Analysis and Approximation of Conservation Laws with Source Terms

Abstract

We consider a conservation law of the form (CL)I>ut + f(u)x = ax , where $a(\cdot)$ 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 $a(\cdot)$ fixed, the semigroup associated with (CL)is an L1 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 $L^\infty$ stable and have stable steady solutions similar in structure to those of (CL). Some numerical tests are reported.