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Periodics Operators
These are gradient, divergence, and interpolation periodic operators. They satisfy the following properties: grad = - div^T and interp from nodes to centers is the transpose of the interp from centers to nodes
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‎src/matlab/div2DPer.m‎

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% SPDX-License-Identifier: GPL-3.0-only
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%
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% Copyright 2008-2024 San Diego State University Research Foundation (SDSURF).
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%
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, version 3.
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%
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% This program is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% LICENSE file or on the web GNU General Public License
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% <https://www.gnu.org/licenses/> for more details.
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%
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% ------------------------------------------------------------------------
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function D = div2DPer(k, m, dx, n, dy)
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% Returns a two-dimensional mimetic divergence operator
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% when the boundary condition is periodic
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%
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% Parameters:
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% k : Order of accuracy
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% m : Number of cells along x-axis
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% dx : Step size along x-axis
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% n : Number of cells along y-axis
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% dy : Step size along y-axis
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D = - grad2DPer(k, m, dx, n, dy);
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end

‎src/matlab/div3DPer.m‎

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% SPDX-License-Identifier: GPL-3.0-only
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%
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% Copyright 2008-2024 San Diego State University Research Foundation (SDSURF).
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%
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, version 3.
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%
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% This program is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% LICENSE file or on the web GNU General Public License
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% <https://www.gnu.org/licenses/> for more details.
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%
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% ------------------------------------------------------------------------
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function D = div3DPer(k, m, dx, n, dy, o, dz)
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% Returns a three-dimensional mimetic divergence operator
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% when the boundary condition is periodic
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%
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% Parameters:
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% k : Order of accuracy
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% m : Number of cells along x-axis
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% dx : Step size along x-axis
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% n : Number of cells along y-axis
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% dy : Step size along y-axis
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% o : Number of cells along z-axis
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% dz : Step size along z-axis
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D = - grad3DPer(k, m, dx, n, dy, o, dz);
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end

‎src/matlab/divPer.m‎

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% SPDX-License-Identifier: GPL-3.0-only
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%
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% Copyright 2008-2024 San Diego State University Research Foundation (SDSURF).
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%
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, version 3.
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%
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% This program is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% LICENSE file or on the web GNU General Public License
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% <https://www.gnu.org/licenses/> for more details.
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%
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% ------------------------------------------------------------------------
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function D = divPer(k, m, dx)
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% Returns a m+2 by m one-dimensional mimetic divergence operator
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% when the boundary condition is periodic
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%
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% Parameters:
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% k : Order of accuracy
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% m : Number of cells
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% dx : Step size
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% Assertions:
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assert(k >= 2, 'k >= 2');
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assert(mod(k, 2) == 0, 'k % 2 = 0');
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assert(m >= 2*k+1, ['m >= ' num2str(2*k+1) ' for k = ' num2str(k)]);
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D = - gradPer(k,m,dx)';
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end

‎src/matlab/grad2DPer.m‎

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% SPDX-License-Identifier: GPL-3.0-only
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%
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% Copyright 2008-2024 San Diego State University Research Foundation (SDSURF).
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%
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, version 3.
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%
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% This program is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% LICENSE file or on the web GNU General Public License
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% <https://www.gnu.org/licenses/> for more details.
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%
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% ------------------------------------------------------------------------
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function G = grad2DPer(k, m, dx, n, dy)
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% Returns a two-dimensional mimetic gradient operator
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% when the boundary condition is periodic
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%
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% Parameters:
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% k : Order of accuracy
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% m : Number of cells along x-axis
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% dx : Step size along x-axis
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% n : Number of cells along y-axis
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% dy : Step size along y-axis
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Gx = gradPer(k, m, dx);
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Gy = gradPer(k, n, dy);
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Im = speye(m, m);
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In = speye(n, n);
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Sx = kron(In, Gx);
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Sy = kron(Gy, Im);
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G = [Sx; Sy];
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end

‎src/matlab/grad3DPer.m‎

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% SPDX-License-Identifier: GPL-3.0-only
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%
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% Copyright 2008-2024 San Diego State University Research Foundation (SDSURF).
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%
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, version 3.
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%
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% This program is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% LICENSE file or on the web GNU General Public License
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% <https://www.gnu.org/licenses/> for more details.
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%
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% ------------------------------------------------------------------------
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function G = grad3DPer(k, m, dx, n, dy, o, dz)
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% Returns a three-dimensional mimetic gradient operator
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% when the boundary condition is periodic
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%
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% Parameters:
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% k : Order of accuracy
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% m : Number of cells along x-axis
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% dx : Step size along x-axis
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% n : Number of cells along y-axis
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% dy : Step size along y-axis
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% o : Number of cells along z-axis
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% dz : Step size along z-axis
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Im = speye(m, m);
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In = speye(n, n);
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Io = speye(o, o);
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Gx = gradPer(k, m, dx);
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Gy = gradPer(k, n, dy);
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Gz = gradPer(k, o, dz);
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Sx = kron(kron(Io, In), Gx);
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Sy = kron(kron(Io, Gy), Im);
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Sz = kron(kron(Gz, In), Im);
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G = [Sx; Sy; Sz];
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end

