A Relaxation Algorithm for Optimal Control Problems Governed by Two-Dimensional Conservation Laws

We develop a class of numerical methods for solving optimal control problems governed by nonlinear conservation laws in two space dimensions. The relaxation approximation is used to transform the nonlinear problem to a semi-linear diagonalizable system with source terms. The relaxing system is hyperbolic and it can be numerically solved without need to either Riemann solvers for space discretization or a non-linear system of algebraic equations solvers for time discretization. In the current study, the optimal control problem is formulated for the relaxation system and at the relaxed limit its solution converges to the relaxed equation of conservation laws. An upwind method is used for reconstruction of numerical fluxes and an implicit-explicit scheme is used for time stepping. Computational results are presented for a two-dimensional inviscid Burgers problem.