|
Identifier |
000354256 |
Title |
Σχήματα πεπερασμένων διαφορών με αυτόματη αναδιαμέριση για υπερβολικούς νόμους διατήρησης |
Alternative Title |
Finite difference schemes on non-uniform meshes for hyperbolic conservation laws |
Author
|
Σφακιανάκης, Νικόλαος Ιωσήφ
|
Thesis advisor
|
Μακριδάκης, Χαράλαμπος
|
Abstract |
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 during 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.
|
Language |
English |
Subject |
Adaptive mesh reconstruction |
|
Finite difference schemes |
|
conservation laws |
|
Αναδιαμέριση |
|
Νόμοι διατήρησης |
|
Σχήματα πεπερασμένων διαφορών |
Issue date |
2009-07-09 |
Collection
|
School/Department--School of Sciences and Engineering--Department of Applied Mathematics--Doctoral theses
|
|
Type of Work--Doctoral theses
|
Permanent Link |
https://elocus.lib.uoc.gr//dlib/c/2/c/metadata-dlib-b236d657944f0d52c3579ddf02bcda72_1267606387.tkl
|
Views |
957 |