Tamtam Proceedings - lamsin

80 Bokanowski et al.4. Numerical examplesWe test our method for the HJB equation (1) with the initial condition{ 0 if (x − 0.4)ϕ(x, y) =2 + (y − 0.4) 2 ≤ 0.1 2 or (x − 0.6) 2 + (y − 0.6) 2 ≤ 0.1 2 ,1 otherwise,the dynamicsf(x, y, u) = (−y + cos(u), x + sin(u)),and the set of controls U = [0, 2π]. In these numerical tests, we take Nu = 8, the domainof computations is D = [0, 1] × [0, 1], and the maximal level of refinement is 6.We visualise the exact solution and the numerical solution computed on each cell.Thenumerical initial condition is the L 1 projection of ϕ on the grid and it is the same asthe projected exact solution (figure.2). In figure.3 the graphic on the right shows the L 1projection of the exact solution on each cell, the one on the left shows the computednumerical solution.Figure 2. Initial condition at t=0 and initial mesh,number of cells=562.Figure 3. Exact solution and numerical solution at t=0.12,number of cells=991.TAMTAM –Tunis– 2005