Anthropogenic emissions of greenhouse gases are enhancing the natural greenhouse effect of the Earth system. If present trends continue, global climate models project that this enhanced greenhouse effect will result in a much warmer climate by 2100. Because of the serious implications of these projections, it is essential that climate models compute the greenhouse effect as accurately as possible. However, the latest assessment by the Intergovernmental Panel on Climate Change (IPCC) shows there are substantial errors in these calculations. To a significant degree, the errors are caused by the ad hoc numerical approximations used for the transmission of electromagnetic radiation in the Earth's atmosphere.

We present a new mathematical method to improve the accuracy of the radiative calculations in climate models. In contrast to previous techniques, the new method provides a geometrical interpretation and a useful error bound for the solutions. The solutions are guaranteed to be globally optimal to any order of approximation. In general, our new approach is applicable to a broad class of functions from the fields of engineering and physical sciences.