A new flux‐limiting approach–based kinetic scheme for the Euler equations of gas dynamics

• Published in: International journal for numerical methods in fluids Vol. 90; no. 1; pp. 22 - 56
• Main Authors: Kumar, Raushan, Dass, Anoop K.
• Format: Journal Article
• Language: English
• Published: Bognor Regis Wiley Subscription Services, Inc 10.05.2019
• Subjects: Accuracy; Boltzmann transport equation; Collisionless Boltzmann equation; compressible flow; Dynamics; Euler equations; Euler-Lagrange equation; Eulers equations; Fluctuations; Fluxes; flux‐limiting approach; Gas dynamics; Kinetic theory; Methods; Resolution; Robustness (mathematics)
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.4713

Summary This paper proposes a new kinetic‐theory‐based high‐resolution scheme for the Euler equations of gas dynamics. The scheme uses the well‐known connection that the Euler equations are suitable moments of the collisionless Boltzmann equation of kinetic theory. The collisionless Boltzmann equation is discretized using Sweby's flux‐limited method and the moment of this Boltzmann level formulation gives a Euler level scheme. It is demonstrated how conventional limiters and an extremum‐preserving limiter can be adapted for use in the scheme to achieve a desired effect. A simple total variation diminishing criteria relaxing parameter results in improving the resolution of the discontinuities in a significant way. A 1D scheme is formulated first and an extension to 2D on Cartesian meshes is carried out next. Accuracy analysis suggests that the scheme achieves between first‐ and second‐order accuracy as is expected for any second‐order flux‐limited method. The simplicity and the explicit form of the conservative numerical fluxes add to the efficiency of the scheme. Several standard 1D and 2D test problems are solved to demonstrate the robustness and accuracy. The collisionless Boltzmann equation discretized using Sweby's flux‐limiting procedure gives a high‐resolution scheme for the Euler equations of gas dynamics when moment is taken. This new formulation called FLKS allows incorporation of conventional, and special purpose limiters as demonstrated by adapting an extremum‐preserving limiter to the scheme. A simple total‐variation‐diminishing (TVD) relaxing parameter introduced improves discontinuity resolution significantly, and the performance of the scheme for several test problems demonstrates entropy‐condition satisfaction, positivity, robustness, and the ability to capture nongrid‐aligned discontinuities sharply.