% The leapfrog scheme on a square membrane.
%
% The two-dimensional version of Chapter 2's scheme, with the five-point
% Laplacian in space:
%
%   u(i,j,n+1) = 2 u(i,j,n) - u(i,j,n-1)
%              + lam^2 (u(i+1,j) + u(i-1,j) + u(i,j+1) + u(i,j-1) - 4 u(i,j)),
%
% on the unit square with u = 0 on the rim.  It is stable exactly when
% lam <= 1/sqrt(2), and there is no Courant number at which it is exact.
%
% u and uold are (N+1)-by-(N+1), rim included, at steps 0 and -1.  The scheme
% advances M steps and returns the displacement at node (ir, jr) after each
% one, and the whole field, as columns u(:), after each of the first K.  The
% `scheme` argument is ignored here; Chapter 5's version of this function
% offers three.

function [y, U] = drum2d(u, uold, lam, M, scheme, ir, jr, K)
n1 = size(u, 1);
l2 = lam ^ 2;
y = zeros(M, 1);
U = zeros(n1 * n1, K);
unew = zeros(n1, n1);
for n = 1:M
    for j = 2:n1 - 1
        for i = 2:n1 - 1
            unew(i, j) = 2 * u(i, j) - uold(i, j) ...
                + l2 * (u(i + 1, j) + u(i - 1, j) + u(i, j + 1) + u(i, j - 1) - 4 * u(i, j));
        end
    end
    uold = u;
    u = unew;
    y(n) = u(ir, jr);
    if n <= K
        U(:, n) = u(:);
    end
end
end