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Nearshore interpolation

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Several Fortran routines in GeoClaw interpolate from the computational grids +to other specified points where output is desired +(typically using the finest AMR resolution available nearby at each desired +output time). This includes:

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  • Gauge output (time series at specified locations), see Gauges,

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  • fgmax grids (maximum of various quantities over the entire simulation at +specified locations), see Fixed grid monitoring,

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  • fgout grids (output of various quantities on a fixed spatial grid at a +sequence of times), see Fixed grid output.

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If the specified location is exactly at the center of a computational cell +at the finest AMR level present, then the value output is simply that cell +value (which in a finite volume method is really a cell average of the +quantity over the cell, but approximates the cell center value to +\(O(\Delta x^2)\) if the quantity is smoothly varying.

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In general, however, the specified points may not lie at cell centers. In +all the cases listed above, the default behavior is to use “zero-order +extrapolation” to determine the value at the point, meaning that the value +throughout each computational cell is approximated by a constant function +(zero degree polynomial) with value equal to the cell average. Hence the +value that is output at any specified point is simply the cell average of +the (finest level) grid cell that the point lies within.

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One might think that a better approximation to the value at a point could be +obtained by using piecewise bilinear approximation (in two +dimensions): Determine what set of four grid centers the point lies within +and construct the bilinear function \(a_0 + a_1x + a_2y + a_3xy\) +that interpolates at these four points, and then evaluate the bilinear +interpolant at the point of interest. If the function being approximated +were smooth then this would be expected to provide an \(O(\Delta x^2)\) +approximation, whereas zero-order extrapolation is only \(O(\Delta x)\) +accurate.

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For GeoClaw simulations, however, we have found that it is best to use +zero-order extrapolation because piecewise bilinear interpolation can cause +problems and misleading results near the coastline, which is often the +region of greatest interest.

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The problem is that interpolating the fluid depth h and the topography B +separately and then computing the surface elevation eta by adding these +may give unrealistic high values. For example if one cell has topo B = -2 +and h = 6 (so eta = B+h = 4) and the neighboring cell has B = 50 +and h = 0, so that eta = B+h = 50. In the latter case, the elevation +eta is simply the elevation of the land and this point is not wet, as +indicated by the fact that h=0. But now if we use linear interpolation +(in 1D for this simple example) to the midpoint between these points, +interpolating the topography gives +B = 24 and interpolating the depth gives h = 3. +Hence we compute eta = B+h = 27 as the surface elevation. +Since the point is apparently wet (h > 0), one might conclude that the sea +surface at this point is 27 meters above the initial sea level, whereas in +fact the sea level is never more than 6 meters above sea level.

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