Geodesic
The structure of a stable manifold for fully implicit schemes
Numerical methods and programming, Tome 14 (2013) no. 1, pp. 44-49

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An analog of the Hadamard-Perron theorem on the existence of a local stable manifold in a neighborhood of a fixed hyperbolic-type point for implicit mappings is proved. This result allows one to constructively study the structure of a manifold for a finite-difference approximation in time in the case of quasilinear parabolic-type equations and to prove that, in terms of the integral metric, the manifold of the nonlinear problem exists in an unbounded ellipsoid. Several theoretical estimates are given. A number of numerical results are discussed.
Keywords: stabilization; numerical algorithms; implicit finite-difference schemes. 
@article{VMP_2013_14_1_a6,
     author = {E. Yu. Vedernikova and A. A. Kornev},
     title = {The structure of a stable manifold for fully implicit schemes},
     journal = {Numerical methods and programming},
     pages = {44--49},
     publisher = {mathdoc},
     volume = {14},
     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2013_14_1_a6/}
}
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E. Yu. Vedernikova; A. A. Kornev. The structure of a stable manifold for fully implicit schemes. Numerical methods and programming, Tome 14 (2013) no. 1, pp. 44-49. http://geodesic.mathdoc.fr/item/VMP_2013_14_1_a6/