Solution of the generalized Riemann problem for advection-reaction equations

We present a method for solving the generalized Riemann problem for partial differ-ential equations of the advection–reaction type. The generalization of the Riemann problem here is twofold. Firstly, the governing equations include nonlinear advection as well as reaction terms and, secondly, the initial condition consists of two arbi-trary but infinitely differentiable functions, an assumption that is consistent with piecewise smooth solutions of hyperbolic conservation laws. The solution procedure, local and valid for sufficiently small times, reduces the solution of the generalized Riemann problem of the inhomogeneous nonlinear equations to that of solving a sequence of conventional Riemann problems for homogeneous advection equations for spatial derivatives of the initial conditions. We illustrate the approach via the model advection–reaction equation, the inhomogeneous Burgers equation and the nonlinear shallow-water equations with variable bed elevation