Implicit Exactly Well-Balanced Finite Volume Schemes for Balance Laws with Singular Source Terms

In previous works, the authors have proposed a methodology to design explicit well-balanced high-order numerical methods for 1d systems of balance laws with singular source terms. These methods rely on the combination of the Generalized Hydrostatic Reconstruction (GHR) technique and a well-balanced reconstruction operator. In this work, first these two ingredients are recalled as well as the family of semi-discrete in space well-balanced high-order methods obtained by applying them. Then the extension to implicit time discretizations are discussed. In particular, two new strategies to design exactly well-balanced implicit finite volume schemes for systems with singular source terms are introduced. Finally, some numerical tests for the Burgers' equation with singular source term are considered to check and compare the resulting implicit numerical methods.