Extensions and investigations of space‐time generalized Riemann problems numerical schemes for linear systems of conservation laws with source terms

The space‐time generalized Riemann problems method allows to obtain numerical schemes of arbitrary high order that can be used with very large time steps for systems of linear hyperbolic conservation laws with source term. They have been introduced in Berthon et al. (J. Sci. Comput. 55 (2013), 268–308.) in 1D and on 2D unstructured meshes made of triangles. The objective of this article is to complement them in order to answer some important questions arising when they are involved. The formulation is described in detail on quadrangle meshes, the choice of approximation basis is discussed and Legendre polynomials are used in practical cases. The addition of a limiter to preserve certain properties without compromising accuracy is also considered. Finally, the asymptotic behavior of the scheme in the diffusion regime is studied.