update

Author: delcmo

paper/paper.md

@@ -62,7 +62,7 @@ The VERTEX-CFD package is an open-source code that is hosted on the ORNL GitHub
 
 ## Governing equations and discretization methods
 
-The VERTEX-CFD solver is still under active development and currently implements a set of partial differential equations (PDEs) discretized with a finite element method (FEM) and a high-order temporal integrators: entropically dumped artificial compressibility (EDAC) Navier-Stokes equations [@Clausen2013], temperature equation, and induction-less magneto-hydrodynamic (MHD) equations. Coupling between the different physics is ensured by source terms that are the Buoyancy force $$f^B$$ and the Lorentz force $$f^L$$. A conservative form of the set of PDEs is implemented in VERTEX-CFD as shown in Equation \autoref{eq:pdes} and solves for the pressure $$P$$, the temperature $$T$$, the velocity $$\mathbf{u}$$ and the electric potential $$\varphi$$. The density $$\rho$, the heat capacity $$C_p$$, the electrical conductivity $$\sigma$$, the thermal conductivity $$k$$, the thermal expansion $$\beta$$, are the fluid properties and are all assumed constant. The external magnetic field, the reference temperature for Buoyancy force and the volumetric heat source are denoted by $$\mathbf{B^0}$$, $$T_0$$ and $$q^{'''}, respectively.
+The VERTEX-CFD solver is still under active development and currently implements a set of partial differential equations (PDEs) discretized with a finite element method (FEM) and a high-order temporal integrators: entropically dumped artificial compressibility (EDAC) Navier-Stokes equations [@Clausen2013], temperature equation, and induction-less magneto-hydrodynamic (MHD) equations. Coupling between the different physics is ensured by source terms that are the Buoyancy force $$f^B$$ and the Lorentz force $$f^L$$. A conservative form of the set of PDEs is implemented in VERTEX-CFD as shown in Equation \autoref{eq:pdes} and solves for the pressure $$P$$, the temperature $$T$$, the velocity $$\mathbf{u}$$ and the electric potential $$\varphi$$. The density $$\rho$$, the heat capacity $$C_p$$, the electrical conductivity $$\sigma$$, the thermal conductivity $$k$$, the thermal expansion $$\beta$$, are the fluid properties and are all assumed constant. The external magnetic field, the reference temperature for Buoyancy force and the volumetric heat source are denoted by $$\mathbf{B^0}$$, $$T_0$$ and $$q^{'''}, respectively.
 
 \begin{align}\label{eq:pdes}
 \left\{