236 research outputs found
Low-diffusivity scalar transport using a WENO scheme and dual meshing
Interfacial mass transfer of low-diffusive substances in an unsteady flow
environment is marked by a very thin boundary layer at the interface and other
regions with steep concentration gradients. A numerical scheme capable of
resolving accurately most details of this process is presented. In this scheme,
the fourth-order accurate WENO method developed by Liu et al. (1994) was
implemented on a non-uniform staggered mesh to discretize the scalar convection
while for the scalar diffusion a fourth-order accurate central discretization
was employed. The discretization of the scalar convection-diffusion equation
was combined with a fourth-order Navier-Stokes solver which solves the
incompressible flow. A dual meshing strategy was employed, in which the scalar
was solved on a finer mesh than the incompressible flow. The solver was tested
by performing a number of two-dimensional simulations of an unstably stratified
flow with low diffusivity scalar transport. The unstable stratification led to
buoyant convection which was modelled using a Boussinesq approximation with a
linear relationship between flow temperature and density. The order of accuracy
for one-dimensional scalar transport on a stretched and uniform grid was also
tested. The results show that for the method presented above a relatively
coarse mesh is sufficient to accurately describe the fluid flow, while the use
of a refined mesh for the low-diffusive scalars is found to be beneficial in
order to obtain a highly accurate resolution with negligible numerical
diffusion
Multidimensional adaptive order GP-WENO via kernel-based reconstruction
This paper presents a fully multidimensional kernel-based reconstruction
scheme for finite volume methods applied to systems of hyperbolic conservation
laws, with a particular emphasis on the compressible Euler equations.
Non-oscillatory reconstruction is achieved through an adaptive order weighted
essentially non-oscillatory (WENO-AO) method cast into a form suited to
multidimensional stencils and reconstruction. A kernel-based approach inspired
by Gaussian process (GP) modeling is presented here. This approach allows the
creation of a scheme of arbitrary order with simply defined multidimensional
stencils and substencils. Furthermore, the fully multidimensional nature of the
reconstruction allows a more straightforward extension to higher spatial
dimensions and removes the need for complicated boundary conditions on
intermediate quantities in modified dimension-by-dimension methods. In
addition, a new simple-yet-effective set of reconstruction variables is
introduced, as well as an easy-to-implement effective limiter for positivity
preservation, both of which could be useful in existing schemes with little
modification. The proposed scheme is applied to a suite of stringent and
informative benchmark problems to demonstrate its efficacy and utility.Comment: Submitted to Journal of Computational Physics April 202
Evolving Neural Network (ENN) Method for One-Dimensional Scalar Hyperbolic Conservation Laws: I Linear and Quadratic Fluxes
We propose and study the evolving neural network (ENN) method for solving
one-dimensional scalar hyperbolic conservation laws with linear and quadratic
spatial fluxes. The ENN method first represents the initial data and the inflow
boundary data by neural networks. Then, it evolves the neural network
representation of the initial data along the temporal direction. The evolution
is computed using a combination of characteristic and finite volume methods.
For the linear spatial flux, the method is not subject to any time step size,
and it is shown theoretically that the error at any time step is bounded by the
representation errors of the initial and boundary condition. For the quadratic
flux, an error estimate is studied in a companion paper. Finally, numerical
results for the linear advection equation and the inviscid Burgers equation are
presented to show that the ENN method is more accurate and cost efficient than
traditional mesh-based methods
Solving 3D relativistic hydrodynamical problems with WENO discontinuous Galerkin methods
Discontinuous Galerkin (DG) methods coupled to WENO algorithms allow high
order convergence for smooth problems and for the simulation of discontinuities
and shocks. In this work, we investigate WENO-DG algorithms in the context of
numerical general relativity, in particular for general relativistic
hydrodynamics. We implement the standard WENO method at different orders, a
compact (simple) WENO scheme, as well as an alternative subcell evolution
algorithm. To evaluate the performance of the different numerical schemes, we
study non-relativistic, special relativistic, and general relativistic
testbeds. We present the first three-dimensional simulations of general
relativistic hydrodynamics, albeit for a fixed spacetime background, within the
framework of WENO-DG methods. The most important testbed is a single TOV-star
in three dimensions, showing that long term stable simulations of single
isolated neutron stars can be obtained with WENO-DG methods.Comment: 21 pages, 10 figure
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