nag_pde_parab_1d_euler_roe (d03puc) calculates a numerical flux function using Roe's Approximate Riemann Solver for the Euler equations in conservative form. It is designed primarily for use with the upwind discretization schemes
nag_pde_parab_1d_cd (d03pfc),
nag_pde_parab_1d_cd_ode (d03plc) or
nag_pde_parab_1d_cd_ode_remesh (d03psc), but may also be applicable to other conservative upwind schemes requiring numerical flux functions.
nag_pde_parab_1d_euler_roe (d03puc) calculates a numerical flux function at a single spatial point using Roe's Approximate Riemann Solver (see
Roe (1981)) for the Euler equations (for a perfect gas) in conservative form. You must supply the
left and
right solution values at the point where the numerical flux is required, i.e., the initial left and right states of the Riemann problem defined below.
In the functions
nag_pde_parab_1d_cd (d03pfc),
nag_pde_parab_1d_cd_ode (d03plc) and
nag_pde_parab_1d_cd_ode_remesh (d03psc), the left and right solution values are derived automatically from the solution values at adjacent spatial points and supplied to the function argument
numflx from which you may call
nag_pde_parab_1d_euler_roe (d03puc).
The Euler equations for a perfect gas in conservative form are:
with
where
is the density,
is the momentum,
is the specific total energy, and
is the (constant) ratio of specific heats. The pressure
is given by
where
is the velocity.
The function calculates the Roe approximation to the numerical flux function
, where
and
are the left and right solution values, and
is the intermediate state
arising from the similarity solution
of the Riemann problem defined by
with
and
as in
(2), and initial piecewise constant values
for
and
for
. The spatial domain is
, where
is the point at which the numerical flux is required. This implementation of Roe's scheme for the Euler equations uses the so-called argument-vector method described in
Roe (1981).
Roe P L (1981) Approximate Riemann solvers, parameter vectors, and difference schemes J. Comput. Phys. 43 357–372
nag_pde_parab_1d_euler_roe (d03puc) must only be used to calculate the numerical flux for the Euler equations in exactly the form given by
(2), with
and
containing the left and right values of
and
, for
, respectively. It should be noted that Roe's scheme, in common with all Riemann solvers, may be unsuitable for some problems (see
Quirk (1994) for examples). In particular Roe's scheme does not satisfy an ‘entropy condition’ which guarantees that the approximate solution of the PDE converges to the correct physical solution, and hence it may admit non-physical solutions such as expansion shocks. The algorithm used in this function does not detect or correct any entropy violation. The time taken is independent of the input arguments.