function plan = plate_fd_plan(params)
% PLATE_FD_PLAN  Grid, time step and cost of the explicit finite-difference scheme.
%
%   plan = plate_fd_plan(params) returns
%     plan.nx, plan.ny, plan.dx, plan.dy   grid (nodes include the edges of the bounding box)
%     plan.dt_max                          largest stable time step [s]
%     plan.dt                              step actually used, params.fd.cfl * dt_max
%     plan.nsteps, plan.ncells, plan.cell_steps   problem size (ncells counts every grid node)
%     plan.nunknowns                       nodes the scheme actually updates
%
%   Stability.  Von Neumann analysis of the scheme in plate_fd for a Fourier
%   mode on which the discrete Laplacian has eigenvalue -lambda gives the
%   amplification polynomial
%     (1 + sigma0 dt) z^2 - (2 - dt^2 kappa^2 lambda^2 - 2 sigma1 dt lambda) z
%                         + (1 - sigma0 dt - 2 sigma1 dt lambda) = 0.
%   Both roots stay inside the unit circle for every lambda up to the largest
%   eigenvalue of the five-point Laplacian, lambda_max = 4/dx^2 + 4/dy^2, iff
%     dt <= 2 / ( lambda_max (sigma1 + sqrt(kappa^2 + sigma1^2)) ),
%   which for a square grid without sigma1 is the familiar dt <= dx^2 / (4 kappa).
%   Because dt scales with dx^2, halving the grid spacing costs eight times
%   as many cell updates for the same simulated time.
%
%   The same bound serves the free and centre-pinned plates and the ellipse:
%   the Dirichlet Laplacian of any sub-domain has its spectrum inside that of
%   the full grid, the clamped correction of plate_matrices does not raise the
%   row sums above the interior stencil's, and the largest eigenvalue of the
%   energy-method operator M \ K lies just below lambda_max^2 (about 0.97 to
%   0.99 of it on the grids used here), so the step is safe and slightly
%   conservative there.

P = plate_setup(params);
plan.nx = P.fd.nx;
plan.ny = P.fd.ny;
if plan.nx < 5 || plan.ny < 5
    error('plate_fd_plan: the grid needs at least 5 nodes in each direction');
end
plan.dx = P.plate.Lx / (plan.nx - 1);
plan.dy = P.plate.Ly / (plan.ny - 1);
lambda_max = 4 / plan.dx^2 + 4 / plan.dy^2;
s1 = P.damping.sigma1;
plan.dt_max = 2 / (lambda_max * (s1 + sqrt(P.kappa^2 + s1^2)));
plan.dt = P.fd.cfl * plan.dt_max;
plan.nsteps = ceil(P.fd.duration / plan.dt);
plan.ncells = plan.nx * plan.ny;
dom = plate_domain(P.plate, plan.nx, plan.ny);
if any(strcmp(P.bc, {'free', 'pinned-center'}))
    plan.nunknowns = dom.nunknowns_free;
else
    plan.nunknowns = dom.nunknowns_dirichlet;
end
plan.cell_steps = plan.ncells * plan.nsteps;
end