namespace regrid_edge_values¶

Overview¶

Edge value estimation for high-order resconstruction. More…

namespace regrid_edge_values {

// global functions

subroutine, public bound_edge_values(N N, h h, u u, edge_val edge_val, h_neglect h_neglect);
subroutine, public average_discontinuous_edge_values(N N, edge_val edge_val);
subroutine, public check_discontinuous_edge_values(N N, u u, edge_val edge_val);
subroutine, public edge_values_explicit_h2(N N, h h, u u, edge_val edge_val, h_neglect h_neglect);
subroutine, public edge_values_explicit_h4(N N, h h, u u, edge_val edge_val, h_neglect h_neglect);
subroutine, public edge_values_implicit_h4(N N, h h, u u, edge_val edge_val, h_neglect h_neglect);
subroutine, public edge_values_implicit_h6(N N, h h, u u, edge_val edge_val, h_neglect h_neglect);

} // namespace regrid_edge_values

Detailed Documentation¶

Edge value estimation for high-order resconstruction.

Global Functions¶

subroutine, public bound_edge_values(
    N N,
    h h,
    u u,
    edge_val edge_val,
    h_neglect h_neglect
    )

Bound edge values by neighboring cell averages.

In this routine, we loop on all cells to bound their left and right edge values by the cell averages. That is, the left edge value must lie between the left cell average and the central cell average. A similar reasoning applies to the right edge values.

Both boundary edge values are set equal to the boundary cell averages. Any extrapolation scheme is applied after this routine has been called. Therefore, boundary cells are treated as if they were local extrama.

Parameters:

n	Number of cells
h	cell widths (size N)
u	cell average properties (size N)
edge_val	Potentially modified edge values, with the same units as u.
h_neglect	A negligibly small width in the same units as h.

subroutine, public average_discontinuous_edge_values(N N, edge_val edge_val)

Replace discontinuous collocated edge values with their average.

For each interior edge, check whether the edge values are discontinuous. If so, compute the average and replace the edge values by the average.

Parameters:

n	Number of cells
edge_val	Edge values that may be modified the second index size is 2.

subroutine, public check_discontinuous_edge_values(N N, u u, edge_val edge_val)

Check discontinuous edge values and replace them with their average if not monotonic.

For each interior edge, check whether the edge values are discontinuous. If so and if they are not monotonic, replace each edge value by their average.

Parameters:

n	Number of cells
u	cell averages (size N)
edge_val	Cell edge values with the same units as u.

subroutine, public edge_values_explicit_h2(
    N N,
    h h,
    u u,
    edge_val edge_val,
    h_neglect h_neglect
    )

Compute h2 edge values (explicit second order accurate) in the same units as h.

Parameters:

n	Number of cells
h	cell widths (size N)
u	cell average properties (size N)
edge_val	Returned edge values, with the same units as u; the second index size is 2.
h_neglect	A negligibly small width

subroutine, public edge_values_explicit_h4(
    N N,
    h h,
    u u,
    edge_val edge_val,
    h_neglect h_neglect
    )

Compute h4 edge values (explicit fourth order accurate) in the same units as h.

Compute edge values based on fourth-order explicit estimates. These estimates are based on a cubic interpolant spanning four cells and evaluated at the location of the middle edge. An interpolant spanning cells i-2, i-1, i and i+1 is evaluated at edge i-1/2. The estimate for each edge is unique. i-2 i-1 i i+1.. o— o— o— o— o.. i-1/2

The first two edge values are estimated by evaluating the first available cubic interpolant, i.e., the interpolant spanning cells 1, 2, 3 and 4. Similarly, the last two edge values are estimated by evaluating the last available interpolant.

For this fourth-order scheme, at least four cells must exist.

Parameters:

n	Number of cells
h	cell widths (size N)
u	cell average properties (size N)
edge_val	Returned edge values, with the same units as u; the second index size is 2.
h_neglect	A negligibly small width

subroutine, public edge_values_implicit_h4(
    N N,
    h h,
    u u,
    edge_val edge_val,
    h_neglect h_neglect
    )

Compute ih4 edge values (implicit fourth order accurate) in the same units as h.

Compute edge values based on fourth-order implicit estimates.

Fourth-order implicit estimates of edge values are based on a two-cell stencil. A tridiagonal system is set up and is based on expressing the edge values in terms of neighboring cell averages. The generic relationship is

\[\alpha u_{i-1/2} + u_{i+1/2} + \beta u_{i+3/2} = a \bar{u}_i + b \bar{u}_{i+1}\]

and the stencil looks like this i i+1.. o— o— o.. i-1/2 i+1/2 i+3/2

In this routine, the coefficients \(\alpha\), \(\beta\), \(a\) and \(b\) are computed, the tridiagonal system is built, boundary conditions are prescribed and the system is solved to yield edge-value estimates.

There are N+1 unknowns and we are able to write N-1 equations. The boundary conditions close the system.

Parameters:

n	Number of cells
h	cell widths (size N)
u	cell average properties (size N)
edge_val	Returned edge values, with the same units as u; the second index size is 2.
h_neglect	A negligibly small width

subroutine, public edge_values_implicit_h6(
    N N,
    h h,
    u u,
    edge_val edge_val,
    h_neglect h_neglect
    )

Compute ih6 edge values (implicit sixth order accurate) in the same units as h.

Sixth-order implicit estimates of edge values are based on a four-cell, three-edge stencil. A tridiagonal system is set up and is based on expressing the edge values in terms of neighboring cell averages.

The generic relationship is

\[\alpha u_{i-1/2} + u_{i+1/2} + \beta u_{i+3/2} = a \bar{u}_{i-1} + b \bar{u}_i + c \bar{u}_{i+1} + d \bar{u}_{i+2}\]

and the stencil looks like this i-1 i i+1 i+2.. o— o— o— o— o.. i-1/2 i+1/2 i+3/2

In this routine, the coefficients \(\alpha\), \(\beta\), a, b, c and d are computed, the tridiagonal system is built, boundary conditions are prescribed and the system is solved to yield edge-value estimates.

Note that the centered stencil only applies to edges 3 to N-1 (edges are numbered 1 to n+1), which yields N-3 equations for N+1 unknowns. Two other equations are written by using a right-biased stencil for edge 2 and a left-biased stencil for edge N. The prescription of boundary conditions (using sixth-order polynomials) closes the system.

CAUTION: For each edge, in order to determine the coefficients of the implicit expression, a 6x6 linear system is solved. This may become computationally expensive if regridding is carried out often. Figuring out closed-form expressions for these coefficients on nonuniform meshes turned out to be intractable.

Parameters:

n	Number of cells
h	cell widths (size N)
u	cell average properties (size N)
edge_val	Returned edge values, with the same units as u; the second index size is 2.
h_neglect	A negligibly small width