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I am using Mathematica 8.

I have two ODEs (in time) and a diffusion PDE (in radius and time) which are all coupled to each other. I solve them using NDSolve by adding a degenerate space dimension to the ODE functions. Let the space dimension be $[a, b]$. As boundary conditions for the diffusion PDE I take a value determined by one of the ODEs at $a$ and a constant 1 at $b$. Initially, I assume a constant 1 besides at $a$, where the value is given by the ODE-function's initial value.

Mathematica can solve the equations, but for some parameter sets, the concentration becomes negative - which should not happen. I want to try to increase the numerical precision of the calculation by

  • StartingStepSize -> small
  • MaxSteps -> high
  • MaxStepSize -> small
  • AccuracyGoal -> high

eventually also by increasing WorkingPrecision if necessary. The problem is that this would drastically increase calculation time and when I tried it, Mathematica crashed. At the same time, all this precision is needed for only about 20% of the spatial interval (which I want to keep large in order to minimise side effects from the box).

Therefore, my question is:

Is it possible in NDSolve to specify different values for above-mentioned options for different parts of the solution interval?

Update

As was pointed out to me in the comments, not all the options I mentioned are relevant for the spatial acccuracy. I therefore want to reformulate my question.

I see numerical errors over the first ~20% of the spatial interval and would like to increase precision there. Which of the above-mentioned options actually do apply to the spatial discretisation and how can I selectively increase them for the critical interval?

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  • $\begingroup$ StartingStepSize , MaxSteps and MaxStepSize are options to change the temporal steps. But you write that you need precision for only 20% of the spatial interval, that is to say you want to control the spatial "steps" (= "points"). Which one do you want to control ? Both ? $\endgroup$
    – andre314
    May 30 '16 at 21:42
  • $\begingroup$ Thank you for the answer. I naively assumed that those options would, in the case of PDEs, also apply to the spatial direction. I would not mind increasing the temporal precision just in case, but my feeling is that the problem lies in too rough a spatial discretisation. I will edit my post. $\endgroup$
    – Sanya
    May 31 '16 at 9:20
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    $\begingroup$ According to Albert Retey comment here, if you want to couple a ODE and a PDE you have to discretize the space and to rewrite the PDE as a set of ODEs "manually" (more precisaly with the tools described in tutorial/NDSolvePDE in the documentation). Have you already done this ? $\endgroup$
    – andre314
    May 31 '16 at 15:25
  • $\begingroup$ This is a good and interesting idea, which would at least help me in choosing the space discretisation steps myself. I will try that out as soon as my present calculation (which has been running for the last ~20h) finishes. What I have done is to follow the idea outlined in the answer of this thread: really just turned the functions from the ODE into functions also depending on space. $\endgroup$
    – Sanya
    May 31 '16 at 15:57