Resolução de equações diferenciais parciais em malhas não-estruturadas via método dos volumes finitos utilizando um método de alta ordem

Abstract

The present work illustrates a method for numerical resolution of partial differential equations based on the finite volume method. This scheme was developed by Carl Olliver-Gooch and it has as main characteristic the reconstruction of the solution at each control volume, with high order precision, in a domain discretized by unstructured meshes of triangles. The reconstructed solution in each control volume is a Taylor’s Series based on polynomial, and the gradients of this polynomial are used to calculate the flows required by the finite volume method on the faces of the control volumes. To give materiality in the illustration of the aforementioned scheme, a code to perform tests involving transport problems, diffusion problems and transport and diffusion problems was implemented in C language using the Linux platform. The computational meshes for the discretized domains were constructed using the software Gmsh. The obtained results corroborated with the references studied proving the effectiveness of the methodology. This proposal has applications on the study of engineering problems that involve partial differential equations which cannot be solved analytically, as some fluids flow problems.