src/DiffEq/Beta/NumberFractionBeta.hpp file

System of number-fraction beta SDEs.

Contents

In a nutshell, the equation integrated governs a set of scalars, $0\!\le\!X_\alpha$ , $\alpha\!=\!1,\dots,N$ , as

\[ \mathrm{d}X_\alpha(t) = \frac{b_\alpha}{2}\left(S_\alpha - X_\alpha\right) \mathrm{d}t + \sqrt{\kappa_\alpha X_\alpha(1-X_\alpha)} \mathrm{d}W_\alpha(t), \qquad \alpha=1,\dots,N \]

with parameter vectors $b_\alpha > 0$ , $\kappa_\alpha > 0$ , and $0 < S_\alpha < 1$ . This is the same as in DiffEq/Beta.h. Here $\mathrm{d}W_\alpha(t)$ is an isotropic vector-valued Wiener process with independent increments. The invariant distribution is the joint beta distribution. This system of SDEs consists of N independent equations. For more on the beta SDE, see https://doi.org/10.1080/14685248.2010.510843.

In addition to integrating the above SDE, there are two additional functions of $ X_\alpha $ are computed as

\[ \begin{aligned} \rho(X_\alpha) & = \rho_{2\alpha} ( 1 - r'_\alpha X_\alpha ) \\ V(X_\alpha) & = \frac{1}{ \rho(X\alpha) } \end{aligned} \]

These equations compute the instantaneous mixture density, $ \rho $ , and instantaneous specific volume, $ V_\alpha $ , for equation $ \alpha $ in the system. These quantities are used in binary mixing of variable-density turbulence between two fluids with constant densities, $ \rho_1, $ and $ \rho_2 $ . The additional parameters, $ \rho_2 $ and $ r' $ are user input parameters and kept constant during integration. Since we compute the above variables, $\rho,$ and $V$ , and call them mixture density and specific volume, respectively, $X$ , governed by the beta SDE is a number (or mole) fraction, hence the name number-fraction beta.

All of this is unpublished, but will be linked in here once published.

Namespaces

namespace walker
Walker declarations and definitions.

Classes

template<class Init, class Coefficients>
class walker::NumberFractionBeta
NumberFractionBeta SDE used polymorphically with DiffEq.