Solution of Steady-State, Two-Dimensional Conservation Laws by Mathematical Programming

Abstract

Solution of steady-state scalar conservation laws $[ f(u) ]_x + \tau [ f(u) ]_y = 0$ ($\tau $ constant) on the unit square is considered for the two cases $f(u) = u$ and $u^2$. Piecewise constant Dirichlet conditions producing shocked solutions are given on the boundary of the unit square. A nonsingular perturbation $2\varepsilon u$ ($\varepsilon $ sufficiently small positive number), rather than a singular perturbation such as $ - \varepsilon (u_{xx} + u_{yy} )$, is added to the conservation law. The perturbed equation is discretized in a finite-volume sense on each of the cells of a grid of $mn$ equal rectangular cells (m in the x-direction and n in the y-direction). Numerical values of u are located at the cell vertices. The arclength and area integrals of the finite-volume formulas are discretized by the trapezoidal rule. A system of $mn$ equations results for the $(m-1)(n-1)$ unknown values of u in the interior of the domain. This system is solved in the $l_1 $ sense; that is, the sum of the absolute values of the residuals of the equations is minimized. An algorithm requiring only $O(mn)$ operations is introduced to solve this mathematical programming problem. The numerical solutions have discontinuities in or near the cells containing the shocks of the physically relevant solutions of the original conservation laws and are $O(\varepsilon )$ approximations of these solutions. A complete theory for the linear case $f(u) = u$ and computational results for the nonlinear case $f(u) = u^2$ are presented. The $l_1 $ procedure captures boundary shocks as well as oblique and even zigzag interior shocks in one cell. The results presented here demonstrate that the $l_1 $ procedure is a robust, efficient, and highly accurate numerical procedure for solving certain two-dimensional scalar conservation laws.