...
As in the rest of Chombo, data are held on rectangular patches. In the mapped multi-block framework, each patch must be contained in only one block. In other words, no patch may straddle two or more blocks.
Examples
| Advanced Tables - Table Plus |
|---|
| columnAttributes | width=40%,width=40% |
|---|
|
Domain in 2D mapped space: five squares | Range in 2D real space: disk |
|---|
 Image Modified
|  Image Modified
|
|
| Advanced Tables - Table Plus |
|---|
| columnAttributes | width=40%,width=40% |
|---|
|
Domain in 2D mapped space: eight squares | Range in 2D real space: single-null edge plasma geometry |
|---|
 Image Modified
|  Image Modified
|
|
| Advanced Tables - Table Plus |
|---|
| columnAttributes | width=40%,width=40% |
|---|
|
Domain in 2D mapped space: six squares | Range in 3D real space: surface of sphere |
|---|
 Image Modified
|  Image Modified
|
|
Advection on a sphere, with one level of refinement
| Multimedia |
|---|
| width | 100%%%%%%%% |
|---|
| name | AdvectCosineBell.polar_lim.10.abs.AMR.64.mpeg |
|---|
| width | 100%%%%%%%% |
|---|
| height | 380 |
|---|
|
Exchanging data between blocks
...
The ghost cells of a patch that lie outside the block containing the patch will be called extra-block ghost cells of the patch. Two layers of extra-block ghost cells of block 2 of the disk example are outlined with dotted blue lines below, in both mapped space and real space. The centers of four of them are marked with blue *. The cells of the interpolation stencils of these four ghost cells are shown with thicker outlines, each with the color of its block.
| Advanced Tables - Table Plus |
|---|
| columnAttributes | height=100% |
|---|
|
In mapped space: four ghost cells with their interpolation stencils |
|---|
 Image Modified
| In real space: four ghost cells with their interpolation stencils |
|---|
 Image Modified
|
|
The function value for each extra-block ghost cell is interpolated from function values for the valid cells in its stencil.
Around each ghost cell, we approximate the function by a Taylor polynomial in real coordinates, of degree P. This polynomial has different coefficients for each ghost cell. In 2D the polynomial is of the form:
| LaTeX Formatting |
|---|
$$f(X, Y) = \sum_{p, q \geq 0: p + q \leq P} a_{pq} \left(\frac{X - X_{\boldsymbol{g}}}{R}\right)^p \left(\frac{Y - Y_{\boldsymbol{g}}}{R}\right)^q$$ |
where (Xg, Yg) is the center of the ghost cell in real coordinates, and R is the mean distance from (Xg, Yg) to the centers in real space of the stencil cells of g.
Starting with averaged values of f over each valid cell, then the coefficients apq come from solving the overdetermined system of equations
| LaTeX Formatting |
|---|
$$\sum_{p, q \geq 0; p + q \leq P} a_{pq} \left\langle \left(\frac{X - X_{\boldsymbol{g}}}{R}\right)^p \left(\frac{Y - Y_{\boldsymbol{g}}}{R}\right)^q \right \rangle_{\boldsymbol{j}} = \left\langle f \right\rangle_{\boldsymbol{j}}$$ |
where there is an equation for every cell j in the selected neighborhood of valid cells of the ghost cell. The notation
| LaTeX Formatting |
|---|
$\langle \cdot \rangle_{\boldsymbol{j}}$ |
indicates an average over cell j.
The average value of f on the ghost cell is then obtained by
| LaTeX Formatting |
|---|
$$\left\langle f \right\rangle_{\boldsymbol{g}} = \sum_{p, q \geq 0; p + q \leq P} a_{pq} \left\langle \left(\frac{X - X_{\boldsymbol{g}}}{R}\right)^p \left(\frac{Y - Y_{\boldsymbol{g}}}{R}\right)^q \right \rangle_{\boldsymbol{g}}$$ |
.
