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A direct Arbitrary Lagrangian Eulerian (ALE) formulation using multi-moment finite volume method for viscous compressible flows on unstructured moving grids has been developed and presented in this paper. High-order polynomials are reconstructed over a compact mesh stencil by making use of both volume integrated average value (VIA) and point values (PV) at the vertices of mesh cells which change in time. By formulating the governing equation in ALE integral form and differential form, the computational variables, VIA and PV, are updated, respectively, by a finite volume scheme and a point-wise discretization using multi-moments (VIA and PV). A simple and efficient formulation is derived for moving mesh which satisfies the geometrical conservation law. In the computations involving moving-body, the PVs of velocity at the vertices of cells adjacent to the body surface are coincided with the motion of the body, which eliminates the numerical approximation to find the cell boundary values on the body surface in conventional finite volume method. Numerical tests are presented to verify the present method as a high-order ALE formulation for compressible flows on both stationary and moving grids.

In the present article, we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with an appropriate predictor−corrector method to achieve higher resolutions. The underlying finite volume scheme is conservative, and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed; thus, unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according to the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.