Numerical modeling time stepping process

Equation 1.3 represents a system of usually several thousand coupled differential equations of second order. It can be solved only numerically in small time steps At via finite-difference methods [16]. There always the situation at t + At is calculated from the situation at t. Considering the very fast oscillations of covalent bonds, At must not be longer than about 1 fs to avoid numerical breakdown connected with problems with energy conservation. This condition imposes a limit of the typical maximum simulation time that for the above-mentioned system sizes is of the order of several ns. The limited possible size of atomistic polymer packing models (cf. above) together with this simulation time limitation also set certain limits for the structures and processes that can be reasonably simulated. Furthermore, the limited model size demands the application of periodic boundary conditions to avoid extreme surface effects. 

At least for a first approach, the active component in the strain-stress relation may be treated in a simple manner. For some strain emax the active stress aa is maximum, and on both sides the stress decreases almost linearly with e — emax. Moreover, the stress is proportional to the muscle tone xjr. By numerically integrating the passive and active contributions across the arteriolar wall, one can establish a relation among the equilibrium pressure Peq, the normalized radius r, and the activation level xjr [19]. This relation is based solely on the physical characteristics of the vessel wall. However, computation of the relation for every time step of the simulation model is time-consuming. To speed up the process we have used the following analytic approximation [12] ... 

The solution to this problem consists in the application of numerical solutions when diagenetic processes are modeled. Such numerical solutions always divide the continuum of reaction space and reaction time into discrete cells and discrete time intervals. If one divides up the continuum of space and time to a sufficient degree into discrete cells and time steps (which is not the decisive problem with the possibilities given by today s computers), one will be able to apply much simpler and better manageable conditions within the corresponding cells, and with regard to the expansion of a time interval, so that, in their entirety, they still will describe a complex system. Thus, it is possible, for example, to apply the two-step-procedure (Schulz and Reardon 1983), in which the individual observation of physical transport (advection, dispersion, diffusion) or any geochemical multiple component reaction is made feasible within one interval of time. 

The differential equations are stiff that is, several processes are going on at the same time, but at widely differing rates. This is a common feature of chemical kinetic equations and makes the numerical solution of the differential equations difficult. A steady state is never reached, so the equations cannot be solved analytically. Traditional methods, such as the Euler method and the Runge-Kutta method, use a time step, which must be scaled to fit the fastest process that is occurring. This can lead to large number of iterations even for small time scales. Hence, the use of Stella to model this oscillatory reaction would lead to an impossible situation. 

This model assumes that the aggregation process itself does not require additional time, i.e. there is always equilibrium between monomers and aggregates in the adsorption layer. To solve this set of equations numerically, first-order finite difference schemes can be applied as described in detail by Aksenenko et al. (1998a). The entire numerical procedure allows step-wise calculation of the time dependencies r(t) or n(t). 

There are three ways to simulate reaction-diffusion system. The traditional method is to solve partial differential equation directly. Another way is to divide system into cells, which is called cell dynamic scheme (CDS). Typical models are cellular automata (CA)[176] and coupled map lattice (CML)[177]. In cellular automata model, each value of the cell (lattice) is digital. On the other hand, in coupled map lattice model, each value of the lattice (cell) is continuous. CA model is microscopic while CML model is mesoscopic. The advantage of the CML is compatibility with the physical phenomena by smaller number of cells and numerical stability. Therefore, the model based on CML is developed. Each cell has continuum state and the time step is discrete. Generally, each cell is static and not deformable. Deformable cell (lattice) is supposed in order to represent deformation process of the gel. Each cell deforms based on the internal state, which is determined by the reaction between the cell and the environment. 

---