NAG CL Interface
d03pwc (dim1_parab_euler_hll)
1
Purpose
d03pwc calculates a numerical flux function using a modified HLL (Harten–Lax–van Leer) Approximate Riemann Solver for the Euler equations in conservative form. It is designed primarily for use with the upwind discretization schemes
d03pfc,
d03plc or
d03psc, but may also be applicable to other conservative upwind schemes requiring numerical flux functions.
2
Specification
The function may be called by the names: d03pwc, nag_pde_dim1_parab_euler_hll or nag_pde_parab_1d_euler_hll.
3
Description
d03pwc calculates a numerical flux function at a single spatial point using a modified HLL (Harten–Lax–van Leer) Approximate Riemann Solver (see
Toro (1992),
Toro (1996) and
Toro et al. (1994)) 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
d03pfc,
d03plc and
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
d03pwc.
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 an 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.
4
References
Toro E F (1992) The weighted average flux method applied to the Euler equations Phil. Trans. R. Soc. Lond. A341 499–530
Toro E F (1996) Riemann Solvers and Upwind Methods for Fluid Dynamics Springer–Verlag
Toro E F, Spruce M and Spears W (1994) Restoration of the contact surface in the HLL Riemann solver J. Shock Waves 4 25–34
5
Arguments
-
1:
– const double
Input
-
On entry: must contain the left value of the component , for . That is, must contain the left value of , must contain the left value of and must contain the left value of .
Constraints:
- ;
- Left pressure, , where is calculated using (3).
-
2:
– const double
Input
-
On entry: must contain the right value of the component , for . That is, must contain the right value of , must contain the right value of and must contain the right value of .
Constraints:
- ;
- Right pressure, , where is calculated using (3).
-
3:
– double
Input
-
On entry: the ratio of specific heats, .
Constraint:
.
-
4:
– double
Output
-
On exit: contains the numerical flux component , for .
-
5:
– Nag_D03_Save *
Communication Structure
-
saved may contain data concerning the computation required by
d03pwc as passed through to
numflx from one of the integrator functions
d03pfc,
d03plc or
d03psc. You should not change the components of
saved.
-
6:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
- NE_REAL
-
Left pressure value : .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
Right pressure value : .
7
Accuracy
d03pwc performs an exact calculation of the HLL (Harten–Lax–van Leer) numerical flux function, and so the result will be accurate to machine precision.
8
Parallelism and Performance
Background information to multithreading can be found in the
Multithreading documentation.
d03pwc is not threaded in any implementation.
d03pwc 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. The time taken is independent of the input arguments.
10
Example
This example uses
d03plc and
d03pwc to solve the Euler equations in the domain
for
with initial conditions for the primitive variables
,
and
given by
This test problem is taken from
Toro (1996) and its solution represents the collision of two strong shocks travelling in opposite directions, consisting of a left facing shock (travelling slowly to the right), a right travelling contact discontinuity and a right travelling shock wave. There is an exact solution to this problem (see
Toro (1996)) but the calculation is lengthy and has, therefore, been omitted.
10.1
Program Text
10.2
Program Data
10.3
Program Results