$K$ | $L$ | $|{I_{K, L}}|$ |
3 | 3 | 64 |
3 | 5 | 144 |
3 | 7 | 256 |
In this paper, we present an application of a Galerkin-Petrov method to the spatially one-dimensional Boltzmann equation. The three-dimensional velocity space is discretised by a spectral method. The space of the test functions is spanned by polynomials, which includes the collision invariants. This automatically insures the exact conservation of mass, momentum and energy. The resulting system of hyperbolic PDEs is solved with a finite volume method. We illustrate our method with two standard tests, namely the Fourier and the Sod shock tube problems. Our results are validated with the help of a stochastic particle method.
Citation: |
Figure 7. Comparison of different sets of basis functions for $ {\rm{Kn}} = 1.0 $. $ K $ is set to 3; $ L $ takes the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $ N_x = 512 $ and $ \tau $ chosen as in equation (17). Figure 7a shows the density at the time $ t_f $, whereas in Figure 7b, the temperature at the time $ t_f $ is shown
Figure 8. Comparison of different sets of basis functions for $ {\rm{Kn}} = 1.0 $ near the walls. $ K $ is set to 3; $ L $ takes the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $ N_x = 512 $ and $ \tau $ chosen as in equation (17). Figures 8a and 8b show the density at the time $ t_f $ near the left and the right wall, respectively. Figures 8c and 8d show the temperature at the time $ t_f $ near the left and the right wall, respectively
Figure 9. Comparison of different sets of basis functions for $ {\rm{Kn}} = 0.25 $. $ K $ is set to 3; $ L $ takes the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $ N_x = 512 $ and $ \tau $ chosen as in equation (17). Figure 9a shows the density at the time $ t_f $, whereas in Figure 9b, the temperature at the time $ t_f $ is shown
Figure 10. Comparison of different sets of basis functions for ${\rm{Kn}} = 0.25$ near the walls. $K$ is set to 3; $L$ takes the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $N_x = 512$ and $\tau$ chosen as in equation (17). Figures 10a and 10b show the density at the time $t_f$ near the left and the right wall, respectively. Figures 10c and 10d show the temperature at the time $t_f$ near the left and the right wall, respectively
Figure 11. Comparison of different sets of basis functions for $ {\rm{Kn}} = 0.025 $. $ K $ is set to 3; $ L $ the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $ N_x = 512 $ and $ \tau $ chosen as in equation (17). Figure 11a shows the density at the time $ t_f $, whereas in Figure 11b, the temperature at the time $ t_f $ is shown
Figure 12. Comparison of different sets of basis functions for ${\rm{Kn}} = 0.025$ near the walls. $K$ is set to 3; $L$ takes the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $N_x = 512$ and $\tau$ chosen as in equation (17). Figures 12a and 12b show the density at the time $t_f$ near the left and the right wall, respectively. Figures 12c and 12d show the temperature at the time $t_f$ near the left and the right wall, respectively
Table 1.
Number of basis and test functions for different choices of parameters. The set
$K$ | $L$ | $|{I_{K, L}}|$ |
3 | 3 | 64 |
3 | 5 | 144 |
3 | 7 | 256 |
Table 2.
Relative
basis functions | $L^2$-error |
$K=1, L=2$ | $5.666437\cdot 10^{-2}$ |
$K=2, L=4$ | $6.201405\cdot 10^{-3}$ |
$K=3, L=6$ | $5.937974\cdot 10^{-4}$ |
$K=4, L=8$ | $7.978434\cdot 10^{-5}$ |
Table 3.
Relative error of mass conservation for
Spatial cells | ||||
64 | 128 | 256 | 512 | |
$K=1, L=2$ | $9.886570 \cdot 10^{-2}$ | $9.875780 \cdot 10^{-2}$ | $9.874720 \cdot 10^{-2}$ | $9.874130 \cdot 10^{-2}$ |
$K=2, L=4$ | $4.842620 \cdot 10^{-2}$ | $4.839010 \cdot 10^{-2}$ | $4.835910 \cdot 10^{-2}$ | $4.834270 \cdot 10^{-2}$ |
$K=3, L=6$ | $2.402240 \cdot 10^{-2}$ | $2.379990 \cdot 10^{-2}$ | $2.367980 \cdot 10^{-2}$ | $2.361850 \cdot 10^{-2}$ |
$K=4, L=8$ | $1.729450 \cdot 10^{-2}$ | $1.706360 \cdot 10^{-2}$ | $1.694290 \cdot 10^{-2}$ | $1.688380 \cdot 10^{-2}$ |
Table 4.
