, Volume 54, Issue 3, pp 271-293

On the numerical solution of the equation \(\frac{{\partial ^2 z}}{{\partial x^2 }}\frac{{\partial ^2 z}}{{\partial y^2 }} - \left( {\frac{{\partial ^2 z}}{{\partial x\partial y}}} \right)^2 = f\) and its discretizations, I

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Summary

The equation indicated in the title is the simplest representative of the class of nonlinear equations of Monge-Ampere type. Equations with such nonlinearities arise in dynamic meteorology, geometric optics, elasticity and differential geometry. In some special cases heuristic procedures for numerical solution are available, but in order for them to be successful a good initial guess is required. For a bounded convex domain, nonnegativef and Dirichlet data we consider a special discretization of the equation based on its geometric interpretation. For the discrete version of the problem we propose an iterative method that produces a monotonically convergent sequence. No special information about an initial guess is required, and to initiate the iterates a routine step is made. The method is self-correcting and is structurally suitable for a parallel computer. The computer program modules and several examples are presented in two appendices.

The work of both authors was supported by AFOSR Grants #83-0319 and 84-0285