The characteristic mapping method uses a computational framework for non-linear advection capable of resolving fine scale fluid phenomena without the necessity of increasing the resolution of the computational grid. By approximating the inverse flow map generated by a velocity field as a composition of submaps, the method generates a discretization with an exponentially increasing polynomial degree of approximation using only a linear increase in the degrees of freedom. This functional spatio-temporal discretization has the capacity of accurately and sparsely representing fine scales globally, substituting the effects of spatial refinement with the operation of composition. As a step towards the application of these techniques to geophysical fluid phenomena, we present a characteristic mapping method for the rotating barotropic vorticity equations. The method is verified using standard test cases demonstrating third-order accuracy in the supremum norm. Numerical experiments illustrating the ability to reproduce the direct energy cascade at finer scales than the computational grid are provided.
翻译:特性绘图方法使用一个非线性对流学的计算框架,能够解决微量流体现象,而不必增加计算网格的分辨率。该方法与速度场生成的逆流图相近,作为子映射的构成,仅使用自由度的线性增长,产生指数性增长的多级近似分解。这种功能空间-时空分解能力准确和稀少地代表全球的微量尺度,以空间改进的效果取代组成功能。作为将这些技术应用于地球物理流体现象的一个步骤,我们为旋转的巴罗托罗诺色多色方程式提供了一种典型的绘图方法。该方法通过标准测试案例来验证,这些案例显示三阶精度的准度。提供了比计算网格在微量尺度上复制直接能源级联动的能力的数值实验。