Analysis and Approximation of Conservation Laws with Source Terms

We consider a conservation law of the form (CL)I>u t + f(u) x = a x , where a(·) is a bounded piecewise smooth source term and f an even convex function. We first characterize the solution to the Riemann problem through a new Lax-type formula. Then we prove that for a(·) fixed, the semigroup associated with (CL)is an L 1 contraction, and we obtain an existence theorem for weak solutions to (CL). We conclude by constructing Godunov-type difference schemes and proving that these schemes are L^∞ stable and have stable steady solutions similar in structure to those of (CL). Some numerical tests are reported.