Finite Difference schemes on non-Uniform meshes for Hyperbolic Conservation Laws

In this work we consider Finite Difference numerical schemes over 1 dimensional non-uniform, adaptively redefined meshes. We combine the basic properties of function approximation over non-uniform mesh with a mesh reconstruction, the spatial solution update over the new mash and with the time evolution with Finite Difference schemes designed for non-uniform meshes. All these steps constitute the Basic Adaptive Scheme. We moreover analyse the Total Variation properties of the Basic Adaptive Scheme and provide the theoretical results of this work. In more details: we investigate the basic notions of Finite Difference approximation over non-uniform meshes. We discuss their properties and compare their qualitative characteristics with the respective approximations over uniform meshes. We then discuss the mesh reconstruction procedure that we use throughout this work. We explain the way the new non-uniform mesh is constructed based on geometric properties of the numerical solution itself, and on the already existing non-uniform mesh. We describe the functionals responsible for this mesh reconstruction and present their properties. Relations with other non-uniform mesh methods are provided in the form of references. Afterwards, we present the process with which the numerical solution is updated/redefined over the new non-uniform mesh. Characteristic properties like, conservation of mass and maximum principle this process are discussed and analysed. Next, we move to the time evolution part of the Basic Adaptive Scheme. We discuss some known and some new numerical schemes, both designed for non-uniform meshes. We notice that some of them reduce to the same numerical scheme when the mesh is uniform; hence we name the numerical schemes under consideration according to their uniform counterparts. We analyse some of their properties like consistency, stability and order of accuracy using, mainly, their modified equations as our tool. Through this process we discover that the usual consistency criterion for Finite Difference scheme is not sufficient when the mesh is non-uniform; hence we provide a generalisation of the consistency criterion valid also for non-uniform meshes. Then a series of numerical tests is conducted. Comparisons between the non-uniform vs uniform mesh case exhibit both the stabilisation properties of the Basic Adaptive Scheme and the higher accuracy that can be achieved when non-uniform mesh is used. These tests are conducted using the numerical schemes that were previously discussed as well as elaborate Entropy Conservative numerical schemes. Finally, we discuss the Total Variation of Basic Adaptive Schemes when oscillatory (either dispersive or anti-diffusive) Finite Difference schemes are used for the time evolution step. We prove under specific assumptions that the Total Variation of such schemes remains bounded, and even more (under more strict assumptions) that the increase of their Total Variation decreases with time. This work has led to the submission of three journal essays.