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Abstract: We present a new time-accurate algorithm for the explicit numerical integration of the compressible Euler equations of gas dynamics. This technique is based on the discrete-event simulation (DES) methodology for nonlinear flux-conservative PDEs [Y.A. Omelchenko, H. Karimabadi, Self-adaptive time integration of flux-conservative equations with sources, J. Comput. Phys. 216 (1) (2006) 179–194]. DES enables adaptive distribution of CPU resources in accordance with local time scales of the underlying numerical solution. It distinctly stands apart from multiple (local) time-stepping algorithms in that it requires neither selecting a global synchronization time step nor pre-determining a sequence of time-integration operations for individual parts of a heterogeneous numerical system. In this paper we extend the DES methodology in three important directions: (i) we apply DES to a system of coupled gas dynamics equations discretized via a central-upwind scheme [A. Kurganov, E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations, J. Comput. Phys. 160 (2000) 241–282; A. Kurganov, S. Noelle, G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations, SIAM J. Sci. Comput. 23 (3) (2001) 707–740]; (ii) we introduce a new Preemptive Event Processing (PEP) technique, which automatically enforces synchronous execution of events with sufficiently close update times; (iii) we significantly improve the accuracy of the previous algorithm [Y.A. Omelchenko, H. Karimabadi, Self-adaptive time integration of flux-conservative equations with sources, J. Comput. Phys. 216 (1) (2006) 179–194] by applying locally second-order-in-time flux-conserving corrections to the solution obtained with the forward Euler scheme. The performance of the new technique is demonstrated in a series of one-dimensional gas dynamics test problems by comparing numerical solutions obtained in event-driven and equivalent time-stepping simulations. [Copyright &y& Elsevier]

The present paper highlights the importance of management of the discontinuous reconstruction in the kinetic schemes for gasdynamic equation systems. Firstly, it is revealed by the analysis of the gas kinetic-BGK scheme [JCP 171 (2001) 289] that a continuous reconstruction created from a discontinuous one is a key to the successful kinetic schemes. When it is applied to a well-resolved region, the numerical flux that takes account of the collision effect correctly becomes Lax–Wendroff-like. When it is applied to an unresolved region, such as a shock layer, an appreciable numerical dissipation, which contributes to the suppression of spurious oscillations, is produced. Secondly, new kinetic schemes for the compressible Navier–Stokes (Euler) equations are developed by using the key. The numerical flux of one of the schemes is computed by using the splitting algorithm, where the effect of the molecular collision is directly taken into account and the undesirable error of the splitting algorithm in the case where the time step is much larger than the mean free time is avoided by a simple modification of the initial data. Although a discontinuous reconstruction is employed in the approximation of the initial data, the continuity is automatically taken into account in the dominant part of the numerical flux. The other schemes are the extensions of the Lax–Wendroff-type scheme to the case of the key reconstruction. Thirdly, the performance of these schemes is tested. It is demonstrated that they work as shock capturing schemes and yield fine boundary-layer profiles with a reasonable number of cells, such as 10 cells in the layer. [Copyright &y& Elsevier]