Jm. Greenberg et al., ANALYSIS AND APPROXIMATION OF CONSERVATION-LAWS WITH SOURCE TERMS, SIAM journal on numerical analysis, 34(5), 1997, pp. 1980-2007
We consider a conservation law of the form (CL) u(t) + f(u)x = a(x), w
here a(.) is a bounded piecewise smooth source term and f an even conv
ex function. We first characterize the solution to the Riemann problem
through a new Lax-type formula. Then we prove that for a(.) fixed, th
e semigroup associated with (CL) is an L-1 contraction, and we obtain
an existence theorem for weak solutions to (CL). We conclude by constr
ucting Godunov-type difference schemes and proving that; these schemes
are L-infinity stable and have stable steady solutions similar in str
ucture to those of (CL). Some numerical tests are reported.