1% Create the laplacian operator. 2% 3% Lap = divand_laplacian(mask,pmn,nu,iscyclic) 4% 5% Form a Laplacian using finite differences 6% assumes that gradient is zero at "coastline" 7% 8% Input: 9% mask: binary mask delimiting the domain. 1 is inside and 0 outside. 10% For oceanographic application, this is the land-sea mask. 11% pmn: scale factor of the grid. 12% nu: diffusion coefficient of the Laplacian 13% field of the size mask or cell arrays of fields 14% 15% Output: 16% Lap: sparce matrix represeting a Laplaciant 17% 18% 19function Lap = divand_laplacian(mask,pmn,nu,iscyclic) 20 21pmn = cat_cell_array(pmn); 22 23% number of dimensions 24n = size(pmn,ndims(pmn)); 25 26sz = size(mask); 27 28if nargin == 2 29 nu = ones(sz); 30end 31 32if isequal(sz,size(nu)) 33 % assume diffusion is the same along all dimensions 34 nu = repmat(nu,[ones(1,n) n]); 35end 36 37% nu: 1st index represents the dimensions 38nu = permute(nu,[n+1 1:n]); 39 40% extraction operator of sea points 41H = sparse_pack(mask==1); 42 43 44sz = size(mask); 45 46%DD = sparse(0); 47DD = sparse(prod(sz),prod(sz)); 48 49Pmn = reshape(pmn,[prod(sz) n]); 50for i=1:n 51 % operator for staggering in dimension i 52 S = sparse_stagger(sz,i,iscyclic(i)); 53 54 % d = 1 for interfaces surounded by water, zero otherwise 55 d = S * mask(:) == 1; 56 57 % metric 58 for j = 1:n 59 tmp = S * Pmn(:,j); 60 61 if j==i 62 d = d .* tmp; 63 else 64 d = d ./ tmp; 65 end 66 end 67 68 % tmp: "diffusion coefficient" along dimension i 69 tmp = nu(i,:); 70 71 %whos tmp d S nu 72 %keyboard 73 d = d .* (S * tmp(:)); 74 75 % Flux operators D 76 % zero at "coastline" 77 78 D = sparse_diag(d) * sparse_diff(sz,i,iscyclic(i)); 79 szt = sz; 80 81 if ~iscyclic(i) 82 % extx: extension operator 83 szt(i) = szt(i)+1; 84 extx = sparse_trim(szt,i)'; 85 86 D = extx * D; 87 else 88 % shift back one grid cell along dimension i 89 D = sparse_shift(sz,i,iscyclic(i))' * D; 90 end 91 92 % add laplacian along dimension i 93 DD = DD + sparse_diff(szt,i,iscyclic(i)) * D; 94end 95 96ivol = prod(pmn,n+1); 97 98% Laplacian on regualar grid DD 99DD = sparse_diag(ivol) * DD; 100 101% Laplacian on grid with on sea points Lap 102Lap = H * DD * H'; 103 104 105 106% Copyright (C) 2014 Alexander Barth <a.barth@ulg.ac.be> 107% 108% This program is free software; you can redistribute it and/or modify it under 109% the terms of the GNU General Public License as published by the Free Software 110% Foundation; either version 2 of the License, or (at your option) any later 111% version. 112% 113% This program is distributed in the hope that it will be useful, but WITHOUT 114% ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 115% FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more 116% details. 117% 118% You should have received a copy of the GNU General Public License along with 119% this program; if not, see <http://www.gnu.org/licenses/>. 120