Geodesic
Explicit expression of grid-characteristic schemes for elasticity equations in 2D and 3D
Matematičeskoe modelirovanie, Tome 18 (2006) no. 6, pp. 96-108

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The article describes (on the example of elasticity equations) obtaining of the grid-characteristic schemes for inner and bound mesh nodes, which do not require solution of linear equations or matrix inversion and simultaneously hold true for 2D and 3D spaces. Such expressions reduce greatly cost of programming and debugging and at the same time provide faster code. A two-phase algorithm is offered for computation of bound nodes, the first phase is independent from boundary conditions, and the second – from the approximation order. In order to obtain explicit expression of schemes, an important subsidiary problem was solved: all eigenvalues and eigenvectors of elasticity equations were found analytically in arbitrary rectilinear coordinate frame. The article contains the results of 3D body with regular structure of internal cavities modeling, in which wavefront becomes wedge-shaped (instead of well-known sphere form) due to numerous reflections of the initial impulse. 
@article{MM_2006_18_6_a7,
     author = {F. B. Chelnokov},
     title = {Explicit expression of grid-characteristic schemes for elasticity equations in {2D} and {3D}},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {96--108},
     publisher = {mathdoc},
     volume = {18},
     number = {6},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2006_18_6_a7/}
}
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F. B. Chelnokov. Explicit expression of grid-characteristic schemes for elasticity equations in 2D and 3D. Matematičeskoe modelirovanie, Tome 18 (2006) no. 6, pp. 96-108. http://geodesic.mathdoc.fr/item/MM_2006_18_6_a7/