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SUMMARY In this paper, a moving mesh BGK scheme (MMBGK) for multi‐material flow computations is proposed. The basic idea of constructing the MMBGK is to couple the Lagrangian method, which tracks material interfaces and keeps the interfaces sharp, with a remapping‐free ALE‐type kinetic method within each single material region, where the kinetic method is based on the BGK (Bhatnagar–Gross–Krook) model. Within each single material region, a numerical flux formulation is developed on moving meshes from motion of microscope particles, and the mesh velocity is determined by requiring both mesh adaptation for accuracy and robustness, such that the grids are moving towards to the regions with high flow gradients in a way of diffusive mechanism (velocity) to adjust the distances between neighboring cells, thus increasing the numerical accuracy. To keep the sharpness of material interfaces, the Lagrangian velocity and flux are constructed at the interfaces only. Consequently, a BGK‐scheme‐based ALE‐type method (i.e., the MMBGK scheme) for multi‐material flows is constructed. Numerical examples in one and two dimensions are presented to illustrate the accuracy and robustness of the MMBGK scheme. Copyright © 2011 John Wiley & Sons, Ltd.

SUMMARYOn the basis of the work [P.‐H. Maire, R. Abgrall, J. Breil, J. Ovadia, SIAM J. Sci. Comput. 29 (2007), 1781–1824], we present an entropy fixed cell‐centered Lagrangian scheme for solving the Euler equations of compressible gas dynamics. The scheme uses the fully Lagrangian form of the gas dynamics equations, in which the primary variables are cell‐centered. And using the nodal solver, we obtain the nodal viscous‐velocity, viscous‐pressures, antidissipation velocity, and antidissipation pressures of each node. The final nodal velocity is computed as a weighted sum of viscous‐velocity and antidissipation velocity, so do nodal pressures, whereas these weights are calculated through the total entropy conservation for isentropic flows. Consequently, the constructed scheme is conservative in mass, momentum, and energy; preserves entropy for isentropic flows, and satisfies a local entropy inequality for nonisentropic flows. One‐ and two‐dimensional numerical examples are presented to demonstrate theoretical analysis and performance of the scheme in terms of accuracy and robustness.Copyright © 2013 John Wiley & Sons, Ltd.

An exact conservative remapping scheme requires overlaps between two tessellations and a reconstruction scheme on the old cells (Lagrangian mesh). While the are intensive discussion on reconstruction schemes, there are relative sparse discussion on how to calculate overlaps of two grids. Computing the exact overlaps was believed complicated and often be avoided. This paper introduces the mathematical formulation of such a problem and tools to solve the problem. We propose methods to calculate the overlaps of two dismissable general quadrilateral mesh of the same logically structure in a planar domain. The quadrilateral polygon intersection problem is reduced to as a problem that how an edge in a new mesh intersects with a local frame which consists at most 7 connected edges in the old mesh. As such, locality of the method is persevered. The alternative direction technique is applied to reduce the dimension of the searching space. It reduces more than 256 possible intersections between a new cell with the old tessellation to 34 (17 when considering symmetry) programmable intersections between an edge and an local frame whenever the intersection between the old and new cell does not degenerate. At the same time, we shall how the computational amount of the overlaps depends on the underlying problem in terms of singular intersection points. A simple and detailed classification on the type of overlaps is presented, according to classification, degeneracy of an overlap can be easily identified.