Physically the even moments represent positive quantities. The same property should be valid in the model. It was observed that the emergence of negative concentrations during a simulation run almost always resulted in non-convergence of the solver.
An obvious idea to avoid negative concentrations is to write the equations in terms of the logarithm of the even moments. Then this problem is eliminated. However, for the discretization the use of logarithmic variables is of disadvantage.
The dilemma can be solved in a similar way in which we tackled the problem of the formulation of the residual function. Again we use linear variables for calculation of the Jacobian. However, when doing the linesearch we use logarithmic variables.
The linesearch
in the space of linear variables proceeds from the
origin along a straight line. Linesearch in the
logarithmic quantities starts from the same
origin, but now we move along a straight line
in the space of logarithmic quantities .
Both curves have the same tangent in the
origin. Hence we can calculate the logarithmic direction
from the linear direction
using the chain rule for differentiation.
The same trick can
be used for any set of independent variables.
The implementation in MINIMOS-NT uses
and
![]() |
(4.7) |
Compared to linear linesearch the logarithmic linesearch has very favorable properties and enhances the robustness of the solver.
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