The report summarizes the work performed at the Department of Mathematics, during the first two years of the European research project GITEC-TWO (contract ENV4-U96-0297). Some of the result will subsequently be published elsewhere in a more complete form. The project work will continue to the end of 1998. The report adresses the following topics
- Development of an optical theory for discrete waves
- Run-up and staircase boundaries
- Finite element techniques for the Boussinesq equations
- Lagrangian finite difference technique for run-up of long waves
- Lagrangian finite element technique for run-up of long waves
- Testing of run-up models by comparison to analytical solutions
- Parallel models
- Domain decomposition; multiphysics
- Bore propagation and ru-up
- Boundary element method for full potential theory
- Discusion of physichal effects and validity of long wave approximations
In the first GITEC project the UiO group performed a series of case studies concerning tsunami events in the Atlantic, the eastern Mediterranean and the Norwegian sea. During the project the focus was slightly shifted towards general model analysis and development. Preliminary Lagrangian run-up models and finite element (FE) techniques for Boussinesq equations were reported. Moreover, tests concerning the convergence and applicability of the standard long wave models were included in the case studies or carried out as separate tasks. Continuing this line of investigation, we have focused mainly on model activities in GITEC-TWO, even though the work on the 1969 tsunami outside Portugal has continued and the study of the Tafjord event (1934) has been renewed. All the model activities rely heavily upon the experience from these and other case studies from the preceding project and elsewhere. Moreover, we have addressed a set of idealized, but challenging test cases, to obtain an improved insight in physics as well as numerics. This will enable us to exploit our new modeling tools in full studies of actual events with better results and control.
A number of test cases have been established, including wave generation and interaction with a shallow seamount, run-up on an idealized headland, slides in fjords, and wave propagation in several two dimensional geometries, usually corresponding to cross sections of actual bathymetries. Some of these problems are also addressed by the LDG group.
The prolonged study of the 1969 tsunami, originating near the Gorringe Bank, is linked to the publication of a joint journal paper (Gjevik et al. 1997) with the ICTE and the LDG groups. Due to the complexity of the problem and the diversity of the subtopics involved, the paper has been substantially revised during GITEC-TWO before being finally accepted. This is described further in section 8. In all simulations, concerning real as well as idealized test cases, emphasis has been put on grid refinement tests. Unless otherwise explicitly noted all the presented simulations herein are close to convergence.
It might seem surprising, but some of the properties of the most standard tsunami models are insufficiently documented in the literature. This is alarming since much of the tsunami work world wide still have to rely on the standard methods. Naturally, many of the features will also be inherited by more advanced models. Facing this problem during the preceding project, we have undertaken a study of the optical properties of finite difference (FD) and FE models, as well as on the influence of so called staircase (sawtooth) boundaries on the coastal response to incident waves. The results are summarized in section 2. Together with section 8 these topics form a fairly broad analysis on the shortcomings and accuracy of linear hydrostatic models. Other investigations concerning properties of finite element discretizations, domain decomposition etc. are described in other sections, the references or in manuscripts in progress.
The FE model for the Boussinesq equations has been upgraded, analyzed and fully documented in the journal paper Langtangen and Pedersen (1998b). Evaluation and verification of the method are partly based on the test cases outlined above. More details are given in section 3.
Particular emphasis has been put the on run-up models, that are described in the sections 4 and 6. The FD Lagrangian model for run-up, initiated under the first GITEC project, has been further developed, tested and documented in journal papers Johnsgard and Pedersen (1997)(ab). Bore treatment and bottom drag have been included as new features. Careful investigations have revealed both conceptual and practical problems concerning bore run-up (section 6). A related, but more general, FE technique has been implemented and compared to analytical solutions as well as to the pre-existing FD method. Both these results and additional simplified case studies are promising. This is elaborated in section 4. Preliminary model description and test results are published in a conference proceedings Langtangen and Pedersen (1998)(a). The FE software has been implemented in C++ using object-oriented design techniques and the Diffpack library.
As is usual in tsunami research, our modeling has been based mainly on hydrostatic and dispersive long wave equations. In order to check the validity of such equations, as well to investigate short wave features of tsunamis, we have developed a model that solves the equations of full nonlinear potential theory as described in section 7.
Two essential themes in the tsunami computation of the near future are parallel computing and automatic coupling of models with different numerical and physical characteristics. Both problems reduce to the development of flexible and general software components for implementing domain decomposition methods on top of existing solvers. Our work in this field, within GITEC-TWO, is sketched in section 5.
In the GITEC project we produced a video cassette describing some results concerning the 1969 Portuguese event and the Storegga tsunami, in collaboration with the Computer Center of the University of Oslo. Also in the new project we will produce animations and videos. However, significant upgrading of both hardware and software has been necessary. In combination with vacancy in the Department's position as computer engineer, with particular responsibility for scientific visualization, this has led to substantial delays. Anyway, sufficient resources is now allocated to the visualization task and the work is in steady progress and should be completed within the extended deadline of the GITEC-TWO project.