Métodos miméticos aplicados em equações diferenciais parciais

Abstract

In this work we will present the Castillo-Grone mimetic method that will be applied to a hyperbolic problem over an uni dimensional, homogeneous and staggered grid. This method is based upon the Finite Differences Method in grid discretization, but stand apart in how the differential operator is built, where the continuum mechanics is preserved. This method is applicable in situations where the standard Finite Difference Method cannot be applied satisfactorily and neither allows a correct handling of boundary conditions. To validate this method we will use an elastic string model with fixed boundaries satisfying the Dirichlet and Neumann boundary conditions. It will be also studied the model behavior while changing the grid resolution in time and space. For the time-space discretization it will be used a Forward-Time Central-Space (FTCS) model. Given the oscillatory behavior, it will be used a Leapfrog model in the time discretization. An analysis will be done, comparing the Castillo-Grone Mimetic Method, the Finite Difference and the analytical solution. In this analysis the Castillo-Grone method has shown slightly better results in most of the analyzed cases. Besides that we also present a computational formulation with algorithms and a source code to be used as reference.