Gridding, adaptive method

Figure 4. An adaptive gridding method, in which the fine resolution is clustered around the flame front, reduces computational time to two days...

Figure 5. The as yet undeveloped injected adaptive gridding method will reduce the computational time to 1100 sec.

Tomlin, A., Berzins, M., Ware, J., Smith, J., Pilling, M.J. On the use of adaptive gridding methods for modelling chemical transport from multi-scale sources. Atmos. Environ. 31, 2945-2959... 

A generalized partial differential equation solver which handles simultaneous parabolic, one dimensional elliptic, ordinary and integral equations and uses B-splines with an adaptive grid was written to solve the model. Further details on the model and solution method can be found in Reference 14. 

Saltzman, J. A Variational Method for Generating Multidimensional Adaptive Grids Courant Mathematics and Computing Laboratory, New York University, 1982. 

Dandekar, H. W., Hlavacek, V and Degreve, J., An explicit 3D finite-volume method for simulation of reactive flows using a hybrid moving adaptive grid. Numer. Heat Trans., B, 24, 1 (1993). 

The challenge is to construct reduced chemistry models which are fast to evaluate, yet which still satisfy the error tolerance Eq. (13). One effective approach to this is the Adaptive Chemistry method (Schwer et al., 2003a, b), where different reduced chemistry models are used under different local reaction conditions. For example, in the 1—d steady premixed flame studied by Oluwole et al. (Oluwole et al., 2006), six different reduced chemistry models were used, and the full chemistry model only had to be used at about 20% of the grid points, Fig. 14. 

As can be inferred from the above discussion, the implementation of adaptive grids is not straightforward, involves additional steps that may make the simulation less efficient and requires the selection of suitable criteria for the location of the moving front. Moreover, the munber of experimental systems that present sharp concentration variations away from the electrode surface is quite limited. Therefore, the universality and value of adaptive methods may be called into question, and their use restricted to the development of general simulation packages where the simulation parameters are optimized automatically. 

For the solution three different methods are used. First, a fully implicit non-adaptive Euler-method, second, a method with constant and equidistant grids and time-error-estimation (LIMEX) [2], third the fully adaptive code PDEXPACK [3, 7]. PDEXPACK was developed to use PDEX for chemical engineering problems and contains special features like internal boundaries, internal sources or state variables, which are not defined on the whole spatial domain etc. 

The comparison of three different solution methods with and without spatial and temporal adaptivity for systems of parabolic differential equations has shown the necessity of using a fully adaptive solution method to achieve authentic solutions. The fully adaptive method liberates the user from choosing grids and time stepsizes. This saves much time because only one solution has to be computed, no comparison of different solutions is necessary. By choosing timesteps and grids which promise optimal performance, the computation time needed for adaptive methods can be optimized. Comparing the solutions of identical quality, the fully adaptive method clearly outperforms its alternatives for the examples considered. 

Petzold, L. R., An adaptive Moving Grid Method for One-Dimensional Systems of Partial Differential Equations and its Numerical Solution, Proc. Workshop on Adaptive Methods for Partial Differential Equations, Renselaer Polytechnic Institute (1988)... 

The numerical solution is performed by the method of lines. Spatial discretization of the partial differential-equation system using finite differences on statically adapted grids leads to large systems of ordinary differential and algebraic equations. This system of coupled equations is solved by an implicit extrapolation method using the software package LIMEX [14]. The code computes species mass-fraction and temperature profiles in the gas phase, fluxes at the gas-surface interface, and surface temperature and coverage as function of time. 

Verwer, J.G., Blom, J.G. and Sanz-Sema, 1981, An adaptive moving grid method for one-dimensional systems of partial differential equations. Journal of Computational Physics, 83(2), 454-486. 

This method was later generalized to systems with nonuniform charge distribution by using an adaptive grid [133]. 

M. L. Gittings, 1992 SAIC s Adaptive Grid Eulerian Code , Defense Nuclear Agency Numerical Methods Symposium, 28-30 (1992). 

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