Implicit and semi-implicit well-balanced finite-volume methods for systems of balance laws

The aim of this work is to design implicit and semi-implicit high-order well-balanced finite-volume numerical methods for 1D systems of balance laws. The strategy introduced by two of the authors in some previous papers for explicit schemes based on the application of a well-balanced reconstruction operator is applied. The well-balanced property is preserved when quadrature formulas are used to approximate the averages and the integral of the source term in the cells. Concerning the time evolution, this technique is combined with a time discretization method of type RK-IMEX or RK-implicit. The methodology will be applied to several systems of balance laws.This work is partially supported by projects RTI2018-096064-B-C21 funded by MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe”, projects P18-RT-3163 of Junta de Andalucía and UMA18-FEDERJA-161 of Junta de Andalucía-FEDER-University of Málaga. G.Russo and S.Boscarino acknowledge partial support from the Italian Ministry of University and Research (MIUR), PRIN Project 2017 (No. 2017KKJP4X) entitled “Innovative numerical methods for evolu-tionary partial differential equations and applications”. I. Gómez-Bueno is also supported by a Grant from “El Ministerio de Ciencia, Innovación y Universidades”, Spain (FPU2019/01541) funded by MCIN/AEI/10.13039/501100011033 and “ESF Invest-ing in your future”. // Funding for open access charge: Universidad de Málaga/CBUA

Well-balanced finite volume schemes for hydrodynamic equations with general free energy

Well balanced and free energy dissipative first- and second-order accurate finite volume schemes are proposed for a general class of hydrodynamic systems with linear and nonlinear damping. The natural Liapunov functional of the system, given by its free energy, allows for a characterization of the stationary states by its variation. An analog property at the discrete level enables us to preserve stationary states at machine precision while keeping the dissipation of the discrete free energy. These schemes allow for analysing accurately the stability properties of stationary states in challeging problems such as: phase transitions in collective behavior, generalized Euler-Poisson systems in chemotaxis and astrophysics, and models in dynamic density functional theories; having done a careful validation in a battery of relevant test cases.Comment: Videos from the simulations of this work are available at https://sergioperezresearch.wordpress.com/well-balance

Relativistic Burgers equations on curved spacetimes. Derivation and finite volume approximation

Within the class of nonlinear hyperbolic balance laws posed on a curved spacetime (endowed with a volume form), we identify a hyperbolic balance law that enjoys the same Lorentz invariance property as the one satisfied by the Euler equations of relativistic compressible fluids. This model is unique up to normalization and converges to the standard inviscid Burgers equation in the limit of infinite light speed. Furthermore, from the Euler system of relativistic compressible flows on a curved background, we derive, both, the standard inviscid Burgers equation and our relativistic generalizations. The proposed models are referred to as relativistic Burgers equations on curved spacetimes and provide us with simple models on which numerical methods can be developed and analyzed. Next, we introduce a finite volume scheme for the approximation of discontinuous solutions to these relativistic Burgers equations. Our scheme is formulated geometrically and is consistent with the natural divergence form of the balance laws under consideration. It applies to weak solutions containing shock waves and, most importantly, is well-balanced in the sense that it preserves steady solutions. Numerical experiments are presented which demonstrate the convergence of the proposed finite volume scheme and its relevance for computing entropy solutions on a curved background.Comment: 19 page