Well-balanced finite difference WENO schemes for the blood flow model

The blood flow model maintains the steady state solutions, in which the flux gradients are non-zero but exactly balanced by the source term. In this paper, we design high order finite difference weighted non-oscillatory (WENO) schemes to this model with such well-balanced property and at the same time keeping genuine high order accuracy. Rigorous theoretical analysis as well as extensive numerical results all indicate that the resulting schemes verify high order accuracy, maintain the well-balanced property, and keep good resolution for smooth and discontinuous solutions

A family of well-balanced WENO and TENO schemes for atmospheric flows

We herein present a novel methodology to construct very high order well-balanced schemes for the computation of the Euler equations with gravitational source term, with application to numerical weather prediction (NWP). The proposed method is based on augmented Riemann solvers, which allow preserving the exact equilibrium between fluxes and source terms at cell interfaces. In particular, the augmented HLL solver (HLLS) is considered. Different spatial reconstruction methods can be used to ensure a high order of accuracy in space (e.g. WENO, TENO, linear reconstruction), being the TENO reconstruction the preferred method in this work. To the knowledge of the authors, the TENO method has not been applied to NWP before, although it has been extensively used by the computational fluid dynamics community in recent years. Therefore, we offer a thorough assessment of the TENO method to evidence its suitability for NWP considering some benchmark cases which involve inertia and gravity waves as well as convective processes. The TENO method offers an enhanced behavior when dealing with turbulent flows and underresolved solutions, where the traditional WENO scheme proves to be more diffusive. The proposed methodology, based on the HLLS solver in combination with a very high-order discretization, allows carrying out the simulation of meso- and micro-scale atmospheric flows in an implicit Large Eddy Simulation manner. Due to the HLLS solver, the isothermal, adiabatic and constant Brunt-Väisälä frequency hydrostatic equilibrium states are preserved with machine accuracy

ARBITRARY ORDER FINITE VOLUME WELL-BALANCED SCHEMES FOR THE EULER EQUATIONS WITH GRAVITY

This work presents arbitrary high order well balanced finite volume schemes for the Euler equations with a prescribed gravitational field. It is assumed that the desired equilibrium solution is known, and we construct a scheme which is exactly well balanced for that particular equilibrium. The scheme is based on high order reconstructions of the fluctuations from equilibrium of density, velocity, and pressure, and on a well-balanced integration of the source terms, while no assumptions are needed on the numerical flux, beside consistency. This technique also allows one to construct well-balanced methods for a class of moving equilibria. Several numerical tests demonstrate the performance of the scheme on different scenarios, from equilibrium solutions to nonsteady problems involving shocks. The numerical tests are carried out with methods up to fifth order in one dimension, and third order accuracy in two dimensions