nag_pde_parab_1d_euler_hll (d03pwc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_pde_parab_1d_euler_hll (d03pwc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_pde_parab_1d_euler_hll (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 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.

2  Specification

#include <nag.h>
#include <nagd03.h>
void  nag_pde_parab_1d_euler_hll (const double uleft[], const double uright[], double gamma, double flux[], Nag_D03_Save *saved, NagError *fail)

3  Description

nag_pde_parab_1d_euler_hll (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 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_hll (d03pwc).
The Euler equations for a perfect gas in conservative form are:
U t + F x =0, (1)
with
U= ρ m e   and  F= m m2ρ+γ-1 e-m22 ρ meρ+mργ-1 e-m22ρ , (2)
where ρ is the density, m is the momentum, e is the specific total energy and γ is the (constant) ratio of specific heats. The pressure p is given by
p=γ-1 e-ρu22 , (3)
where u=m/ρ is the velocity.
The function calculates an approximation to the numerical flux function FUL,UR=FU*UL,UR, where U=UL and U=UR are the left and right solution values, and U*UL,UR is the intermediate state ω0 arising from the similarity solution Uy,t=ωy/t of the Riemann problem defined by
U t + F y =0, (4)
with U and F as in (2), and initial piecewise constant values U=UL for y<0 and U=UR for y>0. The spatial domain is -<y<, where y=0 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:     uleft[3]const doubleInput
On entry: uleft[i-1] must contain the left value of the component Ui, for i=1,2,3. That is, uleft[0] must contain the left value of ρ, uleft[1] must contain the left value of m and uleft[2] must contain the left value of e.
Constraints:
  • uleft[0]0.0;
  • Left pressure, pl0.0, where pl is calculated using (3).
2:     uright[3]const doubleInput
On entry: uright[i-1] must contain the right value of the component Ui, for i=1,2,3. That is, uright[0] must contain the right value of ρ, uright[1] must contain the right value of m and uright[2] must contain the right value of e.
Constraints:
  • uright[0]0.0;
  • Right pressure, pr0.0, where pr is calculated using (3).
3:     gammadoubleInput
On entry: the ratio of specific heats, γ.
Constraint: gamma>0.0.
4:     flux[3]doubleOutput
On exit: flux[i-1] contains the numerical flux component F^i, for i=1,2,3.
5:     savedNag_D03_Save *Communication Structure
saved may contain data concerning the computation required by nag_pde_parab_1d_euler_hll (d03pwc) as passed through to numflx from one of the integrator functions nag_pde_parab_1d_cd (d03pfc), nag_pde_parab_1d_cd_ode (d03plc) or nag_pde_parab_1d_cd_ode_remesh (d03psc). You should not change the components of saved.
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument value 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.
NE_REAL
Left pressure value pl<0.0: pl=value.
On entry, gamma=value.
Constraint: gamma>0.0.
On entry, uleft[0]=value.
Constraint: uleft[0]0.0.
On entry, uright[0]=value.
Constraint: uright[0]0.0.
Right pressure value pr<0.0: pr=value.

7  Accuracy

nag_pde_parab_1d_euler_hll (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  Further Comments

nag_pde_parab_1d_euler_hll (d03pwc) must only be used to calculate the numerical flux for the Euler equations in exactly the form given by (2), with uleft[i-1] and uright[i-1] containing the left and right values of ρ,m and e, for i=1,2,3, respectively. The time taken is independent of the input arguments.

9  Example

This example uses nag_pde_parab_1d_cd_ode (d03plc) and nag_pde_parab_1d_euler_hll (d03pwc) to solve the Euler equations in the domain 0x1 for 0<t0.035 with initial conditions for the primitive variables ρx,t, ux,t and px,t given by
ρx,0=5.99924, ux,0=-19.5975, px,0=460.894,   for ​x<0.5, ρx,0=5.99242, ux,0=-6.19633, px,0=046.095,   for ​x>0.5.
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.

9.1  Program Text

Program Text (d03pwce.c)

9.2  Program Data

Program Data (d03pwce.d)

9.3  Program Results

Program Results (d03pwce.r)

Produced by GNUPLOT 4.4 patchlevel 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 35 Example Program Euler Equation Solution Showing Collision of Two Strong Shocks DENSITY u(x,t) Time x
Produced by GNUPLOT 4.4 patchlevel 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 10 15 20 Euler Equation Solution Showing Collision of Two Strong Shocks VELOCITY u(x,t) Time x
Produced by GNUPLOT 4.4 patchlevel 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 0.2 0.4 0.6 0.8 1 0 200 400 600 800 1000 1200 1400 1600 1800 Euler Equation Solution Showing Collision of Two Strong Shocks PRESSURE u(x,t) Time x

nag_pde_parab_1d_euler_hll (d03pwc) (PDF version)
d03 Chapter Contents
d03 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012