A Roe-like numerical method for weakly hyperbolic systems of equations in conservation and non-conservation form

We describe a scheme for the numerical approximation of uniformly or locally weakly hyperbolic systems of equations in conservation and non-conservation forms. The strategy focuses on generalizations of the Roe scheme for non-conservation systems that can be considered as upstream differencing discretizations. The present work explicitly treats an advection matrix that is not diagonalizable by extending previous ideas that introduce a perturbation into the weakly hyperbolic problem to convert it to a nearby strongly hyperbolic problem. The perturbative expansion allows one to identify and remove the terms that would otherwise result in an apparent singularity and isolate the regular part of the numerical method. The performance of the method is showcased for uniformly and locally weakly hyperbolic systems in one and two dimensions. Generalization to higher dimensional systems can be performed by following the usual practice of dimension-by-dimension splitting.