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We present a class of hybrid FD-FV (finite difference and finite volume)methods for solving general hyperbolic conservation laws written in first-orderform. The presentation focuses on one- and two-dimensional Cartesian grids;however, the generalization to higher dimensions is straightforward. Thesemethods use both cell-averaged values and nodal values as dependent variablesto discretize the governing partial differential equation (PDE) in space, andthey are combined with method of lines for integration in time. This frameworkis absent of any Riemann solvers while it achieves numerical conservationnaturally. This paper focuses on the accuracy and linear stability of theproposed FD-FV methods, thus we suppose in addition that the solutions aresufficiently smooth. In particular, we prove that the spatial-order of theFD-FV method is typically one-order higher than that of the discretedifferential operator, which is involved in the construction of the method. Inaddition, the methods are linearly stable subjected to a Courant-Friedrich-Lewycondition when appropriate time-integrators are used. The numerical performanceof the methods is assessed by a number of benchmark problems in one and twodimensions. These examples include the linear advection equation, nonlinearEuler equations, the solid dynamics problem for linear elastic orthotropicmaterials, and the Buckley-Leverett equation.

A hybrid FD-FV method for first-order hyperbolic conservation laws on Cartesian grids: The smooth problem case