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NAG Toolbox: nag_pde_1d_parab_euler_osher (d03pv)

Purpose

nag_pde_1d_parab_euler_osher (d03pv) calculates a numerical flux function using Osher's Approximate Riemann Solver for the Euler equations in conservative form. It is designed primarily for use with the upwind discretization schemes nag_pde_1d_parab_convdiff (d03pf), nag_pde_1d_parab_convdiff_dae (d03pl) or nag_pde_1d_parab_convdiff_remesh (d03ps), but may also be applicable to other conservative upwind schemes requiring numerical flux functions.

Syntax

[flux, ifail] = d03pv(uleft, uright, gamma, path)
[flux, ifail] = nag_pde_1d_parab_euler_osher(uleft, uright, gamma, path)

Description

nag_pde_1d_parab_euler_osher (d03pv) calculates a numerical flux function at a single spatial point using Osher'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 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_1d_parab_convdiff (d03pf), nag_pde_1d_parab_convdiff_dae (d03pl) and nag_pde_1d_parab_convdiff_remesh (d03ps), 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_1d_parab_euler_osher (d03pv).
The Euler equations for a perfect gas in conservative form are:
(U)/(t) + (F)/(x) = 0,
U t + F x =0,
(1)
with
U =
[ ρ m e ]
  and  F =
[ m (m2)/ρ + (γ − 1) (e − (m2)/(2 ρ)) (me)/ρ + m/ρ(γ − 1) (e − (m2)/(2ρ)) ]
,
U=[ ρ m e ]   and  F=[ m m2ρ+(γ-1) (e-m22 ρ ) meρ+mρ(γ-1) (e-m22ρ ) ] ,
(2)
where ρρ is the density, mm is the momentum, ee is the specific total energy, and γγ is the (constant) ratio of specific heats. The pressure pp is given by
p = (γ1) (e(ρu2)/2) ,
p=(γ-1) (e-ρu22) ,
(3)
where u = m / ρu=m/ρ is the velocity.
The function calculates the Osher approximation to the numerical flux function F(UL,UR) = F(U*(UL,UR))F(UL,UR)=F(U*(UL,UR)), where U = ULU=UL and U = URU=UR are the left and right solution values, and U*(UL,UR)U*(UL,UR) is the intermediate state ω(0)ω(0) arising from the similarity solution U(y,t) = ω(y / t)U(y,t)=ω(y/t) of the Riemann problem defined by
(U)/(t) + (F)/(y) = 0,
U t + F y =0,
(4)
with UU and FF as in (2), and initial piecewise constant values U = ULU=UL for y < 0y<0 and U = URU=UR for y > 0y>0. The spatial domain is < y < -<y<, where y = 0y=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 UU consisting of subpaths which are piecewise parallel to the eigenvectors of the Jacobian of the PDE system. There are two variants of the Osher solver termed O (original) and P (physical), which differ in the order in which the subpaths are taken. The P-variant is generally more efficient, but in some rare cases may fail (see Hemker and Spekreijse (1986) for details). The parameter path specifies which variant is to be used. The algorithm for Osher's solver for the Euler equations is given in detail in the Appendix of Pennington and Berzins (1994).

References

Hemker P W and Spekreijse S P (1986) Multiple grid and Osher's scheme for the efficient solution of the steady Euler equations Applied Numerical Mathematics 2 475–493
Pennington S V and Berzins M (1994) New NAG Library software for first-order partial differential equations ACM Trans. Math. Softw. 20 63–99
Quirk J J (1994) A contribution to the great Riemann solver debate Internat. J. Numer. Methods Fluids 18 555–574

Parameters

Compulsory Input Parameters

1:     uleft(33) – double array
uleft(i)ulefti must contain the left value of the component UiUi, for i = 1,2,3i=1,2,3. That is, uleft(1)uleft1 must contain the left value of ρρ, uleft(2)uleft2 must contain the left value of mm and uleft(3)uleft3 must contain the left value of ee.
Constraints:
  • uleft(1)0.0uleft10.0;
  • Left pressure, pl0.0pl0.0, where plpl is calculated using (3).
2:     uright(33) – double array
uright(i)urighti must contain the right value of the component UiUi, for i = 1,2,3i=1,2,3. That is, uright(1)uright1 must contain the right value of ρρ, uright(2)uright2 must contain the right value of mm and uright(3)uright3 must contain the right value of ee.
Constraints:
  • uright(1)0.0uright10.0;
  • Right pressure, pr0.0pr0.0, where prpr is calculated using (3).
3:     gamma – double scalar
The ratio of specific heats, γγ.
Constraint: gamma > 0.0gamma>0.0.
4:     path – string (length ≥ 1)
The variant of the Osher scheme.
path = 'O'path='O'
Original.
path = 'P'path='P'
Physical.
Constraint: path = 'O'path='O' or 'P''P'.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     flux(33) – double array
flux(i)fluxi contains the numerical flux component iF^i, for i = 1,2,3i=1,2,3.
2:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry,gamma0.0gamma0.0,
orpath'O'path'O' or 'P''P'.
  ifail = 2ifail=2
On entry,the left and/or right density or pressure value is less than 0.00.0.

Accuracy

nag_pde_1d_parab_euler_osher (d03pv) performs an exact calculation of the Osher numerical flux function, and so the result will be accurate to machine precision.

Further Comments

nag_pde_1d_parab_euler_osher (d03pv) must only be used to calculate the numerical flux for the Euler equations in exactly the form given by (2), with uleft(i)ulefti and uright(i)urighti containing the left and right values of ρ,mρ,m and ee, for i = 1,2,3i=1,2,3, respectively. It should be noted that Osher's scheme, in common with all Riemann solvers, may be unsuitable for some problems (see Quirk (1994) for examples). The time taken depends on the input parameter path and on the left and right solution values, since inclusion of each subpath depends on the signs of the eigenvalues. In general this cannot be determined in advance.

Example

function nag_pde_1d_parab_euler_osher_example
uleft = [1;
     0;
     2.5];
uright = [1;
     0;
     2.5];
gamma = 1.4;
path = 'P';
[flux, ifail] = nag_pde_1d_parab_euler_osher(uleft, uright, gamma, path)
 

flux =

         0
    1.0000
         0


ifail =

                    0


function d03pv_example
uleft = [1;
     0;
     2.5];
uright = [1;
     0;
     2.5];
gamma = 1.4;
path = 'P';
[flux, ifail] = d03pv(uleft, uright, gamma, path)
 

flux =

         0
    1.0000
         0


ifail =

                    0



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
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