‎src/matlab/gradPer.m‎

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% SPDX-License-Identifier: GPL-3.0-only
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%
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% Copyright 2008-2024 San Diego State University Research Foundation (SDSURF).
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%
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, version 3.
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%
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% This program is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% LICENSE file or on the web GNU General Public License
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% <https://www.gnu.org/licenses/> for more details.
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%
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% ------------------------------------------------------------------------
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function G = gradPer(k, m, dx)
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% Returns a m by m+2 one-dimensional mimetic gradient operator
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% when the boundary condition is periodic
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%
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% Parameters:
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% k : Order of accuracy
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% m : Number of cells
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% dx : Step size
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% Assertions:
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assert(k >= 2, 'k >= 2');
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assert(mod(k, 2) == 0, 'k % 2 = 0');
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assert(m >= 2*k, ['m >= ' num2str(2*k) ' for k = ' num2str(k)]);
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% constructing circulant matrix
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V = sparse(1, m); % vector of values for circulant matrix
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idx = repmat(-1,m, m); % matrix of indices for circulant matrix
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idx(:,1) = 1:m;
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idx = cumsum(idx, 2);
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idx = rem(idx+m, m) + 1;
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switch k
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case 2
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V(2:3) = [1, -1];
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case 4
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V(1:4) = [-1/24, 9/8, -9/8, 1/24];
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case 6
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V(1:5) = [-25/384, 75/64, -75/64, 25/384, -3/640]; V(end) = 3/640;
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case 8
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V(1:6) = [-245/3072, 1225/1024, -1225/1024, 245/3072, -49/5120, 5/7168]; V(end-1) = -5/7168; V(end) = 49/5120;
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end
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% G constructed as a circulant matrix
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G = V(idx);
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G = (1/dx).*G;
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end

‎src/matlab/interpolCentersToFacesD1DPer.m‎

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% SPDX-License-Identifier: GPL-3.0-only
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%
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% Copyright 2008-2024 San Diego State University Research Foundation (SDSURF).
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%
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, version 3.
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%
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% This program is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% LICENSE file or on the web GNU General Public License
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% <https://www.gnu.org/licenses/> for more details.
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%
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% ------------------------------------------------------------------------
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function I = interpolCentersToFacesD1DPer(k, m)
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% 1D interpolation from centers to faces.
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% when the boundary condition is periodic
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% logical centers are [1 1.5 2.5 ... m-1.5 m-0.5]
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% m is the number of cells in the logic x-axis
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I = interpolFacesToCentersG1DPer(k,m)';
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end

‎src/matlab/interpolCentersToFacesD2DPer.m‎

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% SPDX-License-Identifier: GPL-3.0-only
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%
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% Copyright 2008-2024 San Diego State University Research Foundation (SDSURF).
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%
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, version 3.
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%
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% This program is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% LICENSE file or on the web GNU General Public License
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% <https://www.gnu.org/licenses/> for more details.
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%
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% ------------------------------------------------------------------------
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function I = interpolCentersToFacesD2DPer(k, m, n)
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% 2D interpolation from centers to faces.
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% when the boundary condition is periodic
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% logical centers are [1 1.5 2.5 ... m-1.5 m-0.5 m]x[1 1.5 2.5 ... n-1.5 n-0.5 n]
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% m and n are the number of cells in the logic x-axis and y-axis
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I = interpolFacesToCentersG2DPer(k, m, n)';
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end

‎src/matlab/interpolCentersToFacesD3DPer.m‎

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% SPDX-License-Identifier: GPL-3.0-only
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%
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% Copyright 2008-2024 San Diego State University Research Foundation (SDSURF).
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%
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, version 3.
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%
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% This program is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
11+
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% LICENSE file or on the web GNU General Public License
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% <https://www.gnu.org/licenses/> for more details.
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%
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% ------------------------------------------------------------------------
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function I = interpolCentersToFacesD3DPer(k, m, n, o)
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% 3D interpolation from centers to faces
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% when the boundary condition is periodic
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% logical centers are [1 1.5 ... m-0.5 m]x[1 1.5 ... n-0.5 n]x[1 1.5 ... o-0.5 o]
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% m, n, o, are the number of cells in the logical x-, y-, z- axes
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I = interpolFacesToCentersG3DPer(k, m, n, o)';
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end

‎src/matlab/interpolCentersToNodes1DPer.m‎

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% SPDX-License-Identifier: GPL-3.0-only
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%
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% Copyright 2008-2024 San Diego State University Research Foundation (SDSURF).
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%
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
7+
% the Free Software Foundation, version 3.
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%
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% This program is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% LICENSE file or on the web GNU General Public License
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% <https://www.gnu.org/licenses/> for more details.
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%
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% ------------------------------------------------------------------------
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function I = interpolCentersToNodes1DPer(k, m)
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% interpolation operator from nodal coordinates to staggered centers
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% when the boundary condition is periodic
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% m is the number of cells in the logical x-axis
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% nodal logical coordinates are [1:1:m]
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% centers logical coordinates [1,1.5:m-0.5,m]
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I = interpolCentersToFacesD1DPer(k, m);
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end

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