378 research outputs found
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
Positivity-Preserving Well-Balanced Central Discontinuous Galerkin Schemes for the Euler Equations under Gravitational Fields
This paper designs and analyzes positivity-preserving well-balanced (WB)
central discontinuous Galerkin (CDG) schemes for the Euler equations with
gravity. A distinctive feature of these schemes is that they not only are WB
for a general known stationary hydrostatic solution, but also can preserve the
positivity of the fluid density and pressure. The standard CDG method does not
possess this feature, while directly applying some existing WB techniques to
the CDG framework may not accommodate the positivity and keep other important
properties at the same time. In order to obtain the WB and
positivity-preserving properties simultaneously while also maintaining the
conservativeness and stability of the schemes, a novel spatial discretization
is devised in the CDG framework based on suitable modifications to the
numerical dissipation term and the source term approximation. The modifications
are based on a crucial projection operator for the stationary hydrostatic
solution, which is proposed for the first time in this work. This novel
projection has the same order of accuracy as the standard -projection, can
be explicitly calculated, and is easy to implement without solving any
optimization problems. More importantly, it ensures that the projected
stationary solution has the same cell averages on both the primal and dual
meshes, which is a key to achieve the desired properties of our schemes. Based
on some convex decomposition techniques, rigorous positivity-preserving
analyses for the resulting WB CDG schemes are carried out. Several one- and
two-dimensional numerical examples are performed to illustrate the desired
properties of these schemes, including the high-order accuracy, the WB
property, the robustness for simulations involving the low pressure or density,
high resolution for the discontinuous solutions and the small perturbations
around the equilibrium state.Comment: 57 page
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