Kamont, Zdzisław; Prządka, Katarzyna Difference methods for non-linear partial differential equations of the first order. (English) Zbl 0659.65077 Ann. Pol. Math. 48, No. 3, 227-246 (1988). For the initial value problem (i) \(z_ x(x,y)=f(x,y,z(x,y),z_ y(x,y))\) \(z(x^{(0)},y)=\omega (y),\quad y=(y_ 2,...,y_ n),\quad z_ y(x,y)=z_{y_ 1}(x,y),...,z_{y_ n}(x,y))\) the one step difference method \[ (ii)\quad \Delta_ 0w^{(i,j)}=\Phi (x^{(i)},y^{(j)},Aw^{(i,j)},[w^{(i,j)}],\quad \Delta w^{(i,j)},h_ i,k),\quad w^{(0,j)}=\omega (y^{(j)}) \] is proposed. Sufficient conditions for the convergence of the sequence \(\{u_ m\}\) of solutions of (ii) to a solution \(\bar u\) of (i) are given. An error estimate of the method is obtained, in terms of a power of the step h. Reviewer: L.G.Vulkov Cited in 1 Document 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 35F25 Initial value problems for nonlinear first-order PDEs Keywords:initial value problem; convergence; error estimate PDFBibTeX XMLCite \textit{Z. Kamont} and \textit{K. Prządka}, Ann. Pol. Math. 48, No. 3, 227--246 (1988; Zbl 0659.65077) Full Text: DOI