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idc_step.m
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function sol = idc_step(ode,ti,yi,opts,param)
% get "unscaled" quadrature nodes
switch opts.grid
case 1
% uniform nodes
t = linspace(0,1,opts.nquad);
case 2
% Gaussian (Gauss Legendre) nodes
t = gaussnodes(0,1,opts.nquad);
case 3
% Chebychev
t = chebyshevnodes(0,1,opts.nquad);
case 4
% Gauss-Lobatto nodes
t = gauss_lobatto(0,1,opts.nquad);
otherwise
error('unknown option in opts.grid')
end % switch opts.grid
m = length(yi);
% storing solution at quadrature nodes
m = length(yi);
y{1} = zeros(m,opts.nquad);
h = (t(2:end)-t(1:end-1))*param.dt;
% predictor
level = 1;
switch param.pred.type
case{0}
% explicit
% treat first substep specially since we don't want to pollute
% structure for values at quadrature nodes
K = zeros(m,param.pred.s);
tk = ti;
switch opts.grid
case {1,4}
% uniform node/gauss lobattos (includes left endpoint)
y{1}(:,1) = yi;
otherwise
h0 = param.dt*(t(1)-0);
for s = 1:param.pred.s
temp = yi + K(:,1:s-1)*param.pred.A(s,1:s-1)';
K(:,s) = h0*ode(tk+param.pred.c(s)*h0,temp);
end
y{1}(:,1) = yi + K*param.pred.b';
tk = tk+h0;
end
for n = 2:opts.nquad
K = zeros(m,param.pred.s);
for s = 1:param.pred.s
temp = y{1}(:,n-1) + K(:,1:s-1)*param.pred.A(s,1:s-1)';
K(:,s) = h(n-1)*ode(tk+param.pred.c(s)*h(n-1),temp);
end
y{1}(:,n) = y{1}(:,n-1) + K*param.pred.b';
tk = tk + h(n-1);
end
case{1}
% diagonally implicit
% structure for values at quadrature nodes
K = zeros(m,param.pred.s);
tk = ti;
switch opts.grid
case {1,4}
% uniform node/gauss lobattos (includes left endpoint)
y{1}(:,1) = yi;
otherwise
h0 = param.dt*(t(1)-0);
for s = 1:param.pred.s
temp = yi + K(:,1:s-1)*param.pred.A(s,1:s-1)';
func_hand = @(Ki) Ki - h0*ode(tk+param.pred.c(s)*h0,...
temp+param.pred.A(s,s)*Ki);
k_init = param.dt*ode(tk+param.pred.c(s)*h0,...
yi+param.pred.c(s)*param.dt*ode(tk,yi));
[K(:,s),~,flag] = ...
fsolve(func_hand, k_init, optimset('Display','off',...
'TolFun',1e-9));
if flag > 1
error('did not converge to root')
end
end
y{1}(:,1) = yi + K*param.pred.b';
tk = tk+h0;
end
for n = 2:opts.nquad
K = zeros(m,param.pred.s);
for s = 1:param.pred.s
temp = y{level}(:,n-1) + K(:,1:s-1)*param.pred.A(s,1:s-1)';
func_hand = @(Ki) Ki - h(n-1)*ode(tk+param.pred.c(s)*h(n-1),...
temp+param.pred.A(s,s)*Ki);
k_init = param.dt*ode(tk+param.pred.c(s)*h(n-1),...
y{level}(:,n-1)+...
param.pred.c(s)*param.dt*ode(tk,y{level}(:,n-1)));
[K(:,s),~,flag] = ...
fsolve(func_hand, k_init, optimset('Display','off',...
'TolFun',1e-9));
if flag > 1
error('did not converge to root')
end
end
y{1}(:,n) = y{1}(:,n-1) + K*param.pred.b';
tk = tk + h(n-1);
end
otherwise
error('fully implicit integrators not coded yet')
end
if opts.levels > 1
% there are correction loops!
% in practice, can compute S{} and L{} once and reuse for all
% large intervals
% note, we scale by dt later -- this is the key difference from
% former code
S = generate_integration_matrix(t,t);
% n = 1
Sstage{1} = generate_integration_matrix(t,param.corr.c*t(1));
Lstage{1} = generate_interpolation_matrix(t,param.corr.c*t(1));
for n = 2:opts.nquad
Sstage{n} = generate_integration_matrix(t,t(n-1)+param.corr.c*(t(n)-t(n-1)));
Lstage{n} = generate_interpolation_matrix(t,t(n-1)+param.corr.c*(t(n)-t(n-1)));
end
end
for level = 2:opts.levels
y{level} = zeros(m,opts.nquad);
% correction loops -- can generalize to different tableaus later
% compute f(t,y) at quadrature points
f = zeros(m,opts.nquad);
for n = 1:opts.nquad
f(:,n) = ode(ti+param.dt*t(n),y{level-1}(:,n));
end
switch param.corr.type
case{0}
% explicit
K = zeros(m,param.corr.s);
tk = ti;
switch opts.grid
case {1,4}
% uniform nodes/gauss lobatto (includes left endpoint)
y{level}(:,1) = yi;
otherwise
% again, handle first time step differently
for s = 1:param.corr.s
temp = yi + K(:,1:s-1)*param.corr.A(s,1:s-1)' + ...
