Adaptive grids

P.A. Hookham, D. Hatfield, and M. Rosenblatt, Calculation of the Interaction of Three Spherical Blast Waves Over a Planar Surface with an Adaptive-Grid TVD Code, California Research and Technology Report, Chatsworth, CA, 1991. 

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. 

Adaptive grids may be of assistance in raising the order of accuracy without increasing the total number of nodes. If the subsidiary information... 

Here the constant C takes care of the relative importance of the second derivative influence. Instead of solving a front problem in the coordinates (x,t) (physical space) we perform the calculations in the computational space (C t). For one dimensional problems this adaptive grid transformation proved to be very successful. We can perform a transformation in a similar spirit for a two dimensional domain (x,y,t) -> A general sketch of this transformation... 

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

II. Stable high-order central finite difference schemes on composite adaptive grids with sharp shock resolution,... 

For example, on an evenly spaced grid that covers the allowable region in composition space. Or, even better, on a specially adapted grid covering only the accessed region of composition space. 

In most applications, linear interpolation has been employed. Higher accuracy can be achieved by using an adaptable grid that places more points in regions where the function is changing quickly. 

Calculation of Rovibrational Spectra of Five-Atom Molecules with Three Identical Atoms by Using a Cs(G6) Symmetry-Adapted Grid Applied to CH3D and CHD3. 

In an unstructured mesh each node can have a different number of neighbours and elements have different shapes and sizes. Therefore connectivity information must be explicitly defined and stored. The unstructured grid approach, that has gained popularity with the enormous advancements of computer technology, allows handling complex geometries with a lower number of elements and a much easier realization of local and adaptive grid refinement. [77]... 

Typically, the numerical solutions techniques used are very specific to the problem. Particularly challenging problems include moving front problems where concentration profiles, for example, may vary widely over a short distance but may not change much at other spatial locations. The spatial discretization must be small close to the front for accuracy and numerical stability, but must be larger at other locations to reduce computation time. Various adaptive grid techniques to change the spatial step sizes have been developed for these problems. One of the more common codes to solve fluid-flow-related problems is FLUENT. 

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.

Clearly also, in order to choose a suitable set of parameters, one must know the requirements before the simulation. If some homogeneous rate constant changes during a series of program runs (for example one in which such a rate constant is searched for), then the grid parameters should change. This makes adaptive grids more useful. These are described below. 

The problem of thin reaction layers are described sufficiently in Chap. 5. The solution is to use unequal intervals, that is, a few very small intervals near the electrode, so that there are sample points within the thin profile. This can be done up to a point by a fixed unequal grid such as the exponentially expanding grid described in Chap. 7. A more flexible approach is the moving adaptive grid also described in that chapter. This problem is thus solved and needs no further attention here. 

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). 

Steady-state or transient two-dimensional (2D) and three-dimensional (3D) flows in standard geometries involving Cartesian, cylindrical, or spherical coordinates and complex geometries involving BFCs with adaptive grids... 

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