The large reconstruction stencil has been the major bottleneck problem in developing high order finite volume schemes on unstructured grids. This paper presents a compact reconstruction procedure for arbitrarily high order finite volume method on unstructured grids to overcome this shortcoming. In this procedure, a set of constitutive relations are constructed by requiring the reconstruction polynomial and its derivatives on the control volume of interest to conserve their averages on face-neighboring cells. These relations result in an over-determined linear equation system which is solved using the method of least-squares. In one-dimensional case, the linear equation system can be reduced to a block-tridiagonal system and solved directly; while in two-dimensional case, the linear equation system must be solved iteratively. Implicit time integration schemes are coupled with the implicit multi-dimensional reconstruction to achieve high computational efficiency. The basic formulations of the reconstruction are presented and a Fourier analysis is performed to study the spectral and stability properties. The WBAP limiter based on the secondary reconstruction is used to suppress non-physical oscillations near discontinuities while achieve high order accuracy in smooth regions of the solution. Numerical results demonstrate the method's high order accuracy, efficiency and shock capturing capability.
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