Starting NDSolve from intermediate time step?

Question

I always wondered if I could start NDSolve from an intermediate time step.

What I mean is, in the code sample below, if I were to run my solution from tmin=0 to tmax=2 and then I realize that my solution hasn't converged yet, could I just change my tmin to 2 and then proceed with a larger tmax (in this case the solution coverges [attains steady state in this example] at tmax=5).

This is because very often I solve 4th order Nonlinear PDEs with NDSolve, some of which take over 70-80 minutes to complete (as I run them for long times of a million time steps or so) and then I realize that the solution hasn't converged yet. So can I just, as in my presumed working example, change my tmin to my tmax from the previous run and change my tmax to a larger run?

tmin = 0;
tmax = 5;
sol = NDSolve[{D[u[t, x], t] == D[u[t, x], x, x], u[0, x] == 0, 
    u[t, 0] == Sin[t], u[t, 5] == 0}, u, {t, tmin, tmax}, {x, 0, 5}];
Plot3D[Evaluate[u[t, x] /. sol], {t, tmin, tmax}, {x, 0, 5}, 
 PlotRange -> All]

Answer 1

You could integrate up to some intermediate time, tintermediate, and then feed the result as initial conditions to the solver to propagate from tintermediate to tmax, like so:

tmin = 0;
tintermediate = 2;
tmax = 5;
sol = NDSolve[{D[u[t, x], t] == D[u[t, x], x, x], u[0, x] == 0, 
    u[t, 0] == Sin[t], u[t, 5] == 0}, 
   u, {t, tmin, tintermediate}, {x, 0, 5}];
sol2 = NDSolve[{D[u[t, x], t] == D[u[t, x], x, x],
    u[tintermediate, x] \[Equal] First@(u[tintermediate, x] /. sol), 
    u[t, 0] == Sin[t], u[t, 5] == 0}, 
   u, {t, tintermediate, tmax}, {x, 0, 5}];

Answer 2

As it turns out, the designers of NDSolve[] have precisely anticipated this sort of use; this is where you can use the NDSolve`StateData framework.

To use acl's example:

(* prepare PDE *)
state = First[NDSolve`ProcessEquations[{D[u[t, x], t] == D[u[t, x], x, x], 
 u[0, x] == 0, u[t, 0] == Sin[t], u[t, 5] == 0}, u, t, {x, 0, 5}]];


(* go up to t = 2 *)
NDSolve`Iterate[state, {0, 2}]
NDSolve`ProcessSolutions[state, "Forward"]

(* go up to t = 5 *)
NDSolve`Iterate[state, 5];
NDSolve`ProcessSolutions[state, "Forward"]

See the docs for more details.