## A well-balanced scheme for the numerical processing of source terms in hyperbolic equations.(English)Zbl 0876.65064

A new numerical method is developed for solving the following scalar hyperbolic conservation law (*) $$u_t+f(u)_x+a_x(x)u=0; f(0)=f'(0)=0, f$$ convex, $$a \geq 0.$$ The conservation law (*) is a problem which models systems where the source terms are balanced by internal forces such that multiple stable steady state solutions exist. A typical example is the shallow water equation over a nonuniform ocean bottom. The new scheme is of first order and preserves for positive stepsizes the positive steady state solution of (*). Most classical numerical schemes, e.g. the Godunov scheme, do not have this property. It is proved that the new scheme converges to the unique solution satisfying the Lax entropy condition. Numerical examples are given.

### MSC:

 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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