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High order discretely well-balanced methods for arbitrary hydrostatic atmospheres

by J. P. Berberich and R. Käppeli and P. Chandrashekar and C. Klingenberg

(Report number 2021-45)

Abstract
We introduce novel high order well-balanced finite volume methods for the full compressible Euler system with gravity source term. They require no a priori knowledge of the hydrostatic solution which is to be well-balanced and are not restricted to certain classes of hydrostatic solutions. In one spatial dimension we construct a method that exactly balances a high order discretization of any hydrostatic state. The method is extended to two spatial dimensions using a local high order approximation of a hydrostatic state in each cell. The proposed simple, flexible, and robust methods are not restricted to a specific equation of state. Numerical tests verify that the proposed method improves the capability to accurately resolve small perturbations on hydrostatic states.

Keywords: finite-volume methods, well-balancing, hyperbolic balance laws, compressible Euler equations with gravity

@Techreport{BKCK21_987,
  author = {J. P. Berberich and R. K\"appeli and P. Chandrashekar and C. Klingenberg},
  title = {High order discretely well-balanced methods for arbitrary hydrostatic atmospheres},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2021-45},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2021/2021-45.pdf },
  year = {2021}
}

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