Adaptive grids as parametrized scale-free networks
Gianluca Argentini
TL;DR
The paper addresses building adaptive grids for numerical solutions of differential equations by tying grid resolution to physical or geometrical properties using a scale-free network mechanism. It models a node-field $k(\mathbf{x},t)$ evolving via $\partial k_i/\partial t = m_i \Pi(k_i)$ with $\Pi(k_i)=k_i/\sum_j k_j$, embedding Barabasi–Albert–type growth in the grid, and sets local spacing as $h_{ij} = \frac{2A}{k_i+k_j}$ or, in the continuum limit, $h(\mathbf{x},t) = \frac{A}{k(\mathbf{x},t)}$. It analyzes three regimes for the constitutive rate $m$: (i) $m_i=0$ leads to a time-invariant grid; (ii) constant $m$ yields $k_i(t)=c_i e^{mt/S}$ and $h_{ij}=(2A/(c_i+c_j)) e^{-mt/S}$; (iii) $m$ depends on $|\nabla\mathbf{u}|/\mu$ (as $m_0+m_1|\nabla\mathbf{u}|/\mu+m_2(|\nabla\mathbf{u}|/\mu)^2$) giving $k=c\exp\left(\frac{1}{\mu^2 S}\int_0^t[\mu^2 m_0 + \mu m_1|\nabla\mathbf{u}| + m_2|\nabla\mathbf{u}|^2] ds\right)$ and $h(\mathbf{x},t)=\frac{\mu^2 S A}{c[\mu^2 S + \mu^2 m_0 t + \mu m_1 \|\nabla\mathbf{u}\|_{L^1(0,t)} + m_2 \|\nabla\mathbf{u}\|_{L^2(0,t)}^2]}$. In a 1D diffusion-transport example $-\mu u''(x)+b u'(x)=0$ with $u(0)=0$, $u(1)=1$, the method yields a boundary-layer concentration near $x=1$ when $\mu\ll b$ and, under the stability constraint $Pe=\frac{b h}{2\mu}<1$, an explicit grid construction with $m_0/S=(1/x_i)\log\left(\frac{b h_1}{2\mu}\right)$ and $x_{j+1}=x_j+\frac{h_1}{\exp\left(\frac{x_j}{x_i}\log\left(\frac{b h_1}{2\mu}\right)\right)}$, yielding adaptive grids that concentrate near the boundary layer and improve accuracy for a fixed node count.
Abstract
In this paper we present a possible model of adaptive grids for numerical resolution of differential problems, using physical or geometrical properties, as viscosity or velocity gradient of a moving fluid. The relation between the values of grid step and these entities is based on the mathematical scheme offered by the model of scale-free networks, due to Barabasi, so that the step can be connected to the other variables by a constitutive relation. Some examples and an application are discussed, showing that this approach can be further developed for treatment of more complex situations.