Geodesic
Ray expansions of solutions around fronts as a shock-fitting tool for shock loading simulation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 4, pp. 722-733

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In shock loading computations based on an implicit finite-difference scheme, the surfaces of velocity discontinuity and the discontinuity sizes are determined by computing an asymptotic (ray) expansion of the solution behind the front surfaces at every time step. The method for constructing ray expansions is based on a recurrence formulation of the geometric and kinematic consistency conditions for discontinuities of the derivatives of functions that are discontinuous on a moving surface. The algorithm is illustrated by computing a simple example of the out-of-plane motion of an incompressible elastic medium. 
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     author = {E. A. Gerasimenko and A. V. Zavertan},
     title = {Ray expansions of solutions around fronts as a~shock-fitting tool for shock loading simulation},
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E. A. Gerasimenko; A. V. Zavertan. Ray expansions of solutions around fronts as a shock-fitting tool for shock loading simulation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 4, pp. 722-733. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_4_a13/