Mathematical Aspects of Numerical Solution of Hyperbolic Systems

Abstract

Abstract. A number of physical phenomena are described by nonlinear hyperbolic equations. Presence of discontinuous solutions motivates the necessity of development of reliable numerical methods based on the fundamental mathematical properties of hyperbolic systems. Construction of such methods for systems more complicated than the Euler gas dynamic equations requires the investigation of existence and uniqueness of the self-similar solutions to be used in the development of discontinuity-capturing high-resolution numerical methods. This frequently necessitates the study of the behavior of discontinuities under vanishing viscosity and dispersion. We discuss these problems in the application to the magnetohydrodynamic equations, nonlinear waves in elastic media, and electromagnetic wave propagation in magnetics.

On the Nonuniqueness of Solutions to the Nonlinear Equations of Elasticity Theory

“…Riemann waves in elastic media were considered in [18, 3, 5, Chapter III]. In the case of small-amplitude waves in weakly anisotropic media it follows [15,17,19, Chapter 3; Section 7.4.5] from (1), (7) that…”

“…If u 1 = U 1 , u 2 = U 2 ahead of the discontinuity, then behind the discontinuity u 1 and u 2 belong to a curve on the (u 1 , u 2 )-plane which can be denoted as the shock adiabat [15,Chapter 4], [17,Section 7.4.6], [20,21] Inequality (6) of nondecreasing entropy gives…”

“…Solutions of a piston problem were analyzed in [15,Chapter 5], [17,Section 7.4.7], [22,23]. For fixed initial values U 1 , U 2 , the whole u 1 , u 2 plane is divided into domains in such a way that, if boundary values u * 1 , u * 2 belong to one of them, then a definite combination of Riemann waves and shocks gives the solution.…”

Velocity-Stress Equations for Wave Propagation in Anisotropic Elastic Media

“…Without losing any generality, we let h 1 , h 2 , and h 3 be the three components of a unit vector h, i.e., |h| = 1. Equation (2.3) is hyperbolic if all eigenvalues of B are real, and B can be diagonalized by its eigenvector matrix [11]. 3) can be decoupled into 9 independent scalar convection equations, each of which would propagate a constant profile, i.e., a Riemann invariant, in the direction h by its own wave speed, i.e., the corresponding eigenvalue.…”

“…Alternatively, one can use a modern numerical method [11,12] to solve the first-order velocity-stress equations to obtain the transient solution of wave propagation in elastic solids. For example, Virieux [25] modeled waves in earth crust by using a finite-difference method to solve the velocity-stress equations.…”