Large time step asymptotic preserving evolution Galerkin methods for multidimensional system of hyperbolic balance laws.

• Applicants: Professorin Dr. Maria Lukacova; Professor Dr. Sebastian Noelle
• Subject Area: Mathematics
• Term: from 2008 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 70878602

• Final Report Year: 2017

Final Report Abstract

The aim of proposed project was development and analysis of large time step asymptotic preserving finite volume methods for two- and three-dimensional systems of hyperbolic balance laws with a singular parameter, such as low Froude or low Mach number. In order to approximate efficiently weakly compressible flows that typically arise in geophysical applications we build upon multiscale analysis that yields us suitable splittings of flux functions into fast and slow waves. We have followed two splitting strategies: (i) acoustic/advection splitting that is applied to the equations describing evolution of perturbations of the (hydrostatic) equilibrium and (ii) reference solution splitting that is based on the use a reference solution. The latter can be either the equilibrium solution or the singular limit solution. Our goal was to study asymptotic preserving properties of these methods and show that a suitable splitting into the linear stiff subsystem for fast acoustic/gravity waves and the nonlinear non-stiff subsystem for slow advection combined with the IMEX finite volume discretization yields asymptotic preserving schemes. In order to obtain large time step schemes, where the CFL stability condition is dictated only by the flow velocity, we have approximated fast waves implicitly and slow waves explicitly. This led us to use globally asymptotic accurate IMEX type methods for time discretization. We have studied theoretically asymptotic consistency and stability of the proposed IMEX finite volume methods. It turned out that in order to obtain an asymptotic preserving method, no artificial viscosity should be applied to the spatial approximation of fast waves. Furthermore, asymptotic stability of our IMEX FV methods has been analysed using both the energy and the modified equation method. For the isentropic Navier-Stokes equation asymptotic preserving error estimates have been proven for a combined finite element-finite volume scheme using the relative entropy approach. We have developed computational codes for two- and three-dimensional IMEX FV schemes and applied them to full compressible or isentropic Euler equations with small Mach number and the shallow water equations with small Froude number. Extensive experimental analysis confirm reliability and asymptotic preserving properties of the our new IMEX FV schemes.

Publications

• A weakly asymptotic preserving low Mach number scheme for the Euler equations of gas dynamics, SIAM J. Sci. Comp. (2015), 36(6), B989–B1024
  Noelle S., Bispen G., Arun K.R., Lukáčová-Medvidová M., Munz C.-D.
  
• Finite volume evolution Galerkin methods for the shallow water equations with dry beds, Comm. Comput. Phys. 10(2) (2011), 371–404
  Bollermann A., Noelle S., Lukáčová-Medvidová M.
  
• Adaptive discontinuous evolution Galerkin method for dry atmospheric flow, J. Comput. Phys. (2014), 268, 106-133
  Yelash L., Müller A., Lukáčová-Medvidová M., Giraldo F.X., Wirth V.
  (See online at https://doi.org/10.1016/j.jcp.2014.02.034)
• IMEX large time step finite volume methods for low Froude number shallow water flows, Commun. Comput. Phys. (2014), 16(2), 307-347
  Bispen G., Arun K.R., Lukáčová-Medvidová M., Noelle S.
  (See online at https://doi.org/10.4208/cicp.040413.160114a)
• IMEX Finite Volume Schemes for the Shallow Water Equations, PhD Thesis, 2015
  Bispen G.
  
• A new stable splitting for singularly perturbed ODEs. Applied Numerical Mathematics (2016), 107, 18-33
  Schütz J. and Kaiser K.
  (See online at https://doi.org/10.1016/j.apnum.2016.04.004)
• Asymptotic preserving error estimates for numerical solutions of compressible Navier-Stokes equations in the low Mach number regime, 2016
  Feireisl E.,Lukáčová-Medvidová M., Nečasová S., Novotný A., She B
  
• A new hydrostatic reconstruction scheme based on subcell reconstructions. IGPM report 440 (2015), RWTH Aachen University. SIAM Journal on Numerical Analysis (2017), 55, 758-784
  Chen G. and Noelle S.
  (See online at https://doi.org/10.1137/15M1053074)
• A new stable splitting for the isentropic Euler equations, J. Sci. Comput. (2017), 70(3), 1390-1407
  Kaiser K., Schütz J., Schöbel R., Noelle S.
  (See online at https://doi.org/10.1007/s10915-016-0286-6)
• Asymptotic preserving IMEX finite volume schemes for low Mach number Euler equations with gravitation, J. Comput. Phys. (2017), 335, 222-248
  Bispen G.,Lukáčová-Medvidová M., Yelash, L.
  (See online at https://doi.org/10.1016/j.jcp.2017.01.020)