NAG Library Function Document

nag_pde_parab_1d_euler_hll (d03pwc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

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:

\frac{\partial U}{\partial t} + \frac{\partial F}{\partial x} = 0,

(1)

with

U = [\begin{matrix} ρ \\ m \\ e \end{matrix}] and F = [\begin{matrix} m \\ \frac{m^{2}}{ρ} + (γ - 1) (e - \frac{m^{2}}{2 ρ}) \\ \frac{m e}{ρ} + \frac{m}{ρ} (γ - 1) (e - \frac{m^{2}}{2 ρ}) \end{matrix}],

(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 - \frac{ρ u^{2}}{2}),

(3)

where

u = m / ρ

is the velocity.

The function calculates an approximation to the numerical flux function

F (U_{L}, U_{R}) = F (U^{*} (U_{L}, U_{R}))

, where

U = U_{L}

and

U = U_{R}

are the left and right solution values, and

U^{*} (U_{L}, U_{R})

is the intermediate state

ω (0)

arising from the similarity solution

U (y, t) = ω (y / t)

of the Riemann problem defined by

\frac{\partial U}{\partial t} + \frac{\partial F}{\partial y} = 0,

(4)

with

U

and

F

as in (2), and initial piecewise constant values

U = U_{L}

for

y < 0

and

U = U_{R}

for

y > 0

. The spatial domain is

- \infty < y < \infty

, 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

U_{i}

, 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:

$uleft [0] \geq 0.0$ ;
Left pressure, $pl \geq 0.0$ , where $pl$ is calculated using (3).

2: uright[ $3$ ] – const doubleInput

On entry:

uright [i - 1]

must contain the right value of the component

U_{i}

, 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:

$uright [0] \geq 0.0$ ;
Right pressure, $pr \geq 0.0$ , where $pr$ is calculated using (3).

3: gamma – doubleInput

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

{\hat{F}}_{i}

, for

i = 1, 2, 3

5: saved – Nag_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: fail – NagError *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: $uleft [0] \geq 0.0$ .
On entry, $uright [0] = ⟨value⟩$ .
Constraint: $uright [0] \geq 0.0$ .
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 Parallelism and Performance

Not applicable.

9 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.

10 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

0 \leq x \leq 1

for

0 < t \leq 0.035

with initial conditions for the primitive variables

ρ (x, t)

u (x, t)

and

p (x, t)

given by

\begin{array}{llcrlcrl} ρ (x, 0) = 5.99924, & u (x, 0) = 19.5975, & p (x, 0) = 460.894, & for ​ x < 0.5, \\ ρ (x, 0) = 5.99242, & u (x, 0) = - 6.19633, & p (x, 0) = 46.095, & for ​ x > 0.5 . \end{array}

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.

NAG Library Function Documentnag_pde_parab_1d_euler_hll (d03pwc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_pde_parab_1d_euler_hll (d03pwc)