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### Mass Conservation and Nonnegativity

A simple observation based on (3.1) is

which means that the model is mass-conservative, i. e.\

Let be a numerically computed approximation to (). 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 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

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 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 () be piecewise continuous. Moreover, let those functions be right-sided continuous everywhere. Interpolate A by a piecewise constant function  on a grid containing the points of discontinuity of both A, c, and () and such that for all (). If A is replaced by  in (3.3), the corresponding solution  on the interval can be written as

(), where, as usually, the matrix exponential function is given by

for arbitrary square matrices M. Since the nondiagonal elements of A are nonnegative, all the elements of are nonnegative for all . If (, ) holds, the nonnegativity of the functions () for follows if, in addition, c is nonnegative. But  converges uniformly to the solution m of (3.3) since  and  converge uniformly to A and c when the grid is refined.

Next: Numerics Up: The Standard Model Previous: The Standard Model

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