Convergence of MUSCL relaxing schemes to the relaxed schemes for conservation laws with stiff source terms

We consider the convergence and stability property of MUSCL relaxing schemes applied to conservation laws with stiff source terms. The maximum principle for the numerical schemes will be established. It will be also shown that the MUSCL relaxing schemes are uniformly l¹- and TV-stable in the sense that they are bounded by a constant independent of the relaxation parameter ε, the Lipschitz constant of the stiff source term and the time step Δt. The Lipschitz constant of the l¹ continuity in time for the MUSCL relaxing schemes is shown to be independent of ε and Δt. The convergence of the relaxing schemes to the corresponding MUSCL relaxed schemes is then established.