| While designing numerical methods for linear problems is primarily a struggle for stability and accuracy, nonlinear problems add a new difficulty: even if the sequence of numerical solutions converges as the grid becomes infinitely fine, the limit may be a spurious unwanted solution. This phenomenon is especially common for hyperbolic systems of conservation laws. The classical Lax-Wendroff theorem ensures that conservative and consistent schemes cannot suffer from this pathology: whenever they converge, the limit must be a weak solution. If, in addition, the scheme satisfies a discrete entropy inequality, then the limit must be an entropy solution. Entropy solutions are widely assumed to be the unique physically correct solutions, although this has been verified only in special cases.; The classical Lax-Wendroff theorem applies only to uniform Cartesian grids. In practice it is necessary to consider unstructured grids, often with moving vertices, and sometimes to apply additional techniques like conservative rezoning/remapping or staggered grids. This work presents a corresponding generalization of the Lax-Wendroff theorem. The only remaining significant limitation is the condition that the unstructured grids are quasi-uniform. |
