797 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
ARBITRARY ORDER FINITE VOLUME WELL-BALANCED SCHEMES FOR THE EULER EQUATIONS WITH GRAVITY
This work presents arbitrary high order well balanced finite volume schemes for the Euler equations with a prescribed gravitational field. It is assumed that the desired equilibrium solution is known, and we construct a scheme which is exactly well balanced for that particular equilibrium. The scheme is based on high order reconstructions of the fluctuations from equilibrium of density, velocity, and pressure, and on a well-balanced integration of the source terms, while no assumptions are needed on the numerical flux, beside consistency. This technique also allows one to construct well-balanced methods for a class of moving equilibria. Several numerical tests demonstrate the performance of the scheme on different scenarios, from equilibrium solutions to nonsteady problems involving shocks. The numerical tests are carried out with methods up to fifth order in one dimension, and third order accuracy in two dimensions
Well-balanced finite volume schemes for nearly steady adiabatic flows
We present well-balanced finite volume schemes designed to approximate the
Euler equations with gravitation. They are based on a novel local steady state
reconstruction. The schemes preserve a discrete equivalent of steady adiabatic
flow, which includes non-hydrostatic equilibria. The proposed method works in
Cartesian, cylindrical and spherical coordinates. The scheme is not tied to any
specific numerical flux and can be combined with any consistent numerical flux
for the Euler equations, which provides great flexibility and simplifies the
integration into any standard finite volume algorithm. Furthermore, the schemes
can cope with general convex equations of state, which is particularly
important in astrophysical applications. Both first- and second-order accurate
versions of the schemes and their extension to several space dimensions are
presented. The superior performance of the well-balanced schemes compared to
standard schemes is demonstrated in a variety of numerical experiments. The
chosen numerical experiments include simple one-dimensional problems in both
Cartesian and spherical geometry, as well as two-dimensional simulations of
stellar accretion in cylindrical geometry with a complex multi-physics equation
of state
GPU driven finite difference WENO scheme for real time solution of the shallow water equations
The shallow water equations are applicable to many common engineering problems involving modelling of waves dominated by motions in the horizontal directions (e.g. tsunami propagation, dam breaks). As such events pose substantial economic costs, as well as potential loss of life, accurate real-time simulation and visualization methods are of great importance. For this purpose, we propose a new finite difference scheme for the 2D shallow water equations that is specifically formulated to take advantage of modern GPUs. The new scheme is based on the so-called Picard integral formulation of conservation laws combined with Weighted Essentially Non-Oscillatory reconstruction. The emphasis of the work is on third order in space and second order in time solutions (in both single and double precision). Further, the scheme is well-balanced for bathymetry functions that are not surface piercing and can handle wetting and drying in a GPU-friendly manner without resorting to long and specific case-by-case procedures. We also present a fast single kernel GPU implementation with a novel boundary condition application technique that allows for simultaneous real-time visualization and single precision simulations even on large ( > 2000 × 2000) grids on consumer-level hardware - the full kernel source codes are also provided online at https://github.com/pparna/swe_pifweno3
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
High-Order Fully General-Relativistic Hydrodynamics: new Approaches and Tests
We present a new approach for achieving high-order convergence in fully
general-relativistic hydrodynamic simulations. The approach is implemented in
WhiskyTHC, a new code that makes use of state-of-the-art numerical schemes and
was key in achieving, for the first time, higher than second-order convergence
in the calculation of the gravitational radiation from inspiraling binary
neutron stars Radice et al. (2013). Here, we give a detailed description of the
algorithms employed and present results obtained for a series of classical
tests involving isolated neutron stars. In addition, using the
gravitational-wave emission from the late inspiral and merger of binary neutron
stars, we make a detailed comparison between the results obtained with the new
code and those obtained when using standard second-order schemes commonly
employed for matter simulations in numerical relativity. We find that even at
moderate resolutions and for binaries with large compactness, the phase
accuracy is improved by a factor 50 or more.Comment: 34 pages, 16 figures. Version accepted on CQ
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