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A spectral difference lattice Boltzmann method for solution of inviscid compressible flows on structured grids. (English) Zbl 1357.76056
Summary: In this work, a spectral difference lattice Boltzmann method (SDLBM) is developed and applied for an accurate simulation of two-dimensional inviscid compressible flows on structured grids. The compressible form of the discrete Boltzmann-BGK equation is used in which multiple particle speeds have to be employed to correctly model the compressibility in a thermal fluid. Here, the 2D compressible Lattice Boltzmann (LB) model proposed by Watari is applied. The spectral difference (SD) method is implemented for the solution of the LB equation in which the particle distribution function is stored at the solution points while the fluxes are computed at the flux points for calculating the particle distribution function. For time accurate solutions, the fourth-order Runge-Kutta scheme is used to discretize the temporal term in the LB equation. Note that the procedure of implementing the SD method to solve the LB equation is nearly the same as that procedure developed in the literature for the solution of the Euler equations. The accuracy and robustness of the present solution methodology are demonstrated by simulating different benchmark compressible flow problems. A sensitivity study is also conducted to evaluate the effects of the numerical parameters and the grid size/distribution on the accuracy and performance of the solution. At first, three inviscid compressible flow problems, namely, the stationary isentropic vortex, the shock tube and the shock-vortex interaction are solved by using the SDLBM to demonstrate the accuracy and robustness of the present solution methodology. Results computed for these test cases are in good agreement with the analytical and the available numerical solutions. To more assess the accuracy and robustness of the SDLBM, a third-order finite volume LBM (FVLBM) is also developed and the solutions obtained by these two methods for the test cases simulated are thoroughly compared with each other. It is demonstrated that the SDLBM is more accurate and robust compressible inviscid flow solver that can be used for evaluating the other compressible LB-based flow solvers. As the final problem, the supersonic flow past a bump is simulated to demonstrate the accuracy and robustness of the SDLBM in reaching to a steady-state solution.

##### MSC:
 76M22 Spectral methods applied to problems in fluid mechanics 76M28 Particle methods and lattice-gas methods 76N99 Compressible fluids and gas dynamics
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##### References:
  Bhatnagar, P.; Gross, E. P.; Krook, M., A model for collision processes in gases I: small amplitude processes in charged and neutral one-component systems, Phys. Rev. E, 94, 511-525, (1954) · Zbl 0055.23609  McNamara, G.; Zanetti, G., Use of the Boltzmann equation to simulate lattice-gas automata, Phys. Rev. Lett., 61, 2332-2335, (1988)  He, X.; Luo, L. S., Lattice Boltzmann model for the incompressible Navier-Stokes equation, J. Stat. Phys., 88, 927-944, (1997) · Zbl 0939.82042  Sun, C.; Hsu, A., Multi-level lattice Boltzmann model on square lattice for compressible flows, Comput. & Fluids, 33, 1363-1385, (2004) · Zbl 1113.76435  Kataoka, T.; Tsutahara, M., Lattice Boltzmann method for the compressible Euler equations, Phys. Rev. E, 69, (2004)  Kataoka, T.; Tsutahara, M., Lattice Boltzmann model for the compressible Navier-Stokes equations with flexible specific-heat ratio, Phys. Rev. E, 69, 035701(R), (2004)  Watari, M., Finite difference lattice Boltzmann method with arbitrary specific heat ratio applicable to supersonic flow simulations, Physica A, 382, 502-522, (2007)  Tsutahara, M.; Kataoka, T.; Shikata, K.; Takada, N., New model and scheme for compressible fluids of the finite difference lattice Boltzmann method and direct simulations of aerodynamic sound, Comput. & Fluids, 37, 79-89, (2008) · Zbl 1194.76237  Yan, G.; Zhang, J.; Liu, Y.; Dong, Y., A multi-energy-level lattice Boltzmann model for the compressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids, 55, 41-56, (2007) · Zbl 1119.76049  Qu, K.; Shu, C.; Chew, Y. T., Alternative method to construct equilibrium distribution functions in lattice-Boltzmann method simulation of inviscid compressible flows at high Mach number, Phys. Rev. E, 75, (2007)  Li, Q.; He, Y. L.; Wang, Y.; Tao, W. Q., Coupled double-distribution-function lattice Boltzmann method for the compressible Navier-Stokes equations, Phys. Rev. E, 76, (2007)  Yang, L. M.; Shu, C.; Wu, J., Development and comparative studies of three non-free parameter lattice Boltzmann models for simulation of compressible flows, Adv. Appl. Math. Mech., 4, 454-472, (2012)  Qu, K.; Shu, C.; Chew, Y. T., Simulation of shock-wave propagation with finite volume lattice Boltzmann method, Internat. J. Modern Phys. C, 18, 447-454, (2007) · Zbl 1137.76463  Ji, C. Z.; Shu, C.; Zhao, N., A lattice Boltzmann method-based flux solver and its application to solve shock tube problem, Modern Phys. Lett. B, 