Analysis and Approximation of Conservation Laws with Source Terms

We consider a conservation law of the form$(\text{CL})\quad u_t + f(u)_x = a_x,$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 L1contraction, 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.