Staggered Grid Structure

An orthogonal 2D staggered grid composed of a basic grid and three staggered grids is used to discretize governing equations using a conservative finite difference formulation described by Gerya (2010) (Fig.). The basic grid, denoted by subscript $b$, encompasses the main model domain $(x_{size}, y_{size})$ with $N_{x,b}$ nodes in the $x$-direction and $N_{y,b}$ nodes in the $y$-direction. The x- and y-coordinates of the basic grid are $(x_{b,j}, y_{b,i})$ where $j$ is in the $x$-direction ranging from $1$ to $N_{x,b}$ and $i$ is in the $y$-direction ranging from $1$ to $N_{y,b}$. Both $y_{b,i}$ and $i$ increase with depth. The basic grid spacings in the $x$- and $y$-directions are denoted by $\Delta x_{b,j}$ where $j$ ranges from $1$ to $N_{x,b}-1$ and $\Delta y_{b,i}$ where $i$ ranges from $1$ to $N_{y,b}-1$, respectively. Temperature unknowns $T_{(i,j)_b}$, deviatoric shear stress $\sigma'_{xy(i,j)_b}$, deviatoric shear strain rate $\dot \epsilon'_{xy(i,j)_b}$, and visco-plastic shear viscosity $\eta_{vp(i,j)_b}$ are defined on the nodes of the basic grid.

Grid Indexing

Basic grid cells and unknowns are indexed with respect to the upper-left node of basic-grid cells that have a global cell index defined as follows Fig.:

eq:basic-cell-index

\[I_{cell} = (j-1)(N_{y,b}-1) + i\]

where $i$ and $j$ are indices of the upper-left basic grid node associated with a cell. Unknowns $v_x$, $v_y$, and $P$ are numbered with the increasing cell index $I_{cell}$ as follows:

eq:global-indices-of-stokes-unknowns

\[\begin{split} I_{v_x} = 3I_{cell} - 2 & \quad \text{for x-velocity unknown} \\ I_{v_y} = 3I_{cell} - 1 & \quad \text{for y-velocity unknown} \\ I_{P} = 3I_{cell} & \quad \text{for pressure unknown} \end{split}\]

Temperature unknowns $T$ are numbered using the global index $I_{T}$, which is defined as follows:

eq:global-indices-of-heat-unknowns

\[I_{T} = (j-1)N_{y,b} + i\]

where $i$ and $j$ are indices of the basic grid Fig..

fig:staggered-grid

Staggered grid used to discretize governing equations. Nodes of the basic grid are shown as black circles. Nodes of the velocity-x grid are denoted by red symbols, nodes of the velocity-y grid are denoted by light blue symbols, and nodes of the pressure grid are denoted by green symbols. Node index pairs for the basic, velocity-x, velocity-y and pressure grids are denoted with subscripts $b$, $v_x$, $v_y$ and $p$, respectively. The global index for temperature unknowns $I_T$ is denoted with bold font, and global cell index $I_{cell}$ used to discretize the Stokes-continuity equations are denoted by bold font in parentheses. The global index of Stoke-continuity unknowns is shown with numbers inside of colored symbols. Colored circles represent normal unknowns where finite-difference stencils are applied. Ghost nodes located outside of the main model domain, which are used to approximate derivatives in the finite difference stencils, are shown as squares. Boundary nodes are shown with diamonds with thin outlines. Boundary unknown nodes are denoted with diamonds with thick outlines. The unknowns defined at boundary unknown nodes have prescribed values leading to large matrix coefficients equal to one and right-hand side terms equal prescribed velocity values.