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Summary: PSEUDO TIME CONTINUATION AND TIME MARCHING
METHODS FOR MONGE-AMP`ERE TYPE EQUATIONS
GERARD AWANOU
Abstract. We discuss the performance of three numerical methods for the fully
nonlinear Monge-Amp`ere equation. The first two are pseudo time continuation
methods while the third is a pure pseudo time marching algorithm. The pseudo
time continuation methods are shown to converge for smooth data on a uniformly
convex domain. We give numerical evidence that they perform well for the non-
degenerate Monge-Amp`ere equation. The pseudo time marching method applies in
principle to any nonlinear equation. Numerical results with this approach for the
degenerate Monge-Amp`ere equation are given as well as for the Pucci and Gauss-
curvature equations.
1. Introduction
We are interested in numerical solutions of equations of type
(1.1) F(x, u(x), Du(x), D2
u(x)) = 0,
on a convex bounded domain of Rn
with boundary and Dirichlet boundary con-
ditions u = g and F real valued. Here u is a real valued function and Du(x), D2
u(x)
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