The purpose of this book is to lay out a mathematical framework for the
convergence and error analysis of the finite volume method for the
discretization of hyperbolic conservation laws on manifolds. Finite
Volume Method (FVM) is a discretization approach for the numerical
simulation of a wide variety physical processes described by
conservation law systems. It is extensively employed in fluid mechanics,
meteorology, heat and mass transfer, electromagnetic, models of
biological processes and many other engineering applications formed by
conservative systems. In this book, from one point of view, we provide a
brief description for the convergence of the FVM by approaches based on
metric and differential forms. The latter can be viewed as a
generalization of the formulation and convergence of the method for
general conservation laws on curved manifolds. On the other hand, we
carried over the error estimate for FVM that is established for the
Euclidean setting to the curved manifolds and obtained an expected rate
of error in the L1-norm.