3,366 research outputs found
Mesh-Free Semi-Lagrangian Methods for Transport on a Sphere Using Radial Basis Functions
We present three new semi-Lagrangian methods based on radial basis function (RBF) interpolation for numerically simulating transport on a sphere. The methods are mesh-free and are formulated entirely in Cartesian coordinates, thus avoiding any irregular clustering of nodes at artificial boundaries on the sphere and naturally bypassing any apparent artificial singularities associated with surface-based coordinate systems. For problems involving tracer transport in a given velocity field, the semi-Lagrangian framework allows these new methods to avoid the use of any stabilization terms (such as hyperviscosity) during time-integration, thus reducing the number of parameters that have to be tuned. The three new methods are based on interpolation using 1) global RBFs, 2) local RBF stencils, and 3) RBF partition of unity. For the latter two of these methods, we find that it is crucial to include some low degree spherical harmonics in the interpolants. Standard test cases consisting of solid body rotation and deformational flow are used to compare and contrast the methods in terms of their accuracy, efficiency, conservation properties, and dissipation/dispersion errors. For global RBFs, spectral spatial convergence is observed for smooth solutions on quasi-uniform nodes, while high-order accuracy is observed for the local RBF stencil and partition of unity approaches
A Characteristic Mapping Method for Transport on the Sphere
A semi-Lagrangian method for the solution of the transport equation on a
sphere is presented. The method evolves the inverse flow-map using the
Characteristic Mapping (CM) [1] and Gradient-Augmented Level Set (GALS) [2]
frameworks. Transport of the advected quantity is then computed by composition
with the numerically approximated inverse flow-map. This framework allows for
the advection of complicated sets and multiple quantities with arbitrarily
fine-features using a coarse computational grid. We discuss the CM method for
linear transport on the sphere and its computational implementation. Standard
test cases for solid body rotation, deformational and divergent flows, and
numerical mixing are presented. The unique features of the method are
demonstrated by the transport of a multi-scale function and a fractal set in a
complex flow environment
Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes
We present a new family of high order accurate fully discrete one-step
Discontinuous Galerkin (DG) finite element schemes on moving unstructured
meshes for the solution of nonlinear hyperbolic PDE in multiple space
dimensions, which may also include parabolic terms in order to model
dissipative transport processes. High order piecewise polynomials are adopted
to represent the discrete solution at each time level and within each spatial
control volume of the computational grid, while high order of accuracy in time
is achieved by the ADER approach. In our algorithm the spatial mesh
configuration can be defined in two different ways: either by an isoparametric
approach that generates curved control volumes, or by a piecewise linear
decomposition of each spatial control volume into simplex sub-elements. Our
numerical method belongs to the category of direct
Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation
formulation of the governing PDE system is considered and which already takes
into account the new grid geometry directly during the computation of the
numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a
posteriori sub-cell finite volume limiter method, in which the validity of the
candidate solution produced in each cell by an unlimited ADER-DG scheme is
verified against a set of physical and numerical detection criteria. Those
cells which do not satisfy all of the above criteria are flagged as troubled
cells and are recomputed with a second order TVD finite volume scheme. The
numerical convergence rates of the new ALE ADER-DG schemes are studied up to
fourth order in space and time and several test problems are simulated.
Finally, an application inspired by Inertial Confinement Fusion (ICF) type
flows is considered by solving the Euler equations and the PDE of viscous and
resistive magnetohydrodynamics (VRMHD).Comment: 39 pages, 21 figure
A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces
In this paper we present a high-order kernel method for numerically solving
diffusion and reaction-diffusion partial differential equations (PDEs) on
smooth, closed surfaces embedded in . For two-dimensional
surfaces embedded in , these types of problems have received
growing interest in biology, chemistry, and computer graphics to model such
things as diffusion of chemicals on biological cells or membranes, pattern
formations in biology, nonlinear chemical oscillators in excitable media, and
texture mappings. Our kernel method is based on radial basis functions (RBFs)
and uses a semi-discrete approach (or the method-of-lines) in which the surface
derivative operators that appear in the PDEs are approximated using
collocation. The method only requires nodes at "scattered" locations on the
surface and the corresponding normal vectors to the surface. Additionally, it
does not rely on any surface-based metrics and avoids any intrinsic coordinate
systems, and thus does not suffer from any coordinate distortions or
singularities. We provide error estimates for the kernel-based approximate
surface derivative operators and numerically study the accuracy and stability
of the method. Applications to different non-linear systems of PDEs that arise
in biology and chemistry are also presented
Study of the convergence of the Meshless Lattice Boltzmann Method in Taylor-Green and annular channel flows
The Meshless Lattice Boltzmann Method (MLBM) is a numerical tool that
relieves the standard Lattice Boltzmann Method (LBM) from regular lattices and,
at the same time, decouples space and velocity discretizations. In this study,
we investigate the numerical convergence of MLBM in two benchmark tests: the
Taylor-Green vortex and annular (bent) channel flow. We compare our MLBM
results to LBM and to the analytical solution of the Navier-Stokes equation. We
investigate the method's convergence in terms of the discretization parameter,
the interpolation order, and the LBM streaming distance refinement. We observe
that MLBM outperforms LBM in terms of the error value for the same number of
nodes discretizing the domain. We find that LBM errors at a given streaming
distance and timestep length are the asymptotic lower
bounds of MLBM errors with the same streaming distance and timestep length.
