Irradiation of solid-state targets by a high-power beam of accelerated charged particles (electrons or ions) with energy-flux density >t 107 W/cm 2 is accompanied by generation of compression and expansion waves in the radiation-free part of the material due to intense warming up of some volume of the target caused by stoppage of particles. Wave propagation over a solid body causes deformations leading to the formation of various defects [1][2][3][4], to changes in the mechanical properties of the material, and even to failure [5].The system of equations describing the generation and propagation of elastoplastic waves in a material, which are induced by a charged-particle beam, includes a kinetic equation for high-velocity charged particles, equations of continuum mechanics, and a wide-range equation of state.Experience on the numerical solution of this problem has shown that many calculations deal with the equations of continuum mechanics. To simulate the effect of high-power beams of charged particles on a material, these equations are solved by various methods: Leshkevich and Khalikov et al. in [6, 7] used the method of macroparticles [8], Akkerman et al. in [9, 10] used the Godunov method [11], and Val'chuk in [12] used the Wilkins method [13].Application of the method of macroparticles to the problem considered here involves two difficulties. First, this method is unstable for weak flows. This does not make it possible to simulate the effect of the beams that are of technological interest and, second, in the free-surface problems the free surfaces are difficult to approximate.The main drawback of the Godunov method is a strong smoothing of the solution in the vicinity of a contact discontinuity.The method for solving the equations of continuum mechanics which was developed by Wilkins [13] avoids the drawbacks listed above, although there are some difficulties in its application due to the use of Lagrangian variables to simulate strong deformations of the material.One more problem, which is common for all schemes containing artificial viscosity, is to select a value of the coefficient of artificial viscosity. Numerical experiments show that with an acceptable smoothness of the solution, variation in this coefficient changes the shock-wave amplitude by 10-20%. In addition, in calculation of flow of a medium that is in the liquid-gas equilibrium region, the schemes with artificial viscosity become unstable because of a large decrease in the sound velocity (by 2-3 orders of magnitude).In the present paper, we find analytical solutions of the equations of continuum mechanics for a material component by means of which the motion of the entire system is simulated. In some cases, the use of the analytical solutions obtained makes it possible to increase the time step of calculation. This increase can be considerable in comparison to the time step found from the Courant condition, as in the example below.The equations are solved in Lagrangian variables. The algorithm proposed for reconstruction of the Lagrangian ...
