Tamtam Proceedings - lamsin

A method for optimal control problems 79Although we don’t have yet a proof of convergence of the UltraBee generalized to HJBequations (we are investigating in this direction), numerical tests give encouraging resultsand computations are simple to implement.3. Linear quadtreesAs we deal with adaptative grids, we look for a technique that facilitates stocking andfinding data relative to each cell of the grid. This technique is explained by Gargantini [5]and called linear quadtree. If we represent our final adapted grid by a tree, each cell is aleaf (final node of the tree) and the initial big quadrant (before adaptation) is the root ofthe tree. The method of stocking data using quadtrees is based on coding each leaf of thetree with a quaternary function. This code representation is implicitly the path from theroot to the concerned leaf.In fact every code is composed of 0,1,2,3. when dividing a cell into four subcells, the NWquadrant is indexed by 0, the NE by 1,the SW by 2 and the SE by 3 (as shown in figure.1).0 123Figure 1. Refinement of a cell by quadtreesOur contribution consists in finding suitable criterion based on mean values (welladapted to the fact that UltraBee scheme transports mean values on each cell) to refineand adapt the initial computation domain. The refinement test deals with a comparisonbetween the value on a given cell jand its neighboring cells values: j is refined if its valueis not equal to all of the values of its neighbors.A regularization step is introduced to get an “homogeneous” grid where the differenceof level of refinement between two neighboring cells is 1 at most. We also introduce amaximal level of refinement (which depends on the required precision). Usual stoppingtests for continuous functions rely on a tolerance of approximation. This is not suitable inour case: in the neighborhood of discontinuities the tolerance would never be reachedTAMTAM –Tunis– 2005