Geodesic
Approximation of solutions of the Monge–Ampère equations by surfaces reduced to developable surfaces
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 1, pp. 205-236
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We consider an approximate construction of the surface $S$ being the graph of a $C^2$-smooth solution $z=z(x,y)$ of the parabolic Monge–Ampère equation $$ (z_{xx}+a)(z_{yy}+b)-z_{xy}^2=0 $$ of a special form with the initial conditions $$ z(x,0)=\varphi(x),\quad q(x,0)=\psi(x), $$ where $a=a(y)$ and $b=b(y)$ are given functions. In the method proposed, the desired solution is approximated by a sequence of $C^1$-smooth surfaces $\{S_n\}$ each of which consists of parts of surfaces reduced to developable surfaces. In this case, the projections of characteristics of the surface $S$ being curved lines in general are approximated by characteristic projections of the surfaces $S_{n}$ being polygonal lines composed of $n$ links. The results of these constructions are formulated in the theorem. Sufficient conditions for the convergence of the family of surfaces $S_{n}$ to the surface $S$ as $n\to\infty$ are presented; this allows one to construct a numerical solution of the problem with any accuracy given in advance. 
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L. B. Pereyaslavskaya. Approximation of solutions of the Monge–Ampère equations by surfaces reduced to developable surfaces. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 1, pp. 205-236. http://geodesic.mathdoc.fr/item/FPM_2006_12_1_a6/

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