Solution of the generalized Riemann problem for advection-reaction equations. (English) Zbl 1019.35061

The authors present a method for solving the generalized Riemann problem for advection-reaction problems, e.g. nonlinear shallow water equation with variable bed elevation. The initial condition for the generalized Riemann problem consists of two arbitrary but infinitely differentiable functions, which are consistent with smooth solution to the hyperbolic system.
Two steps are needed in order to evaluate the solution. In the first step all time derivatives are replaced by the space derivatives. In the second step it is shown that these space derivatives obey corresponding homogeneous advection problems. For the space derivatives conventional Riemann problems are consequently solved. The authors illustrate this approach via the model advection-reaction equation, the inhomogeneous Burgers equation, and the nonlinear shallow water equations with geometrical source term arising from bed elevation.
In principle, this technique can be applied to problems of advection-reaction type with general piecewise smooth initial conditions. The solution is then valid only for a short time, i.e. until propagating discontinuities interact. These local solutions can be used to construct higher-order Godunov methods, see e.g. [V. A. Titarev and E. F. Toro, J. Sci. Comput. 17, 609-618 (2002; Zbl 1024.76028)].


35L60 First-order nonlinear hyperbolic equations
35L45 Initial value problems for first-order hyperbolic systems


Zbl 1024.76028


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