Picture child's feet next to pens, pencils and paper

Open Access research that is helping to improve educational outcomes for children

Strathprints makes available scholarly Open Access content by researchers in the School of Education, including those researching educational and social practices in curricular subjects. Research in this area seeks to understand the complex influences that increase curricula capacity and engagement by studying how curriculum practices relate to cultural, intellectual and social practices in and out of schools and nurseries.

Research at the School of Education also spans a number of other areas, including inclusive pedagogy, philosophy of education, health and wellbeing within health-related aspects of education (e.g. physical education and sport pedagogy, autism and technology, counselling education, and pedagogies for mental and emotional health), languages education, and other areas.

Explore Open Access education research. Or explore all of Strathclyde's Open Access research...

Deterministic numerical solutions of the Boltzmann equation using the fast spectral method

Wu, Lei and White, Craig and Scanlon, Thomas and Reese, Jason and Zhang, Yonghao (2013) Deterministic numerical solutions of the Boltzmann equation using the fast spectral method. Journal of Computational Physics, 250. pp. 27-52. ISSN 0021-9991

[img]
Preview
PDF
White_C_et_al_Deterministic_numerical_solutions_of_the_Boltzmann_equation_using_the_fast_spectral_method_May_2013.pdf
Preprint

Download (1MB) | Preview

Abstract

The Boltzmann equation describes the dynamics of rarefied gas flows, but the multidimensional nature of its collision operator poses a real challenge for its numerical solution. In this paper, the fast spectral method [36], originally developed by Mouhot and Pareschi for the numerical approximation of the collision operator, is extended to deal with other collision kernels, such as those corresponding to the soft, Lennard–Jones, and rigid attracting potentials. The accuracy of the fast spectral method is checked by comparing our numerical solutions of the space-homogeneous Boltzmann equation with the exact Bobylev–Krook–Wu solutions for a gas of Maxwell molecules. It is found that the accuracy is improved by replacing the trapezoidal rule with Gauss–Legendre quadrature in the calculation of the kernel mode, and the conservation of momentum and energy are ensured by the Lagrangian multiplier method without loss of spectral accuracy. The relax-to-equilibrium processes of different collision kernels with the same value of shear viscosity are then compared; the numerical results indicate that different forms of the collision kernels can be used as long as the shear viscosity (not only the value, but also its temperature dependence) is recovered. An iteration scheme is employed to obtain stationary solutions of the space-inhomogeneous Boltzmann equation, where the numerical errors decay exponentially. Four classical benchmarking problems are investigated: the normal shock wave, and the planar Fourier/Couette/force-driven Poiseuille flows. For normal shock waves, our numerical results are compared with a finite difference solution of the Boltzmann equation for hard sphere molecules, experimental data, and molecular dynamics simulation of argon using the realistic Lennard–Jones potential. For planar Fourier/Couette/force-driven Poiseuille flows, our results are compared with the direct simulation Monte Carlo method. Excellent agreements are observed in all test cases, demonstrating the merit of the fast spectral method as a computationally efficient method for rarefied gas dynamics.