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On a class of high resolution total-variational-stable finite-difference schemes (with appendix by Peter D. Lax). (English) Zbl 0547.65062
For the numerical solution of hyperbolic conservation laws $$(*)\quad u_ t+f(u)_ x=0,\quad u(.,0)=u_ 0$$ the author considers numerical schemes of type $$Lv^{n+1}=Rv^ n,\quad t=n\tau$$ where L and R are centered finite-difference operators. It is well known [cf. the author and P. D. Lax, ibid. 18, 289-315 (1981; Zbl 0467.65038)] that for bounded initial data with bounded total variation explicit consistent schemes satisfying an entropy inequality converge in $$L^ 1_{loc}$$ to the (unique) entropy solution of (*) if the schemes are total-variation stable. First, the author shows that this result also holds true in the implicit case assuming periodicity of initial data. Then, for scalar conservation laws he considers finite-difference operators of the form $$(Lv)_ j=v_ j+\eta\lambda (\bar f_{j+1/2}-\bar f_{j-1/2}),\quad (Rv)_ j=v_ j-(1-\eta)\lambda (\bar f_{j+1/2}-\bar f_{j- 1/2}),\quad j=xh,\quad\lambda =\tau /h,\quad 0\leq\eta \leq 1,$$ with Lipschitz continuous numerical flux given by $$\bar f_{j+1/2}=(f_ j+f_{j+1})/2-q(v_ j,v_{j+1})(v_{j+1}-v_ j)/(2\lambda)$$ (q some bounded function), and he proves that the associated scheme is total- variation diminishing (TVD) under a Courant-Friedrichs-Lewy (CFL) type condition. The basic idea is to modify these first order schemes in such a way that they become order ones while preserving the TVD property. This can be done by introducing a new flux $$f+(1/\lambda)g$$ with appropriately chosen g. It is shown that the resulting highly nonlinear schemes are consistent and thus possess a subsequence converging to a weak solution of (*). There is numerical evidence that they also satisfy the entropy inequality although so far a rigorous proof for that doesn’t exist. The second order schemes can be extended to systems of conservation laws and it is proved that in the constant coefficient case and under a CFL- condition they are TVD and convergent. The author also shows how to apply the newly developed schemes to steady-state calculations, and in an appendix P. Lax provides a general criterion for a linear difference operator to be TVD.
Reviewer: R.H.W.Hoppe

##### 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 76L05 Shock waves and blast waves in fluid mechanics 35L67 Shocks and singularities for hyperbolic equations
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