## D03pvc 1.3

nag_pde_parab_1d_euler_osher (d03pvc) calculates a numerical ﬂux function using Osher’s ApproximateRiemann Solver for the Euler equations in conservative form. It is designed primarily for use with theupwindschemes requiring numerical ﬂux functions.

#include <nag.h>#include <nagd03.h>
void nag_pde_parab_1d_euler_osher (const
nag_pde_parab_1d_euler_osher (d03pvc) calculates a numerical ﬂux function at a single spatial point usingOsher’s Approximate Riemann Solver (see Hemker and Spekreijse (1986) and Pennington and Berzins(1994)) for the Euler equations (for a perfect gas) in conservative form. You must supply the left and rightsolution values at the point where the numerical ﬂux is required, i.e., the initial left and right states of theRiemann
solution values are derived automatically from the solution values at adjacent spatial points and supplied tothe function argument numﬂx from which you may call nag_pde_parab_1d_euler_osher (d03pvc).

The Euler equations for a perfect gas in conservative form are:
where is the density, m is the momentum, e is the speciﬁc total energy, and is the (constant) ratio ofspeciﬁc heats. The pressure p is given by
ð L; URÞÞ, where U ¼ UL and U ¼ UR are the left and right solution values, and
ð Þ arising from the similarity solution U y
with U and F as in and initial piecewise constant values U ¼ U L for y < 0 and U ¼ U R for y > 0.

The spatial domain is À1 < y < 1, where y ¼ 0 is the point at which the numerical ﬂux is required.

Osher’s solver carries out an integration along a path in the phase space of U consisting of subpaths whichare piecewise parallel to the eigenvectors of the Jacobian of the PDE system. There are two variants of theOsher solver termed O (original) and P (physical), which differ in the order in which the subpaths aretaken.

The P-variant is generally more efﬁcient, but in some rare cases may fail (see Hemker and
Spekreijse (1986) for details). The arwhich variant is to be used. The algorithm forOsher’s solver for the Euler equations is given in detail in the Appendix of Pennington and Berzins (1994).

Hemker P W and Spekreijse S P (1986) Multiple grid and Osher’s scheme for the efﬁcient solution of thesteady Euler equations Applied Numerical Mathematics 2 475–493
Pennington S V and Berzins M (1994) New NAG Library software for ﬁrst-order partial differentialequations ACM Trans. Math. Softw. 20 63–99
Quirk J J (1994) A contribution to the great Riemann solver debate Internat. J. Numer. Methods Fluids 18555–574
On entry: ulefti À 1 must contain the left value of the component U i, for i ¼ 1; 2; 3. That is,uleft0 must contain the left value of , uleft1 must contain the left value of m and uleft2 mustcontain the left value of e.

½0 ! 0:0;Left pressure, pl ! 0:0, where pl is calculated using
On entry: urighti À 1 must contain the right value of the component U i, for i ¼ 1; 2; 3. That is,uright0 must contain the right value of , ½1 must contain the right value of m anduright2 must contain the right value of e.

uright0 ! 0:0;Right pressure, pr ! 0:0, where pr is calculated using
On entry: the ratio of speciﬁc heats, .

On entry: the variant of the Osher scheme.

Constraint: path ¼ Nag_OsherOriginal or Nag_OsherPhysical.

On exit: fluxi À 1 contains the numerical ﬂux component

*^*
deﬁned type (see Essential Introduction).

contain data concerning the computation required by nag_pde_parab_1d_euler_osher(d03pvc) as passed through to numﬂx from one of the integrator functions nag_pde_parab_1d_cdshould not change the
The NAG error argument (see the Essential Introduction).

An internal error has occurred in this function. Check the function call and any array sizes. If thecall is correct then please consult NAG for assistance.

Left pressure value pl < 0:0: pl ¼ value
On entry, uright0 < 0:0: uright0 ¼ value
Right pressure value pr < 0:0: pr ¼ value
nag_pde_parab_1d_euler_osher (d03pvc) performs an exact calculation of the Osher numerical ﬂuxfunction, and so the result will be accurate to machine precision.

nag_pde_parab_1d_euler_osher (d03pvc) must only be used to calculate the numerical ﬂux for the Eulerequations in exactly the form given by with ulefti À 1 and urighti À 1 containing the left and rightvalues of ; m and e, for i ¼ 1; 2; 3, respectively. It should be noted that Osher’s scheme, in common withall Riemann solvers, may be unsuitable for some problems (see Quirk (1994) for examples). The timetaken depends on the input the left and right solution values, since inclusion of eachsubpath depends on the signs of the eigenvalues. In general this cannot be determined in advance.

Advice on Replacement Calls for Withdrawn/Superseded Routines
Implementation-specific Information
c05 - Roots of One or More Transcendental Equations
d02 - Ordinary Differential Equations
e04 - Minimizing or Maximizing a Function
f06 - Linear Algebra Support Functions
f08 - Least-squares and Eigenvalue Problems (LAPACK)
g01 - Simple Calculations on Statistical Data
g02 - Correlation and Regression Analysis
s - Approximations of Special Functions
Source: http://www.nag.co.uk/numeric/cl/nagdoc_cl08/pdf/D03/d03pvc.pdf

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