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A variety of numerical schemes operates on , that is a pair of meshes of the same computational domain whose interior nodes (in 1D), edges (in 2D) and faces (in 3D) do not coincide. While the shape of a staggered grid is canonical on an uniformly refined Cartesian mesh it becomes more complicated for an underlying adaptively refined grid, in particular in higher spatial dimensions. Here we present both a construction technique for staggered exploiting the structure of the adaptively refined Cartesian in 2D and 3D, and discuss the necessary modifications of a standard Finite Volume scheme which is originally formulated on uniform meshes.

This report summarizes our works on hyperbolic systems of first-order partial differential equations with source terms. We discuss the introduction of our structural stability and entropy dissipation conditions for initial or initial-boundary value problems. For initial value problems, several systematic results are reviewed. These include the non-existence of (linearly stable) relaxation approximations to non-strongly hyperbolic systems of equations, the justification of the formal zero relaxation limit, the existence of relaxation shock profiles, and the existence of global smooth solutions for balance laws.