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On convergence of monotone finite difference schemes with variable spatial differencing. (English) Zbl 0533.65061
Explicit and implicit difference schemes and the method of lines used for solution of the Cauchy problem $u_ t+\sum^{2}_{i=1}f_ i(u)_{x_ i}=0,\quad u(x,0)=u_ 0(x)$ on a nonuniform spatial mesh are considered. In all methods the operator $$D_ x$$ approximating the space derivatives is monotone and consistent. Theorems on the convergence of solutions of difference schemes and the method of lines to the unique solution of the original equation satisfying the entropy condition for $$\delta\to 0$$ are proved. Here $$\delta$$ is a parameter characterizing the maximum mesh size $$\Omega_{j,k}=[x_ j,x_{j+1}]\times [y_ k,y_{k+1}],$$ where $$x=x_ 1$$, $$y=x_ 2$$.
Reviewer: Yu.Shokin

MSC:
 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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