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Difference methods for non-linear partial differential equations of the first order. (English) Zbl 0659.65077

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

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
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