The Solver of Boltzmann equation on unstructured spatial grids

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The purpose of this work is to develop a universal computer program (solver) which solves kinetic Boltzmann equation for simulations of rarefied gas flows in complexly shaped devices. The structure of the solver is described in details. Its efficiency is demonstrated on an example of calculations of a modern many tubes Knudsen pump. The kinetic Boltzmann equation is solved by finite-difference method on discrete grid in spatial and velocity spaces. The differential advection operator is approximated by finite difference method. The calculation of the collision integral is based on the conservative projection method.

In the developed computational program the unstructured spatial mesh is generated using GMSH and may include prisms, tetrahedrons, hexahedrons and pyramids. The mesh is denser in areas of flow with large gradients of gas parameters. A three-dimensional velocity grid consists of cubic cells of equal volume.

A huge amount of calculations requires effective parallelization of the algorithm which is implemented in the program with the use of Message Passing Interface (MPI) technology. An information transfer from one node to another is implemented as a kind of boundary condition. As a result, every MPI node contains the information about only its part of the grid.

The main result of the work is presented in the graph of pressure difference in 2 reservoirs connected by a multitube Knudsen pump from Knudsen number. This characteristic of the Knudsen pump obtained by numerical methods shows the quality of the pump. Distributions of pressure, temperature and gas concentration in a steady state inside the pump and the reservoirs are presented as well.

The correctness of the solver is checked using two special test solutions of more simple boundary problems — test with temperature distribution between 2 planes with different temperatures and test with conservation of total gas mass.

The correctness of the obtained data for multitube Knudsen pump is checked using denser spatial and velocity grids, using more collisions in collision integral per time step.

Keywords: Boltzmann equation, Knudsen effect, unstructured grid, micropump, distribution function, collision integral, projection method
Citation in English: Gasparyan M.M., Samonov A.S., Sazykina T.A., Ostapov E.L., Sakmarov A.V., Shahatarov O.K. The Solver of Boltzmann equation on unstructured spatial grids // Computer Research and Modeling, 2019, vol. 11, no. 3, pp. 427-447
Citation in English: Gasparyan M.M., Samonov A.S., Sazykina T.A., Ostapov E.L., Sakmarov A.V., Shahatarov O.K. The Solver of Boltzmann equation on unstructured spatial grids // Computer Research and Modeling, 2019, vol. 11, no. 3, pp. 427-447
DOI: 10.20537/2076-7633-2019-11-3-427-447
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