Finally, we suggest an expression for the MLBM error that consists of the LBM
error and other terms related to the semi-Lagrangian nature of the discussed
method itself
A Comparison of Two Shallow Water Models with Non-Conforming Adaptive Grids: classical tests
In an effort to study the applicability of adaptive mesh refinement (AMR)
techniques to atmospheric models an interpolation-based spectral element
shallow water model on a cubed-sphere grid is compared to a block-structured
finite volume method in latitude-longitude geometry. Both models utilize a
non-conforming adaptation approach which doubles the resolution at fine-coarse
mesh interfaces. The underlying AMR libraries are quad-tree based and ensure
that neighboring regions can only differ by one refinement level.
The models are compared via selected test cases from a standard test suite
for the shallow water equations. They include the advection of a cosine bell, a
steady-state geostrophic flow, a flow over an idealized mountain and a
Rossby-Haurwitz wave. Both static and dynamics adaptations are evaluated which
reveal the strengths and weaknesses of the AMR techniques. Overall, the AMR
simulations show that both models successfully place static and dynamic
adaptations in local regions without requiring a fine grid in the global
domain. The adaptive grids reliably track features of interests without visible
distortions or noise at mesh interfaces. Simple threshold adaptation criteria
for the geopotential height and the relative vorticity are assessed.Comment: 25 pages, 11 figures, preprin
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Modeling single-phase flow and solute transport across scales
textFlow and transport phenomena in the subsurface often span a wide range of length (nanometers to kilometers) and time (nanoseconds to years) scales, and frequently arise in applications of CO₂ sequestration, pollutant transport, and near-well acid stimulation. Reliable field-scale predictions depend on our predictive capacity at each individual scale as well as our ability to accurately propagate information across scales. Pore-scale modeling (coupled with experiments) has assumed an important role in improving our fundamental understanding at the small scale, and is frequently used to inform/guide modeling efforts at larger scales. Among the various methods, there often exists a trade-off between computational efficiency/simplicity and accuracy. While high-resolution methods are very accurate, they are computationally limited to relatively small domains. Since macroscopic properties of a porous medium are statistically representative only when sample sizes are sufficiently large, simple and efficient pore-scale methods are more attractive. In this work, two Eulerian pore-network models for simulating single-phase flow and solute transport are developed. The models focus on capturing two key pore-level mechanisms: a) partial mixing within pores (large void volumes), and b) shear dispersion within throats (narrow constrictions connecting the pores), which are shown to have a substantial impact on transverse and longitudinal dispersion coefficients at the macro scale. The models are verified with high-resolution pore-scale methods and validated against micromodel experiments as well as experimental data from the literature. Studies regarding the significance of different pore-level mixing assumptions (perfect mixing vs. partial mixing) in disordered media, as well as the predictive capacity of network modeling as a whole for ordered media are conducted. A mortar domain decomposition framework is additionally developed, under which efficient and accurate simulations on even larger and highly heterogeneous pore-scale domains are feasible. The mortar methods are verified and parallel scalability is demonstrated. It is shown that they can be used as “hybrid” methods for coupling localized pore-scale inclusions to a surrounding continuum (when insufficient scale separation exists). The framework further permits multi-model simulations within the same computational domain. An application of the methods studying “emergent” behavior during calcite precipitation in the context of geologic CO₂ sequestration is provided.Petroleum and Geosystems Engineerin
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