We propose a well-balanced stable generalized Riemann problem (GRP) scheme for the shallow water equations with irregular bottom topography based on moving, adaptive, unstructured, triangular meshes. In order to stabilize the computations near equilibria, we use the Rankine-Hugoniot condition to remove a singularity from the GRP solver. Moreover, we develop a remapping onto the new mesh (after grid movement) based on equilibrium variables. This, together with the already established techniques, guarantees the well-balancing. Numerical tests show the accuracy, efficiency, and robustness of the GRP moving mesh method: lake at rest solutions are preserved even when the underlying mesh is moving (e.g., mesh points are moved to regions of steep gradients), and various comparisons with fixed coarse and fine meshes demonstrate high resolution at relatively low cost. Copyright (c) 2013 John Wiley & Sons, Ltd.
This article uses the Z -transform to develop a method for solving the linearised multidimensional discrete-time systems, which can be used to discuss the effects of policies on economy (including the welfare gains and initial effects on economy) raised by multi-sector perfect-foresight-discrete-time models. Our method is not restricted to the dimension of the dynamic system, and it can not only analyse the effect of permanent policy change on the economy but also can be used to analyse the effect of temporal policy change on the economy. As an application example, we analyse the effects of fiscal policy on the initial economy and social welfare in the discrete-time Uzawa--Lucas model.