On dynamics of multi-phase elastic-plastic media

The paper is concerned with dynamics of multi-phase media consisting of a solid permeable material and a compressible Newtonian fluid. Governing macroscopic equations are derived starting from the space-averaged microscopic mass and momentum balances. The Reynolds stress models (i.e., momentum dispersive fluxes) are discussed, and a suitable model is developed. In the case of granular media the solid constituent is considered as an elastic-plastic matrix, and the yield condition is approximated by Coulomb friction law. It is revealed that the classical principle of maximum plastic work is not, in general, valid for granular media, and an appropriate variational principle is developed. This novel principle coincides with the maximum plastic work principle for the case of cohesionless and free from internal friction granular media.Comment: 16 page

On stability of difference schemes. Central schemes for hyperbolic conservation laws with source terms

The stability of difference schemes for, in general, hyperbolic systems of conservation laws with source terms are studied. The basic approach is to investigate the stability of a non-linear scheme in terms of its cor- responding scheme in variations. Such an approach leads to application of the stability theory for linear equation systems to establish stability of the corresponding non-linear scheme. It is established the notion that a non-linear scheme is stable if and only if the corresponding scheme in variations is stable. A new modification of the central Lax-Friedrichs (LxF) scheme is developed to be of the second order accuracy. A monotone piecewise cubic interpolation is used in the central schemes to give an accurate approximation for the model in question. The stability of the modified scheme are investigated. Some versions of the modified scheme are tested on several conservation laws, and the scheme is found to be accurate and robust. As applied to hyperbolic conservation laws with, in general, stiff source terms, it is constructed a second order nonstaggered central scheme based on operator-splitting techniques.Comment: 33 pages, 7 figure

On monotonicity, stability, and construction of central schemes for hyperbolic conservation laws with source terms (Revised Version)

The monotonicity and stability of difference schemes for, in general, hyperbolic systems of conservation laws with source terms are studied. The basic approach is to investigate the stability and monotonicity of a non-linear scheme in terms of its corresponding scheme in variations. Such an approach leads to application of the stability theory for linear equation systems to establish stability of the corresponding non-linear scheme. The main methodological innovation is the theorems establishing the notion that a non-linear scheme is stable (and monotone) if the corresponding scheme in variations is stable (and, respectively, monotone). Criteria are developed for monotonicity and stability of difference schemes associated with the numerical analysis of systems of partial differential equations. The theorem of Friedrichs (1954) is generalized to be applicable to variational schemes with non-symmetric matrices. A new modification of the central Lax-Friedrichs (LxF) scheme is developed to be of the second order accuracy. A monotone piecewise cubic interpolation is used in the central schemes to give an accurate approximation for the model in question. The stability and monotonicity of the modified scheme are investigated. Some versions of the modified scheme are tested on several conservation laws, and the scheme is found to be accurate and robust. As applied to hyperbolic conservation laws with, in general, stiff source terms, it is constructed a second order scheme based on operator-splitting techniques.Comment: 48 pages, 6 figures. Found inexactitudes are corrected. A second order scheme based on operator-splitting techniques is constructe