Energy balanced numerical schemes with very high order. The Augmented Roe Flux ADER scheme. Application to the shallow water equations

In this work, an ADER type finite volume numerical scheme is proposed as an extension of a first order solver based on weak solutions of RPs with source terms. The type of source terms considered here are a special but relevant type of source terms: their spatial integral is discontinuous. The relevant difference with other previously defined ADER schemes is that it considers the presence of the source term in the solutions of the DRP. Unlike the original ADER schemes, the proposed numerical scheme computes the RPs of the high order terms of the DRP departing from time derivatives of the fluxes as initial conditions for these RPs. Weak solutions of the RPs defined for the DRP are computed using an augmented version of the Roe solver that includes an extra wave that accounts for the contribution of the source term. The discretization done over the source term leads to an energy balanced numerical scheme that allows to obtain the exact solution for steady cases with independence of the grid refinement. In unsteady problems, the numerical scheme ensures the convergence to the exact solution. The numerical scheme is constructed with an arbitrary order of accuracy, and has no theoretical barrier. Numerical results for the Burger's equation and the shallow water equations are presented in this work and indicate that the proposed numerical scheme is able to converge with the expected order of accuracy.