Constructing PDFs of spatially dependent fields using finite elements

TL;DR

This work presents a finite-element framework for estimating the PDF $f_Y(y)$ of spatially varying fields by projecting the integral of the indicator $\mathbb{I}_{Y(x)<y}$ onto a discontinuous Galerkin space, yielding a robust CDF $F_Y(y)$ and a distributional PDF $\mathsf{f}_Y$. It decomposes the FE PDF into smooth and singular components $\mathsf{f}_0 \in L^2(\Omega_Y)$ and $\mathsf{f}_1 \in L^2(\Gamma)$ to capture jumps, and prescribes monotone reconstruction via slope limiting. The CDF inversion is carried out element-wise on a nonuniform mesh to obtain the inverse $\mathsf{F}^{-1}_B$ and compute the APE via $\beta^*(z) = F_B^{-1} \circ F_Z(z)$, with application to a Kelvin-Helmholtz instability illustrating accuracy and efficiency gains over histogram methods. The method is extensible to 3D and non-rectangular domains and is implemented in the NumDF package with documentation and data available publicly.

Abstract

A probability density function (PDF) of a spatially dependent field provides a means of calculating moments of the field or, equivalently, the proportion of a spatial domain that is mapped to a given set of values. This paper describes a finite element approach to estimating the PDF of a spatially dependent field and its numerical implementation in the Python package NumDF.

Figures (2)

• Figure 1: The CDF $F_Y$ of $Y(x)$ is constructed by projecting the integral of the indicator function \ref{['eq:CDF']} onto the discontinuous piecewise linear finite element space (DG1), which in this schematic has 5 elements. (left) The extended function $\hat{Y}(x,y) = Y(x)$, (middle) the indicator function $\mathbb{I}_{Y(x) < y}$ induced by $Y(x)$ and (right) the extended CDF $\hat{F}_Y(x,y) = F_Y(y)$ are defined on the domain $\Omega_X \times \Omega_Y$ which is discretised in $y$ but not in $x$. By representing $F_Y$ using DG1, regions of width $\Delta$ where $Y(x) = y_0$ is constant are correctly described as jumps of height $\Delta$ at $y_0$ in the CDF.
• Figure 2: (Top) As the two dimensional Kelvin-Helmholtz instability evolves, the stably stratified interface of negative/positive buoyancy (blue/red) is deformed. This process first creates APE by displacing negatively buoyant fluid upward and positively buoyant fluid downward. As the Kelvin-Helmholtz billows break kinetic energy increases and APE is destroyed. (Bottom) The finite element approximation of the PDF of buoyancy in terms of its continuous $\mathsf{f}_0$ (filled red) and singular $\mathsf{f}_1$ (dashed blue) measures becomes more uniform as time evolves and APE reduces, while the finite element approximation of the CDF (solid black) which initially approximates a step function becomes more tilted.