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differential equations - How to modify NDSolve`StateData without crashing the kernel?


Probably a hard question, but it's better to cry out loud.



Reminded by Chris K, I noticed my fix function has been broken since v11.3. After some checking, I found NDSolve`StateData[…], which is not an atom in v9 and becomes an atom now, can no longer be modified with pattern matching.


To keep fix up to date, I tried the solution in this excellent post about modifying data inside atom, sadly the modified NDSolve`StateData crashes the kernel in the NDSolve`ProcessSolutions stage even in v9 and I can't figure out a workaround.


The following is a simplified example attempting to modify the difference order to 2. The last line crashes the kernel of course so please save your work before testing.


tmax = 10; lb = 0; rb = 5;
system = With[{u = u[t, x]}, {D[u, t] == D[u, x, x], u == 0 /. t -> 0, 
                              u == Sin[t] /. x -> lb, u == 0 /. x -> rb}];

{state} = NDSolve`ProcessEquations[system, u, {t, 0, tmax}, {x, lb, rb}];

teststate = 

  state /. a_NDSolve`FiniteDifferenceDerivativeFunction :> 
    RuleCondition@
     NDSolve`FiniteDifferenceDerivative[a@"DerivativeOrder", a@"Coordinates", 
      "DifferenceOrder" -> 2, PeriodicInterpolation -> a@"PeriodicInterpolation"];

Head[#]["DifferenceOrder"] & /@ 
 teststate["NumericalFunction"]["FunctionExpression"][[2, 1]]
(* Should give {{2}} if the replacement succeeds. *)

(*Failed attempt: *)

ml = LinkCreate[LinkMode -> Loopback];
LinkWrite[ml, With[{e = state}, Hold[e]]];
holdstate = LinkRead[ml];

newstate = holdstate /. 
   a_NDSolve`FiniteDifferenceDerivativeFunction :> 
    RuleCondition@
     NDSolve`FiniteDifferenceDerivative[a@"DerivativeOrder", a@"Coordinates", 
      "DifferenceOrder" -> 2, PeriodicInterpolation -> a@"PeriodicInterpolation"] // 
  ReleaseHold


NDSolve`Iterate[newstate, tmax]

(*Warning: the following line crashes the kernel.*)
u /. NDSolve`ProcessSolutions@newstate

Any way to avoid the crashing and fix my fix function?



A bit of spelunking shows this seems to be (at least partly) related to Internal`Bag. A minimal example:


bag = Internal`Bag[]

ml = LinkCreate[LinkMode -> Loopback];
LinkWrite[ml, With[{e = bag}, Hold[e]]]
holdbag = LinkRead[ml]
LinkClose[ml]
ReleaseHold[holdbag] === bag
(* False *)


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