TVB Boundary Treatment for Numerical Solutions of Conservation Laws

In the computation of hyperbolic conservation laws ${u_t} + f{(u)_x} = 0$, TVD (total-variation-diminishing) and TVB (total-variation-bounded) schemes have been very successful for initial value problems. But most of the existing boundary treatments are only proved to be linearly stable, hence the combined initial-boundary scheme may not be TVB. In this paper we describe a procedure of boundary treatment which uses the original high-order scheme up to the boundary, plus extrapolation and upwind treatment at the boundary. The resulting scheme is proved to be TVB for the scalar nonlinear case and for linear systems.