• Applied mathematics: the backward diffusion problem.
  
  • Introduction:
    This part focuses on the backward diffusion problem, also called backward heat conduction problem, which consists in a partial differential equation in space and time with border and initial conditions. In the case of Neumann boundary condition, the complete system writes:
    
    Any attempt of solving this equation directly reveals a contamination of the solution, as illustrated with the example of a gaussian initial condition:
    
    At small times (10^(-5)), the solution is contaminated by instabilities of growing amplitude, to such stage that the solution looses its meaning. To better understand the origin of this instability, let us have a look over the dispersion relation:
    
    On the example of a periodic initial condition (in blue on the next plot), in the Fourier space, the main mode of the function (wavenumber k=6) can be identified, the other modes being truncated at the roundoff precision of the machine (double precision ~10^(-16)).
    
    The logarithmic representation clearly exhibits the quadratical growth of wavenumbers (in red), whose amplitude becomes larger than that of the main mode: the solution has lost its sense. Some techniques may be employed to limit the action of instabilities: filtering and regularization.
  
  
  • Methods:
    Filtering consists in eliminating the unstable modes, as shown in the next figure, on another example.
    
    The regularization method we use consists in introducing a regularization parameter into the partial differential equation in the aim of controlling the growth of unstable modes. In our particular case, a four order spatial derivative term has been chosen (note that another boundary condition must be introduced). The system finally becomes:
    
    with the following dispersion relation:
    
    The growth of unstable modes is then reversed, as illustrated on the next exmaple (periodic initial condition):
    
    The black curve clearly features the effect of the regularizing term with an reversion at the wavenumber k=26.
  
  
  • Results:
    The main consequence that can be expected from these techniques is the increase of the simulation time, as revealed by the next plots obtained at T=0.01 with the filtering technique:
    
    The solution to retrieve is plotted in blue, the computed solution in black. Globally, the error estimate is of the magnitude of 10^(-2).
    The same observation is done with regularization, whose result is presented hereafter for T=0.01.
    
  
  
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© 2009 Fabien Ternat