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Numerical simulations of multi-scale phenomena are commonly used for modeling purposes in many applications such as combustion, chemical vapor deposition, or air pollution modeling. In general, all these models raise several difficulties created by the large number of unknowns and the wide range of temporal scales due to large and detailed chemical kinetic mechanisms, as well as steep spatial gradients associated with very localized fronts of high chemical activity. Furthermore, a natural stumbling block to perform 3D simulations is the unreasonable memory requirements of standard numerical strategies for time dependent problems.
An alternative numerical strategy is then to combine implicit and explicit schemes to discretize nonlinear evolution problems in time. Further studies settled the appropriate numerical background for these methods called IMEX, which in particular might be conceived to solve stiff nonlinear problems [2, 3]. These methods are usually very efficient. Nevertheless, on the one hand, the feasibility of utilizing dedicated implicit solvers over a discretized domain becomes soon critical when treating large computational domains. And on the other hand, the time steps globally imposed over partial regions or the entire domain are strongly limited by either the stability restrictions of the explicit solver or by the fastest scales treated by the implicit scheme.
In the context of multi-scale waves, the dedicated methods chosen for each subsystem are then responsible for dealing with the fastest scales associated with each one of them, in a separate manner; then, the composition of the global solution based on the splitting scheme should guarantee the good description of the global physical coupling. A rigorous numerical analysis is therefore required to better establish the conditions for which the latter fundamental constraint is verified. As a matter of fact, several works [4, 5, 1] proved that the standard numerical analysis of splitting schemes fails in the presence of scales much faster than the splitting time step and motivated more rigorous studies for these stiff configurations [6, 7] and in the case where spatial multi-scale phenomena arise as a consequence of steep spatial gradients [8].
Higher order splitting configurations are also possible. Nevertheless, the order conditions for such composition methods state that either negative time substeps or complex coefficients are necessary (see [9]). The formers imply normally important stability restrictions and more sophisticated numerical implementations. In the particular case of negative time steps, they are completely undesirable for PDEs that are ill-posed for negative time progression.
The standard orders achieved with a Lie or Strang scheme are no longer valid when we consider very stiff reactive or diffusive terms (see [6]). Furthermore, if the fastest time scales play a leading role in the global physics of the phenomenon, then the solution obtained by means of a splitting composition scheme will surely fail to capture the global dynamics of the phenomenon, unless we consider splitting time steps small enough to resolve such scales.
In the opposite case where these fast scales are not directly related to the physical evolution of the phenomenon, larger splitting time steps might be considered, but order reductions may then appear due to short-life transients associated with the fastest variables. This is usually the case for propagating reaction waves where for instance, the speed of propagation is much slower than the chemical scales. In this context, it has been proved in [6] that better performances are expected while ending the splitting scheme by the time integration of the reaction part (4):
On the other hand, one must also take into account possible order reductions coming this time from space multi-scale phenomena due to steep spatial gradients whenever large splitting time steps are considered, as analyzed in [8]:
Radau5 [9] is a high order method (formally of order , which at worst might be reduced to ) that is not only an A-stable method, but also L-stable, so that very stiff systems of ODEs might be solved without any stability problem. It considers also an adaptive time stepping strategy which guarantees a requested accuracy of the numerical integration and at the same time, allows to discriminate stiff zones from regular ones; hence, smaller time steps correspond to stiffer behaviors.
Another important feature of this strategy is that precious computation time is saved because we adapt the time integration step only at nodes where the reaction phenomenon takes place. For multi-scale reaction waves, this happens in a very low percentage of the spatial domain, normally only in the neighborhood of the wavefront. Therefore, larger time steps are considered at nodes with a chemistry at (partial) equilibrium. This would not be possible if we integrated the entire reaction-diffusion system (2) at once.
We are concerned with the propagation of reacting wavefronts, hence important reactive activity as well as steep spatial gradients are localized phenomena. This implies that if we consider the resolution of reactive problem (4), a considerable amount of computing time is spent on nodes without any chemical activity (see for example a precise computing time evaluation in [15]). Moreover, there is no need to represent these quasi-stationary regions with the same spatial discretization needed to describe the reacting wavefront, so that the diffusion problem (3) might also be solved over a smaller number of nodes. An adapted mesh obtained by a multiresolution process [12, 13] then turn out to be a very convenient solution to overcome these difficulties.
The dynamical system associated with this system models reactive excitable media with a large time scale spectrum (see [18] for more details). Moreover, the spatial configuration with addition of diffusion generates propagating wavefronts with steep spatial gradients. Hence, this model presents all the difficulties associated with a stiff multi-scale configuration. Even though the system structure allows an eventual discrimination of slow and fast variables, and thus other numerical strategies as partitioning methods might be considered [19], in this work we are not interested in performing such decomposition in order to remain consistent with more complex models for which such decomposition becomes complicated, as it was discussed in the Introduction. The advantages of applying a splitting strategy to these models have already been studied and presented in [20]. In what follows, we will briefly consider a 1D case of (13) in order to illustrate the errors of splitting schemes for stiff problems, then 2D and 3D configurations will be implemented.
In this part, we perform a short illustrating study of the behavior of splitting schemes when dealing with stiff problems as it was explained in 2.2. In the BZ model, stiffness is given by fast time scales as well as steep spatial gradients; we consider then a 1D configuration of problem (13) with homogeneous Neumann boundary conditions and the following parameters, taken from [18]: , , and , with diffusion coefficients , and , for a space region of . A uniform mesh of points is considered while exact solution is approximated by a reference or quasi-exact solution of coupled reaction-diffusion problem (13) performed by Radau5 with very fine tolerances. Splitting schemes consider Radau5 and ROCK4 as resolution methods for reaction and diffusion problems as it was presented in 2.2.
Figure 1 shows that both Lie and Strang schemes have asymptoticly local order 2 and 3 for small time steps. Nevertheless, for larger time steps, previous studies in [6] and [8] describe better the numerical behavior of these schemes. For , order drops to as predicted by (7); whereas for , we see the influence of spatial gradients as predicted by (11) and thus, order is recovered after some transition phase. Same conclusions are drawn for Strang schemes, order of drops from to according to (8), while for , we see the influence of steep spatial gradients that alter the order given by (10). Maximum error considers the maximum value between computed normalized local errors for , and variables; in these numerical tests, it corresponds to variable . Finally, in all cases the reaction ending schemes show better behaviors for larger splitting time steps, according to [6]. In particular, behaves even better than , whereas is the best alternative for all time steps.
We have focused our attention on reaction-diffusion systems in order to settle the foundations for simulation of more complex phenomena with fully convection-reaction-diffusion systems or more detailed models. In particular, we search to extend this strategy to multi-scale problems in combustion domain, where other numerical techniques have already been proposed (see for instance [21, 22]) and where there is a continuous research of new and more efficient methods. Therefore, an important amount of work is still in progress concerning on the one hand, programming features such as data structures, optimized routines and parallelization strategies. And on the other hand, numerical analysis of theoretical aspects, which may surely allow the introduction of dynamical error estimators and even adaptive splitting time steps for more general multi-scale phenomena [23, 24]; these are particular topics of our current research. 041b061a72