py.

Referenced by plot_convergence(), plot_instability(), and plot_profile().

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◆ patch_file()

12_higher-order-solvers.patch_file	(	directory,
		rk_order
	)

Definition at line 117 of file 12_higher-order-solvers.py.

References solver_name().

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◆ plot_convergence()

12_higher-order-solvers.plot_convergence ( runs )

Max error at t=ConvergenceEndTime over the grid cells per axis.

Parameters

runs	dict Maps rk_order (None for Finite Volumes) onto a list of (grid cells per axis, directory).

Definition at line 269 of file 12_higher-order-solvers.py.

References errors(), and label().

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◆ plot_instability()

12_higher-order-solvers.plot_instability ( runs )

Max error over time for the explicit Euler time stepping vs the others.

Parameters

runs	list List of (rk_order, directory) on the same mesh.

Definition at line 372 of file 12_higher-order-solvers.py.

References density_along_x_axis(), errors(), exact_density(), and label().

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◆ plot_profile()

12_higher-order-solvers.plot_profile ( runs )

Density along y=0 after one period on the coarsest mesh.

Parameters

runs	list List of (rk_order, directory).

Definition at line 334 of file 12_higher-order-solvers.py.

References density_along_x_axis(), exact_density(), and label().

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◆ snap()

12_higher-order-solvers.snap	(	offset,
		size
	)

Snap a patch's offset and size onto the regular mesh over the domain.

The patch files hold offsets and sizes with seven significant digits only. Below an error of roughly 1e-7, this would dominate the error.

Definition at line 203 of file 12_higher-order-solvers.py.

Referenced by density_along_x_axis(), and errors().

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◆ solver_name()

12_higher-order-solvers.solver_name ( rk_order )

Name of the solver: EulerFV for rk_order None, EulerFD4 otherwise.

Definition at line 112 of file 12_higher-order-solvers.py.

Referenced by build_and_run(), and patch_file().

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Variable Documentation

◆ args

12_higher-order-solvers.args = parser.parse_args()

Definition at line 437 of file 12_higher-order-solvers.py.

◆ choices

12_higher-order-solvers.choices

Definition at line 434 of file 12_higher-order-solvers.py.

◆ ConvergenceEndTime

float 12_higher-order-solvers.ConvergenceEndTime = 0.05

Definition at line 98 of file 12_higher-order-solvers.py.

◆ d

12_higher-order-solvers.d

Initial value:

=  directory(
                    "convergence-{}-{}".format(
                        "fv" if rk_order is None else "rk{}".format(rk_order), cells
                    )
                )

Definition at line 451 of file 12_higher-order-solvers.py.

◆ default

12_higher-order-solvers.default

Definition at line 422 of file 12_higher-order-solvers.py.

◆ DomainOffset

float 12_higher-order-solvers.DomainOffset = -0.1

Definition at line 91 of file 12_higher-order-solvers.py.

◆ DomainWidth

float 12_higher-order-solvers.DomainWidth = 0.2

Definition at line 92 of file 12_higher-order-solvers.py.

◆ Flux

str 12_higher-order-solvers.Flux

Initial value:

=  """
  const double irho  = 1.0 / Q[0];
  constexpr double gamma = 1.4;
  const double p = (gamma - 1.0) * (Q[3] - 0.5 * irho * (Q[1] * Q[1] + Q[2] * Q[2]));
  switch (normal) {
    case 0:
      F[0] = Q[1];
      F[1] = irho * Q[1] * Q[1] + p;
      F[2] = irho * Q[1] * Q[2];
      F[3] = irho * Q[1] * (Q[3] + p);
      break;
    case 1:
      F[0] = Q[2];
      F[1] = irho * Q[2] * Q[1];
      F[2] = irho * Q[2] * Q[2] + p;
      F[3] = irho * Q[2] * (Q[3] + p);
      break;
  }
"""

Definition at line 51 of file 12_higher-order-solvers.py.

◆ help

12_higher-order-solvers.help

Definition at line 423 of file 12_higher-order-solvers.py.

◆ Here

12_higher-order-solvers.Here = os.path.dirname(os.path.abspath(__file__))

Definition at line 36 of file 12_higher-order-solvers.py.

◆ InitPointWisely

str 12_higher-order-solvers.InitPointWisely

Initial value:

=  """
  constexpr double gamma = 1.4;
  constexpr double p     = 1.0;
  constexpr double u     = 1.0;
  const double     rho   = 1.0 + 0.2 * std::sin(2.0 * M_PI * (x[0] + 0.1) / 0.2);
  Q[0] = rho;
  Q[1] = rho * u;
  Q[2] = 0.0;
  Q[3] = p / (gamma - 1) + 0.5 * rho * u * u;
"""

Definition at line 80 of file 12_higher-order-solvers.py.

◆ MaxEigenvalue

str 12_higher-order-solvers.MaxEigenvalue

Initial value:

=  """
  const double irho  = 1.0 / Q[0];
  constexpr double gamma = 1.4;
  const double p = (gamma - 1.0) * (Q[3] - 0.5 * irho * (Q[1] * Q[1] + Q[2] * Q[2]));
  const double c = std::sqrt(gamma * p * irho);
  const double u = Q[1 + normal] * irho;
  lambdaMax[0] = std::max(std::abs(u + c), std::abs(u - c));
"""

Definition at line 71 of file 12_higher-order-solvers.py.

◆ nargs

12_higher-order-solvers.nargs

Definition at line 432 of file 12_higher-order-solvers.py.

◆ parser

12_higher-order-solvers.parser

Initial value:

=  argparse.ArgumentParser(
        description="Create the PNGs of the tutorial 12_higher-order-solvers.ipynb"
    )

Definition at line 417 of file 12_higher-order-solvers.py.

◆ PeanoRoot

12_higher-order-solvers.PeanoRoot = os.path.abspath(os.path.join(Here, "..", "..", ".."))

Definition at line 37 of file 12_higher-order-solvers.py.

◆ PeriodEndTime

float 12_higher-order-solvers.PeriodEndTime = 0.2

Definition at line 99 of file 12_higher-order-solvers.py.

◆ RungeKuttaColours

dict 12_higher-order-solvers.RungeKuttaColours = {1: "C3", 2: "C1", 3: "C2", 4: "C0"}

Definition at line 102 of file 12_higher-order-solvers.py.

◆ runs

list 12_higher-order-solvers.runs = {}

Definition at line 447 of file 12_higher-order-solvers.py.

◆ VolumeSizes

list 12_higher-order-solvers.VolumeSizes = [(0.0075, 27), (0.0025, 81), (0.00083, 243)]

Definition at line 96 of file 12_higher-order-solvers.py.

◆ work_directory

12_higher-order-solvers.work_directory

Initial value:

=  args.work_directory or tempfile.mkdtemp(
        prefix="12_higher-order-solvers-"
    )

Definition at line 439 of file 12_higher-order-solvers.py.

Functions

Variables

Function Documentation

◆ build_and_run()

◆ density_along_x_axis()

◆ directory()

◆ errors()

◆ exact_density()

◆ label()

◆ patch_file()

◆ plot_convergence()

◆ plot_instability()

◆ plot_profile()

◆ snap()

◆ solver_name()

Variable Documentation

◆ args

◆ choices

◆ ConvergenceEndTime

◆ d

◆ default

◆ DomainOffset

◆ DomainWidth

◆ Flux

◆ help

◆ Here

◆ InitPointWisely

◆ MaxEigenvalue

◆ nargs

◆ parser

◆ PeanoRoot

◆ PeriodEndTime

◆ RungeKuttaColours

◆ runs

◆ VolumeSizes

◆ work_directory