Asymptotically and exactly energy balanced augmented flux-ADER schemes with application to hyperbolic conservation laws with geometric source terms

Abstract

In this work, an arbitrary order HLL-type numerical scheme is constructed using the flux-ADER methodology. The proposed scheme is based on an augmented Derivative Riemann solver that was used for the first time in [A. Navas-Montilla, J. Murillo, Energy balanced numerical schemes with very high order. The Augmented Roe Flux ADER scheme. Application to the shallow water equations, J. Comput. Phys. 290 (2015) 188-218]. Such solver, hereafter referred to as Flux-Source (FS) solver, was conceived as a high order extension of the augmented Roe solver and led to the generation of a novel numerical scheme called AR-ADER scheme. Here, we provide a general definition of the FS solver independently of the Riemann solver used in it. Moreover, a simplified version of the solver, referred to as Linearized-Flux-Source (LFS) solver, is presented. This novel version of the FS solver allows to compute the solution without requiring reconstruction of derivatives of the fluxes, nevertheless some drawbacks are evidenced. In contrast to other previously defined Derivative Riemann solvers, the proposed FS and LFS solvers take into account the presence of the source term in the resolution of the Derivative Riemann Problem (DRP), which is of particular interest when dealing with geometric source terms. When applied to the shallow water equations, the proposed HLLS-ADER and AR-ADER schemes can be constructed to fulfill the exactly well-balanced property, showing that an arbitrary quadrature of the integral of the source inside the cell does not ensure energy balanced solutions. As a result of this work, energy balanced flux-ADER schemes that provide the exact solution for steady cases and that converge to the exact solution with arbitrary order for transient cases

Modelos de simulación de alto orden para la resolución de fenómenos de propagación de ondas en flujos de lámina libre con turbulencia

“…donde la condición inicial viene dada por la reconstrucción WENO (Navas-Montilla y Murillo, 2016). El término Si+1/2 representa una aproximación de la integral del término fuente en la pared, que será cero cuando el término fuente no sea de tipo geométrico (término de fondo en ecuaciones de aguas poco profundas).…”

“…Siempre se garantizará el equilibrio hidrostático (propiedad well-balanced), y en otros casos particulares, también se puede garantizar la conservación de la energía o incluso el equilibrio geostrófico, si el dominio está en rotación. Los métodos propuestos están basados en un solver de Riemann de orden arbitrario, denominado LFS solver, que incluye la variación de los términos fuente a través de las paredes de las celdas computacionales en la solución al Problema de Riemann Derivativo (Navas-Montilla y Murillo, 2016;Navas-Montilla y Murillo, 2018). En particular se utilizará el solver de Roe aumentado (ARoe).…”

“…λ ± y ! e m son los valores y vectores propios del jacobiano (Navas-Montilla yMurillo, 2016).RESULTADOSEn este apartado, el modelo de simulación descrito anteriormente se utiliza para reproducir un experimento de laboratorio. En 1997, Lloyd y Stansby (1997) llevaron a cabo distintos experimentos que involucraban un flujo de aguas poco profundas sobre islas cónicas de baja pendiente con diferentes geometrías y relación entre la altura de la isla y el calado.…”

A comprehensive explanation and exercise of the source terms in hyperbolic systems using Roe type solutions. Application to the 1D-2D shallow water equations

“…The well balanced numerical property in cases of quiescent equilibrium is a particular case. In [20,21,22,23] exactly well-balanced methods, named energy balanced numerical schemes and hereafter referred as E-schemes, able to reproduce exactly steady solutions with independence of the mesh refinement, were presented. On the other hand, non-physical negative water height becomes problematic when computing simulations as the eigenvalues do not determine the time step size as a result of the not pure hyperbolic characteristic of the system of equations [24,25,15].…”

“…Although the ARoe solver detailed here is only first order with an explicit Euler time-stepping, it is worth pointing out that a relevant feature of this solver is that it can be directly applied to flux-ADER schemes with arbitrary order [22,23] without losing any of its desired properties.…”