The growth of the computing capacities makes it possible to obtain more and more precise simulation results. These results are often calculated in binary64 with the idea that round-off errors are not significant. However, exascale is pushing back the known limits and the problems of accumulating round-off errors could come back and require increasing further the precision. But working with extended precision, regardless of the method used, has a significant cost in memory, computation time and energy and would not allow to use the full performance of HPC computers. It is therefore important to measure the robustness of the binary64 by anticipating the future computing resources in order to ensure its durability in numerical simulations. For this purpose, numerical experiments have been performed and are presented in this article. Those were performed with weak floats which were specifically designed to conduct an empirical study of round-off errors in hydrodynamic simulations and to build an error model that extracts the part due to round-off error in the results. This model confirms that errors remain dominated by the scheme errors in our numerical experiments.
Numerical simulations are carefully-written programs, and their correctness is based on mathematical results. Nevertheless, those programs rely on floating-point arithmetic and the corresponding round-off errors are often ignored. This article deals with a specific simple scheme applied to advection, that is a particular equation from hydrodynamics dedicated to the transport of a substance. It shows a tight bound on the round-off error of the 1D and 2D upwind scheme, with an error roughly proportional to the number of steps. The error bounds are generic with respect to the format and exceptional behaviors are taken into account. Some experiments give an insight of the quality of the bounds.
Numerical simulations are carefully-written programs, and their correctness is based on mathematical results. Nevertheless, those programs rely on floating-point arithmetic and the corresponding round-off errors are often ignored. This article deals with a specific simple scheme applied to advection, that is a particular equation from hydrodynamics dedicated to the transport of a substance. It shows a tight bound on the round-off error of the 1D and 2D upwind scheme, with an error roughly proportional to the number of steps. The error bounds are generic with respect to the format and exceptional behaviors are taken into account. Some experiments give an insight of the quality of the bounds.
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