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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.

Natural neighbour co-ordinates (Sibson co-ordinates) is a well-known interpolation scheme for multivariate data ÿtting and smoothing. The numerical implementation of natural neighbour co-ordinates in a Galerkin method is known as the natural element method (NEM). In the natural element method, natural neighbour co-ordinates are used to construct the trial and test functions. Recent studies on NEM have shown that natural neighbour co-ordinates, which are based on the Voronoi tessellation of a set of nodes, are an appealing choice to construct meshless interpolants for the solution of partial di erential equations. In Belikov et al. (Computational Mathematics and Mathematical Physics 1997; 37(1):9-15), a new interpolation scheme (non-Sibsonian interpolation) based on natural neighbours was proposed. In the present paper, the non-Sibsonian interpolation scheme is reviewed and its performance in a Galerkin method for the solution of elliptic partial di erential equations that arise in linear elasticity is studied. A methodology to couple ÿnite elements to NEM is also described. Two signiÿcant advantages of the non-Sibson interpolant over the Sibson interpolant are revealed and numerically veriÿed: the computational e ciency of the non-Sibson algorithm in 2-dimensions, which is expected to carry over to 3-dimensions, and the ability to exactly impose essential boundary conditions on the boundaries of convex and non-convex domains.

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