**Problem C1.5.**** Radial Expansion Wave (2D or 3D)**

**ATTENTION!**

**It has been
brought to our attention that the flow described below in 2D on fine grids
develops a kink before t=3, so that it ceases to be a suitable test problem for
evaluating the high order of accuracy of a method. It is recommended to do the
error analysis at t=2 instead.**

**In the 3D
case, moving the analysis to the earlier time helps somewhat to limit the loss
of accuracy, but is not a full remedy. At present it is not quite clear what
causes the breakdown of accuracy in 3D; it is being investigated further.
Possibly, nonradial instabilities are involved.**

**One
suggestion is to run codes at gamma = 3. This nonphysical specific-heat ratio
tends to decouple the acoustic modes and may postpone nonlinear complications.**

**Overview **

This
time-dependent radial flow is entirely determined by its initial values. These
show a cylindrical (2D) or spherical (3D) expansion wave ready to move toward
the origin and interact with itself. There are no
walls and there is no influence of the domain boundary, owing to supersonic
outflow everywhere.

**Governing
Equations**

The
governing equation is the 2D or 3D Euler equations with a constant ratio of
specific heats of g = g = 1.4.

**Flow Conditions**

The initial condition is defined
everywhere. The flow is cylindrically (2D) or spherically (3D) symmetric but
computed on a Cartesian grid. The flow field is purely radial. The initial
distribution of the radial velocity *q* is infinitely differentiable:

The
Cartesian velocity components are related to *q* at any time and place by

The
initial distribution of the speed of sound *a *derives from *q**(**r, **0):*

here the ratio of
specific heats may be chosen by
the user from the interval 1 < g < 3, e. g. = 7/5 (aero), = 5/3
(astro), = 2 (civil, atmospheric). For this
workshop, g = 7/5. The density initially equals g in the origin and further
follows from assuming uniform entropy:

The
initial values of any other flow quantity derive from the above equations and
the perfect-gas law, e. g., pressure:

.

Because
the flow is entirely determined by the initial values, these should be
carefully discretized. For instance, if the numerical
method is a finite-volume scheme that uupdates the
cell-averages of the conserved flow quantities, these must be computed from the
analytical initial values by a sufficiently accurate Gaussian quadrature. The
same holds for the integrals needed to compute the weights of the basis functions in a Discontinuous Galerkin
discretization. If superconvergence
is anticipated, the order of accuracy of the Gaussian quadrature must be taken
high enough so as not to obscure the truncation error of the scheme with an
initialization error.

At
outflow the Mach number is 2, which means supersonic outflow normal to all
domain boundaries, in both 2D and 3D.

**Geometry**

The
computational domain is a square (cube) [−4, 4] ×[−4,
4] (×[−4, 4]), uniformly divided into cells with Dx = Dy
(=Dz) = h, where h
takes the values 1/8 (grid 1), 1/16 (grid 2), 1/32 (grid 3), etc. The simulations should be run from t=0
to t=3. The largest wave speed appearing in the problem is 3/g, which may be
helpful in setting the time step.
However, tests should be performed to ensure that the time step is small
enough (or the order of time integration is high enough) that temporal
resolution has minimal effect on the results.

**Boundary
Conditions**

Supersonic outflow everywhere.

**Requirements**

1.
At
the initial time (t=0) and at the final time (t=3), compute the L2-norm of the
error in the cell-average of the entropy. Compute the local entropy error as

In
each cell, compute the average of this error by Gaussian quadrature of
its subcell distribution. Note that for a
finite-volume method you must first perform the method’s standard subcell interpolation, as if starting a new time-step. The error
norm is defined as usual:

where the sum is
taken over all (*N*) cells.

2.
Study
the numerical order of accuracy according to vs.

3.
Submit
the following data to the workshop for this case

a.
error vs. work units for different *h* and *p* at t=0 and t=3

b.
error vs for different *h* and *p* at t=0 and t=3

c.
A time history figure showing (log scale) versus time for
different *h* and *p*.

**Further Error
Analysis**

Because
the solution is infinitely smooth and the grid is Cartesian, the numerical
solution to this problem lends itself especially well to Richardson extrapolation.
That is, solutions obtained on a sequence of grids may serve, after
prolongation of the finer-grid solutions to the coarsest grid, to find an
expansion of the local truncation on the coarsest grid in powers of the mesh
width. A good choice of flow quantity for this process is the density, since
all FV and FE codes update directly the cell-averaged density; no Gaussian
quadrature is needed. This type of error analysis is elective.