Study and development of methods for the numerical solution of boundary value
problems that correspond to the determination of curves of minimum length.
The aim of WP 3 is to study and propose new numerical
methods that are adapted to the nature of the second order ordinary differential
equations with boundary conditions arising in WP 2. The exact form of these
equations depends on the data and their
distribution. There are several types of numerical approximation approaches the
solution of boundary value problems (BVPs) such as methods of finite
differences, finite elements, collocation and others. These methods will be
studied and adapted so to be applicable to BVPs arising in WP 2. Novel ideas and
modifications of existing methods appropriate for the specific differential
equations will be proposed.
A number of sensitivity-evaluation tests will be
organized for the obtained algorithms based on test cases with known analytic
solutions. Moreover, a benchmark of the prevailed methods based on the accuracy
the corresponding computational cost will reveal the optimum
methodology/algorithm for the problems arising in WP2.
Methodology
An extensive literature study on the "state of the art" will be followed by
detailed tuning and efficiency tests for the appropriate numerical methods
adopted
in this workpackage. Some practical problems, arising in WP 2, and other test
problems will be solved based on different methods and targeting to the
selection of the optimum one.
The BVP problem we study has a special quadratic form. So, in the study of the numerical methods this feature has to be taken into consideration. The problem was solved using
Finite difference methods, where a modification of the LU procedure in the solution by Newton's iteration of the resulted nonlinear system and a study on the use of Quasi Newton methods. The numerical solution was implemented in Matlab. A presentation in this subject.
A shooting method, using as an integrator a Runge Kutta specially constructed for quadratic IVPs. This solution was implemented in Mathematica. A presentation in this subject.
We have used A-stable Symmetric Mono Implicit Runge-Kutta schemes and their continuous extensions modifying the popular MIRKDC Fortran code. A presentation in this subject.
These numerical methods where applied to 24 test
problems. The cases of multiple shooting method and MIRK method where both
considered ideal for the solution of our problems.
Deliverables
·
Internal Technical Report.
·
Computer code and programs.
·
A meeting-seminar in TEI of Athens. The main purpose of the meeting will be the
diffusion of the results of the project among the research members of the team
and other scientists. The seminar will be organized in the middle of the project
and issues relative to the project planning and monitoring will be discussed
based on the results at that time.
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