Geodesic
A three-time-level explicit difference scheme of the second order of accuracy for parabolic equations
Matematičeskie zametki, Tome 60 (1996) no. 5, pp. 751-759

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The difference schemes of Richardson [1] and of Crank–Nicolson [2] are schemes providing second-order approximation. Richardson's three-time-level difference scheme is explicit but unstable and the Crank–Nicolson two-time-level difference scheme is stable but implicit. Explicit numerical methods are preferable for parallel computations. In this paper, an explicit three-time-level difference scheme of the second order of accuracy is constructed for parabolic equations by combining Richardson's scheme with that of Crank–Nicolson. Restrictions on the time step required for the stability of the proposed difference scheme are similar to those that are necessary for the stability of the two-time-level explicit difference scheme, but the former are slightly less onerous. 
@article{MZM_1996_60_5_a8,
     author = {A. S. Shvedov},
     title = {A three-time-level explicit difference scheme of the second order of accuracy for parabolic equations},
     journal = {Matemati\v{c}eskie zametki},
     pages = {751--759},
     publisher = {mathdoc},
     volume = {60},
     number = {5},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_60_5_a8/}
}
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A. S. Shvedov. A three-time-level explicit difference scheme of the second order of accuracy for parabolic equations. Matematičeskie zametki, Tome 60 (1996) no. 5, pp. 751-759. http://geodesic.mathdoc.fr/item/MZM_1996_60_5_a8/