A special place in climatology is taken by the so-called conceptual climate models. These relatively simple sets of differential equations can successfully describe single mechanisms of climate. We focus on one family of such models based on the global energy balance. This gives rise to a degenerate nonlocal parabolic nonlinear partial differential equation for the zonally averaged temperature. We construct a fully discrete numerical method that has an optimal spectral accuracy in space and second order in time. Our scheme is based on the Galerkin formulation of the Legendre basis expansion, which is particularly convenient for this setting. By using extrapolation, the numerical scheme is linear even though the original equation is nonlinear. We also test our theoretical results during various numerical simulations that support the aforementioned accuracy of the scheme.

翻译：所谓的概念气候模型在气候学中占有特殊位置。 这些相对简单的差异方程式可以成功地描述单一的气候机制。 我们侧重于基于全球能源平衡的一组此类模型。 这导致地区平均温度的非本地抛物线非线性非线性部分差异方程式退化。 我们构建了一个完全离散的数字方法,该方法在空间中具有最佳的光谱精度,在时间上具有第二顺序。 我们的计划基于Galerkin 的图伦特尔基扩展公式, 这对于这一环境特别方便。 通过外推, 数字法是线性的, 尽管原始方程式是非线性方程式。 我们还在支持上述计划精确度的各种数字模拟中测试我们的理论结果。