Relative error of mass conservation for
Spatial cells | ||||
64 | 128 | 256 | 512 | |
$K=1, L=2$ | $3.226070 \cdot 10^{-2}$ | $3.226070 \cdot 10^{-2}$ | $3.226070 \cdot 10^{-2}$ | $3.226070 \cdot 10^{-2}$ |
$K=2, L=4$ | $1.175180 \cdot 10^{-2}$ | $1.176170 \cdot 10^{-2}$ | $1.174380 \cdot 10^{-2}$ | $1.172670 \cdot 10^{-2}$ |
$K=3, L=6$ | $2.435140 \cdot 10^{-3}$ | $2.343420 \cdot 10^{-3}$ | $2.277850 \cdot 10^{-3}$ | $2.233940 \cdot 10^{-3}$ |
$K=4, L=8$ | $2.415870 \cdot 10^{-3}$ | $2.309910 \cdot 10^{-3}$ | $2.242550 \cdot 10^{-3}$ | $2.197230 \cdot 10^{-3}$ |
Table 5.
Relative error of mass conservation for
Spatial cells | ||||
64 | 128 | 256 | 512 | |
$K=1, L=2$ | $3.226070 \cdot 10^{-2}$ | $3.226070 \cdot 10^{-2}$ | $3.226070 \cdot 10^{-2}$ | $3.226070 \cdot 10^{-2}$ |
$K=2, L=4$ | $7.244610 \cdot 10^{-3}$ | $7.324720 \cdot 10^{-3}$ | $7.358720 \cdot 10^{-3}$ | $7.393080 \cdot 10^{-3}$ |
$K=3, L=6$ | $8.215370 \cdot 10^{-4}$ | $8.215370 \cdot 10^{-4}$ | $8.215370 \cdot 10^{-4}$ | $8.215370 \cdot 10^{-4}$ |
$K=4, L=8$ | $2.432010 \cdot 10^{-4}$ | $2.791020 \cdot 10^{-4}$ | $2.729200 \cdot 10^{-4}$ | $2.519770 \cdot 10^{-4}$ |
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Generalised spectrum of
Approximation of the solution by a piecewise constant function. To fulfil boundary conditions, the dashed cells, called ghost cells, are added to the discretisation
Sketch of the one-dimensional Fourier problem. We seek the particle density function along the axis labeled by
Course of the total mass for
Sketch of the initial situation of the shock tube problem. Two areas of same bulk velocities and temperatures but different densities are separated by diaphragm (dashed line), which is removed at
Contour plot of the final particle density function for
Comparison of different sets of basis functions for
Comparison of different sets of basis functions for
Comparison of different sets of basis functions for
Comparison of different sets of basis functions for
Comparison of different sets of basis functions for
Comparison of different sets of basis functions for
Comparison of the final particle density function for
Numerical solution of the shock tube problem at
Numerical solution of the shock tube problem at
Numerical solution of the shock tube problem at $t_f$ obtained with DSMC and the Galerkin--Petrov method for ${\rm{Kn}} = 0.01$. The exact solution of the Euler equations is shown by dashed lines
Particle density functions for
Particle density functions for ${\rm{Kn}} = 0.1$ when the shock discontinuity reaches the third evaluation point
Particle density functions for ${\rm{Kn}} = 0.1$ when the contact discontinuity reaches the third evaluation point at time $t_f$
Particle density functions for
Particle density functions for ${\rm{Kn}} = 0.01$ when the shock discontinuity reaches the third evaluation point
Particle density functions for ${\rm{Kn}} = 0.01$ when the contact discontinuity reaches the third evaluation point at time $t_f$
Particle density functions for ${\rm{Kn}} = 0.001$ shortly after the diaphragm is removed
Particle density functions for ${\rm{Kn}} = 0.001$ when the shock discontinuity reaches the third evaluation point
Particle density functions for ${\rm{Kn}} = 0.001$ when the contact discontinuity reaches the third evaluation point at time $t_f$