23, 313-316, (2009) · Zbl 1419.76520  Qu, K.; Shu, C.; Chew, Y. T., Lattice Boltzmann and finite volume simulation of inviscid compressible flows with curved boundary, Adv. Appl. Math. Mech., 2, 573-586, (2010)  Watari, M.; Tsutahara, M., Supersonic flow simulations by a three-dimensional multispeed thermal model of the finite difference lattice Boltzmann method, Physica A, 364, 129-144, (2006)  Gan, Y.; Xu, A.; Zhang, G.; Yu, X.; Li, Y., Two-dimensional lattice Boltzmann model for compressible flows with high Mach number, Physica A, 387, 1721-1732, (2008)  So, R. M.; Fu, S. C.; Leung, R. C., Finite difference lattice Boltzmann method for compressible thermal fluids, AIAA J., 48, 1059-1071, (2010)  Hiraishi, M.; Tsutahara, M.; Leung, R. C.K., Numerical simulation of sound generation in a mixing layer by the finite difference lattice Boltzmann method, Comput. Math. Appl., 59, 2403-2410, (2010) · Zbl 1193.76110  He, Y. L.; Liu, Q.; Li, Q., Three-dimensional finite-difference lattice Boltzmann model and its application to inviscid compressible flows with shock waves, Physica A, 392, 4884-4896, (2013) · Zbl 1395.76050  Nejat, A.; Abdollahi, V., A critical study of the compressible lattice Boltzmann methods for Riemann problem, J. Sci. Comput., 54, 1-20, (2013) · Zbl 1426.76610  Shi, X.; Lin, J.; Yu, Z., Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element, Internat. J. Numer. Methods In Fluids, 42, 1249-1261, (2003) · Zbl 1033.76046  Min, M.; Lee, T., A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows, J. Comput. Phys., 230, 245-259, (2011) · Zbl 1427.76189  Uga, K. C.; Min, M.; Lee, T.; Fischer, P. F., Spectral-element discontinuous Galerkin lattice Boltzmann simulation of flow past two cylinders in tandem with an exponential time integrator, Comput. Math. Appl., 65, 239-251, (2013) · Zbl 1268.76049  Patel, S. S.; Min, M.; Uga, K. C.; Lee, T., A spectral-element discontinuous Galerkin lattice Boltzmann method for simulating natural convection heat transfer in a horizontal concentric annulus, Comput. & Fluids, 95, 197-209, (2014) · Zbl 1391.76628  Hejranfar, K.; Ezzatneshan, E., A high-order compact finite-difference lattice Boltzmann method for simulation of steady and unsteady incompressible flows, Internat. J. Numer. Methods Fluids, 75, 713-746, (2014)  Hejranfar, K.; Ezzatneshan, E., Implementation of a high-order compact finite-difference lattice Boltzmann method in generalized curvilinear coordinates, J. Comput. Phys., 267, 28-49, (2014) · Zbl 1349.76475  Hejranfar, K.; Hajihassanpour, M., Chebyshev collocation spectral lattice Boltzmann method for simulation of low-speed flows, Phys. Rev. E, 91, (2015) · Zbl 1390.76719  Liu, Y.; Vinokur, M.; Wang, Z. J., Spectral difference method for unstructured grids I: basic formulation, J. Comput. Phys., 216, 780-801, (2006) · Zbl 1097.65089  Wang, Z. J.; Liu, Y.; May, G.; Jameson, A., Spectral difference method for unstructured grids II: extension to the Euler equations, J. Sci. Comput., 32, 45-71, (2007) · Zbl 1151.76543  Sun, Y.; Wang, Z. J.; Liu, Y., High-order multidomain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids, Commun. Comput. Phys., 2, 310-333, (2007) · Zbl 1164.76360  Liang, C.; Jameson, A.; Wang, Z. J., Spectral difference method for compressible flow on unstructured grids with mixed elements, J. Comput. Phys., 228, 2847-2858, (2009) · Zbl 1159.76029  Wang, Z. J., Spectral (finite) volume method for conservation laws on unstructured grids basic formulation, J. Comput. Phys., 178, 210-251, (2002) · Zbl 0997.65115  Wang, Z. J.; Liu, Y., Extension of the spectral volume method to high-order boundary representation, J. Comput. Phys., 211, 154-178, (2006) · Zbl 1161.76536  Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comp., 52, 411-435, (1989) · Zbl 0662.65083  Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141, 199-224, (1998) · Zbl 0920.65059  Farrashkhalvat, M.; Miles, J. P., Basic structured grid generation: with an introduction to unstructured grid generation, (2003), Butterworth Heinemann Oxford  Hirsch, C., Internal and External Flows, vol. 1, (2007), John Wiley and Sons Oxford  Visbal, M. R.; Gaitonde, D. V., On the use of higher-order finite-difference schemes on curvilinear and deforming meshes, J. Comput. Phys., 181, 155-185, (2002) · Zbl 1008.65062  S. Premasuthan, C. Liang, A. Jameson, A Spectral Difference Method for Viscous Compressible Flows with Shocks, 19th AIAA Computational Fluid Dynamics, June 2009, Texas, USA.  S. Premasuthan, C. Liang, A. Jameson, Computation of flows with shocks using spectral difference scheme with artificial viscosity, in: 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, January 2010, Florida, USA.  Mahmoodi Darian, H.; Esfahanian, V.; Hejranfar, K., A shock-detecting sensor for filtering of high-order compact finite difference schemes, J. Comput. Phys., 230, 494-514, (2011) · Zbl 1283.76045  Huerta, A.; Casoni, E.; Peraire, J., A simple shock-capturing technique for high-order discontinuous Galerkin methods, Internat. J. Numer. Methods Fluids, 69, 1614-1632, (2012) · Zbl 1253.76058
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