Space-time boundary element methods for the heat equation

The temperature distribution of an object in a time-dependent process can be computed by solving the heat equation. Conventional simulation methods compute the changes in small time steps. This approach requires many sequential computations. In this way, it is not possible to take full advantage of the huge resources of recent supercomputers. Therefore such computations usually take a lot of time. Space-time methods deal with the considered time interval as a whole. Therefore, a very large problem has to be solved instead of a sequence of smaller problems. This seems to be more difficult at first glance but offers enormous potential. Dealing with the full time interval as a whole makes possible an additional parallelization in time. Thus, the computation can be distributed to more processors and the simulation result can be computed much faster. In addition, it is possible to adapt the approximations of the temperature much better to the specific problem. In boundary element methods it is sufficient to compute the considered data on the surface initially. Then the temperature can easily be evaluated in the interior. Although boundary element methods for the heat equation are space-time methods inherently, the related problem is usually still solved in small time steps by forward elimination. In doing so, the potential of the methods mentioned above is not exploited. In this project, we develop fast methods for the solution of space-time boundary element methods to compute the temperature distribution of an object. These fast methods are specifically tailored to the space-time equations. For this purpose suitable preconditioning techniques and solution procedures are developed, as it is not straightforward to compute the current state without knowing the previous one. In addition, specifically designed representations of the wanted temperature are implemented to adopt the computation much better to the temperature distribution of the considered problem. We put an extra emphasis on the parallel implementation of the methods on modern supercomputers. Modern processors enable the same computation on several sets of data at the same time by vectorization. In addition, they are equipped with several cores which can be used simultaneously. Supercomputers have a large number of computers which can be used in a cooperating way. We will utilize all three levels of parallelization in this project. This has to be taken into account in the development of the methods. By the end of the project, it will be possible to compute the temperature distribution of an object much more accurately and faster than by currently used conventional methods.

The temperature distribution of an object in a time-dependent process can be computed by solving the heat equation. Conventional simulation methods compute the changes in small time steps. This approach requires many sequential computations. In this way, it is not possible to take full advantage of the huge resources of modern supercomputers. Therefore such computations usually take a lot of time. We have developed a highly parallel space-time boundary element method based on the parabolic fast multipole method. The space-time approach allows us an additional parallelization in time. When a purely spatial parallelization reaches its speedup limit, the space-time parallel method can reduce the computational times further. In our algorithm, we use distributed parallelization regarding time and shared memory parallelization and vectorization within every computing node. Our examples showed close to optimal scalability for up to 256 computing nodes (6144 cores). In addition, space-time methods allow for adaptive temporal refinement and locally refined spatial resolutions. This results in smaller linear systems and possibly faster computations than uniform discretizations. But standard fast boundary element methods may fail in compressing the related dense matrices efficiently. Thus we have developed substantial enhancements of the considered fast method. Our new method can efficiently deal with widely varying time step sizes as well as with much finer spatial resolutions and spatially adaptive meshes. The enhancements provide substantial improvements over the standard fast method and one can notice the superior performance of the adaptive method over the standard method with the uniform time steps and uniform spatial meshes in related examples. Our highly parallel fast space-time boundary element method and our enhancements for widely varying time step sizes and locally refined spatial meshes are important steps towards adaptive space-time boundary element methods. Without this progress, it would be hardly possible to run adaptive algorithms for space-time boundary element methods for the heat equation in three dimensions.