On the difference approximations of overdetermined hyperbolic equations of classical mathematical physics

Solid–fluid diffuse interface model in cases of extreme deformations

“…In pure solid phase ða s ¼ 1Þ or in a homogeneous case ða s ¼ constÞ we find usual geometric equations (3) and (4). The numbers n b characterize the mixture anisotropy.…”

“…A numerical strategy based on the parabolic regularization of (6) that control the gauge constraint w b ¼ 0 in solids was developed in Miller and Colella [23] and Miller and Colella [24]. In the paper of Babii et al [4] a different strategy has been proposed to resolve, in particular, the stationary constraints for the Maxwell equations, linear elasticity and magnetohydrodynamics. Analytically, the method consists in a resolution of an overdetermined system of equations where new unknowns (called ''potentials") and the corresponding evolution equations are introduced.…”

“…We refer to Godunov and Romenskii [17] for a general criterion of hyperbolicity of equations of hyperelasticity with geometric constraint (4). It can be shown that with the equations of state (10) and (11), the one-dimensional equations of motion are hyperbolic in a full domain of variables.…”

On Approximations for Overdetermined Hyperbolic Equations

“…Numerical examples for the acoustic equations in a more general form to illustrate the efficiency of this approach are shown in [7].…”

Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid–fluid transitions

“…Overdetermined systems of hyperbolic-elliptic type are well known in computational continuum mechanics mainly due to the fact that inevitable errors in numerical calculations cause the constraints to be violated and consequently numerical solutions become physically meaningless unless some sophisticated constraints-treatment procedures (see for example Refs. [46,47,39,48] and references therein) are implemented. This difficulty is one of the reasons why we shall attempt (in the next section) to lift (23) to a larger system that is free of constraints and all equations in the system are local conservation laws.…”