param.dt * f * Sstage{1}(s,:)';
K(:,s) = h0*ode(tk+param.corr.c(s)*h0,temp) - ...
h0*f*Lstage{1}(s,:)';
end
y{level}(:,1) = yi + K*param.corr.b' + param.dt*f*S(1,:)';
tk = tk+h0;
end
for n = 2:opts.nquad
K = zeros(m,param.corr.s);
for s = 1:param.corr.s
temp = y{level}(:,n-1) + K(:,1:s-1)*param.corr.A(s,1:s-1)' + ...
param.dt * f * (Sstage{n}(s,:)' - S(n-1,:)');
K(:,s) = h(n-1)*ode(tk+param.corr.c(s)*h(n-1),temp) - ...
h(n-1)*f*Lstage{n}(s,:)';
end
y{level}(:,n) = y{level}(:,n-1) + K*param.corr.b' + param.dt*f*(S(n,:)'-S(n-1,:)');
tk = tk + h(n-1);
end
case{1}
% diagonally implicit
K = zeros(m,param.corr.s);
tk = ti;
switch opts.grid
case {1,4}
% uniform nodes/gauss lobatto (includes left endpoint)
y{level}(:,1) = yi;
otherwise
% again, handle first time step differently
for s = 1:param.corr.s
temp = yi + K(:,1:s-1)*param.corr.A(s,1:s-1)' + ...
param.dt * f * Sstage{1}(s,:)';
func_hand = @(Ki) Ki - ...
h0*ode(tk+param.corr.c(s)*h0,...
temp+param.corr.A(s,s)*Ki) + ...
h0*f*Lstage{1}(s,:)';
k_init = h0*ode(tk+param.corr.c(s)*h0,...
yi+param.corr.c(s)*param.dt*ode(tk,yi));
[K(:,s),~,flag] = ...
fsolve(func_hand, k_init, optimset('Display','off',...
'TolFun',1e-9));
if flag > 1
error('did not converge to root')
end
end
y{level}(:,1) = yi + K*param.corr.b' + param.dt*f*S(1,:)';
tk = tk+h0;
end % switch
for n = 2:opts.nquad
K = zeros(m,param.corr.s);
for s = 1:param.corr.s
temp = y{level}(:,n-1) + K(:,1:s-1)*param.corr.A(s,1:s-1)' + ...
param.dt * f * (Sstage{n}(s,:)' - S(n-1,:)');
func_hand = @(Ki) Ki - ...
h(n-1)*ode(tk+param.corr.c(s)*h(n-1),...
temp+param.corr.A(s,s)*Ki) + ...
h(n-1)*f*Lstage{n}(s,:)';
k_init = h(n-1)*ode(tk+param.corr.c(s)*h(n-1),...
y{level}(:,n-1)+param.corr.c(s)*param.dt*ode(tk,yi));
[K(:,s),~,flag] = ...
fsolve(func_hand, k_init, optimset('Display','off',...
'TolFun',1e-9));
if flag > 1
error('did not converge to root')
end
end
y{level}(:,n) = y{level}(:,n-1) + K*param.corr.b' + param.dt*f*(S(n,:)'-S(n-1,:)');
tk = tk + h(n-1);
end
otherwise
% implicit
error('fully implicit integrators not coded yet');
end
end
if ~isfield(opts,'interp')
% if opts.interp is not set, default to collocation solution
opts.interp = 0;
end
switch opts.grid
case{1,4} % uniform or gauss--lobatto
sol = y{opts.levels}(:,end);
otherwise % chebychev or gauss--legendre
if opts.interp == 1
% use interpolation!
Lfinal = generate_interpolation_matrix(t,1);
sol = y{opts.levels}*Lfinal';
else
% generate picard/collocation solution
S_final = generate_integration_matrix(t,1);
sol = yi;
for n = 1:opts.nquad
sol = sol + param.dt*ode(ti + t(n)*param.dt,y{opts.levels}(:,n))*S_final(n);
end
end
end