On the conservation and convergence to weak solutions of global schemes (Q1403974)

The aim of the paper is to study some difference schemes for hyperbolic conservation laws, written in the one-dimensional case as \[ u_t+ f(u)_x= 0,\qquad u(x, 0)= u^0(x),\qquad -1\leq x\leq 1. \] The classical Lax-Wendroff theorem asserts that the function to which the difference solution converges is the weak solution of the conservative law. Unfortunately the conditions of this theorem exclude the so-called global schemes. The authors extend the Lax-Wendroff theorem to schemes of this category such as: Compact schemes, Legendre spectral collocation schemes and Chebyshev spectral collocation schemes. Multi-domain Legendre methods for interface boundary conditions are also discussed.