Geodesic
Implicit difference methods for Hamilton Jacobi functional differential equations
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 12 (2009) no. 1, pp. 57-70

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Classical solutions of initial boundary value problems are approximated in this paper by solutions of associated implicit difference functional equations. The stability is proved by using a comparison technique with nonlinear estimates of the Perron type for given functions. The Newton method is used for numerical solving of nonlinear equations generated by implicit difference schemes. It is shown that there are implicit difference schemes which are convergent and the corresponding explicit difference methods are not convergent. The results can be applied to differential integral problems and differential equations with deviated variables. 
@article{SJVM_2009_12_1_a4,
     author = {Z. Kamont and W. Czernous},
     title = {Implicit difference methods for {Hamilton} {Jacobi} functional differential equations},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {57--70},
     publisher = {mathdoc},
     volume = {12},
     number = {1},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2009_12_1_a4/}
}
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Z. Kamont; W. Czernous. Implicit difference methods for Hamilton Jacobi functional differential equations. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 12 (2009) no. 1, pp. 57-70. http://geodesic.mathdoc.fr/item/SJVM_2009_12_1_a4/