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The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme. (English) Zbl 0598.65067
The following monotonicity preserving schemes are considered: \[ (1)\quad v_{\nu}(t+k)=v_{\nu}(t)-(k/\Delta x)[h(v_{\nu}(t),v_{\nu +1}(t))- h(v_{\nu -1}(t),v_{\nu}(t))] \] which serve as consistent approximations to the scalar conservation law \((2)\quad \partial u/\partial t+\partial f/\partial x (u(x,t))=0\) with the initial conditions \[ v_{\nu}(t)|_{t=0}=u(x_{\nu},0),\quad u(x,0)\in L^ 1\cap L^{\infty}\cap BV \] where h(.,.) is the Lipschitz continuous numerical flux, \(h(v,v)=f(v)\). Identifying 3-point conservative schemes according to their numerical viscosity coefficient the author gives the following characterization: monotonicity-preserving schemes (compactness property) are exactly those having a numerical dissipation no more than the Lax-Friedrichs scheme (LFS), no less than Murman’s scheme, and entropy-satisfying schemes are those having no less dissipation than Godunov’s scheme. In the case of strictly convex f in (2), i.e. if \(\ddot f\geq \dot a_*>0\), it is shown, that for the LFS the divided differences of the numerical solution at time t not exceed \(2(t\dot a_*)^{-1}\) which guarantees the entropy compactness of the scheme.
Reviewer: Gy.Molnárka

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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