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Mass Conservation and Nonnegativity

A simple observation based on (3.1) is
displaymath3691
which means that the model is mass-conservative, i. e.\
 equation207
Let tex2html_wrap_inline2831 be a numerically computed approximation to tex2html_wrap_inline2833 (tex2html_wrap_inline2785). In view of property (3.4), it makes sense to impose either mass conservation (in time) on the computed approximation or nonnegativity, i. e. the restriction tex2html_wrap_inline2837 for all k, or both. Mass conservation is a natural concern for environmental simulations [30, 41]. If nonnegativity does not hold, physically meaningless mass values are calculated for a time t, and this error might propagate in later time steps. Moreover, the cytodynamic module in use might not necessarily be able to handle those erroneous values. In such a case, a filtering technique has to be used to "redistribute" negative masses such that nonnegativity or mass-consistency holds [40]. Consequently, implementations based on such techniques are not very transparent. Numerical issues with respect to nonnegativity and mass conservation will be discussed in detail in the next subsection. First, let us focus on the nonnegativity of the exact solution of (3.3).

Denote by
displaymath3692
the system matrix of (3.3), which is also the Jacobian of the right hand side with respect to m. The nonnegativity of the solution tex2html_wrap_inline2845 of (3.3) under quite weak conditions can now be seen as follows. Let A be piecewise uniform continuous and c as well as all tex2html_wrap_inline2851 (tex2html_wrap_inline2785) be piecewise continuous. Moreover, let those functions be right-sided continuous everywhere. Interpolate A by a piecewise constant function tex2html_wrap_inline2857 on a grid tex2html_wrap_inline2859 containing the points of discontinuity of both A, c, and tex2html_wrap_inline2851 (tex2html_wrap_inline2785) and such that tex2html_wrap_inline2869 for all tex2html_wrap_inline2871 (tex2html_wrap_inline2873). If A is replaced by tex2html_wrap_inline2857 in (3.3), the corresponding solution tex2html_wrap_inline2879 on the interval tex2html_wrap_inline2881 can be written as
displaymath3693
(tex2html_wrap_inline2873), where, as usually, the matrix exponential function is given by
displaymath3694
for arbitrary square matrices M. Since the nondiagonal elements of A are nonnegative, all the elements of tex2html_wrap_inline2889 are nonnegative for all tex2html_wrap_inline2891. If tex2html_wrap_inline2893 (tex2html_wrap_inline2785, tex2html_wrap_inline2873) holds, the nonnegativity of the functions tex2html_wrap_inline2899 (tex2html_wrap_inline2785) for tex2html_wrap_inline2891 follows if, in addition, c is nonnegative. But tex2html_wrap_inline2879 converges uniformly to the solution m of (3.3) since tex2html_wrap_inline2857 and tex2html_wrap_inline2913 converge uniformly to A and c when the grid is refined.


next up previous contents
Next: Numerics Up: The Standard Model Previous: The Standard Model

Joerg Fliege
Wed Dec 22 12:25:31 CET 1999