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Can EDIT the space: chombo-admin ,  chombo-editors , 
Can ADMINISTER the space: chombo-admin , 
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Can VIEW the space: [email protected][email protected]
Can EDIT the space: [email protected][email protected]
Can ADMINISTER the space: [email protected][email protected]

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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
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Domain in 2D mapped space: five squares

Range in 2D real space: disk

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Advanced Tables - Table Plus
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Domain in 2D mapped space: eight squares

Range in 2D real space: single-null edge plasma geometry

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Advanced Tables - Table Plus
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Domain in 2D mapped space: six squares

Range in 3D real space: surface of sphere

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Advection on a sphere, with one level of refinement

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Exchanging data between blocks

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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
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In mapped space: four ghost cells with their interpolation stencils

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In real space: four ghost cells with their interpolation stencils

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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:

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$$f(X, Y) = \sum_{p, q \geq 0: p + q \leq P} a_{pq} \left(\frac{X - X_{\boldsymbol{g}}}{R}\right)^p \left(\frac{Y - Y_{\boldsymbol{g}}}{R}\right)^q$$

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

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$\langle \cdot \rangle_{\boldsymbol{j}}$

indicates an average over cell j.
The average value of f on the ghost cell is then obtained by

LaTeX Formatting
$$\left\langle f \right\rangle_{\boldsymbol{g}} = \sum_{p, q \geq 0; p + q \leq P} a_{pq} \left\langle \left(\frac{X - X_{\boldsymbol{g}}}{R}\right)^p \left(\frac{Y - Y_{\boldsymbol{g}}}{R}\right)^q \right \rangle_{\boldsymbol{g}}$$

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