28 research outputs found
Embedded discontinuous Galerkin transport schemes with localised limiters
Motivated by finite element spaces used for representation of temperature in
the compatible finite element approach for numerical weather prediction, we
introduce locally bounded transport schemes for (partially-)continuous finite
element spaces. The underlying high-order transport scheme is constructed by
injecting the partially-continuous field into an embedding discontinuous finite
element space, applying a stable upwind discontinuous Galerkin (DG) scheme, and
projecting back into the partially-continuous space; we call this an embedded
DG scheme. We prove that this scheme is stable in L2 provided that the
underlying upwind DG scheme is. We then provide a framework for applying
limiters for embedded DG transport schemes. Standard DG limiters are applied
during the underlying DG scheme. We introduce a new localised form of
element-based flux-correction which we apply to limiting the projection back
into the partially-continuous space, so that the whole transport scheme is
bounded. We provide details in the specific case of tensor-product finite
element spaces on wedge elements that are discontinuous P1/Q1 in the horizontal
and continuous P2 in the vertical. The framework is illustrated with numerical
tests
A conservative implicit multirate method for hyperbolic problems
This work focuses on the development of a self adjusting multirate strategy
based on an implicit time discretization for the numerical solution of
hyperbolic equations, that could benefit from different time steps in different
areas of the spatial domain. We propose a novel mass conservative multirate
approach, that can be generalized to various implicit time discretization
methods. It is based on flux partitioning, so that flux exchanges between a
cell and its neighbors are balanced. A number of numerical experiments on both
non-linear scalar problems and systems of hyperbolic equations have been
carried out to test the efficiency and accuracy of the proposed approach
High-order Discretization of a Gyrokinetic Vlasov Model in Edge Plasma Geometry
We present a high-order spatial discretization of a continuum gyrokinetic
Vlasov model in axisymmetric tokamak edge plasma geometries. Such models
describe the phase space advection of plasma species distribution functions in
the absence of collisions. The gyrokinetic model is posed in a four-dimensional
phase space, upon which a grid is imposed when discretized. To mitigate the
computational cost associated with high-dimensional grids, we employ a
high-order discretization to reduce the grid size needed to achieve a given
level of accuracy relative to lower-order methods. Strong anisotropy induced by
the magnetic field motivates the use of mapped coordinate grids aligned with
magnetic flux surfaces. The natural partitioning of the edge geometry by the
separatrix between the closed and open field line regions leads to the
consideration of multiple mapped blocks, in what is known as a mapped
multiblock (MMB) approach. We describe the specialization of a more general
formalism that we have developed for the construction of high-order,
finite-volume discretizations on MMB grids, yielding the accurate evaluation of
the gyrokinetic Vlasov operator, the metric factors resulting from the MMB
coordinate mappings, and the interaction of blocks at adjacent boundaries. Our
conservative formulation of the gyrokinetic Vlasov model incorporates the fact
that the phase space velocity has zero divergence, which must be preserved
discretely to avoid truncation error accumulation. We describe an approach for
the discrete evaluation of the gyrokinetic phase space velocity that preserves
the divergence-free property to machine precision
A seamless, extended DG approach for advection-diffusion problems on unbounded domains
We propose and analyze a seamless extended Discontinuous Galerkin (DG)
discretization of advection-diffusion equations on semi-infinite domains. The
semi-infinite half line is split into a finite subdomain where the model uses a
standard polynomial basis, and a semi-unbounded subdomain where scaled Laguerre
functions are employed as basis and test functions. Numerical fluxes enable the
coupling at the interface between the two subdomains in the same way as
standard single domain DG interelement fluxes. A novel linear analysis on the
extended DG model yields unconditional stability with respect to the P\'eclet
number. Errors due to the use of different sets of basis functions on different
portions of the domain are negligible, as highlighted in numerical experiments
with the linear advection-diffusion and viscous Burgers' equations. With an
added damping term on the semi-infinite subdomain, the extended framework is
able to efficiently simulate absorbing boundary conditions without additional
conditions at the interface. A few modes in the semi-infinite subdomain are
found to suffice to deal with outgoing single wave and wave train signals more
accurately than standard approaches at a given computational cost, thus
providing an appealing model for fluid flow simulations in unbounded regions.Comment: 27 pages, 8 figure
Flexible and efficient discretizations of multilayer models with variable density
We show that the semi-implicit time discretization approaches previously
introduced for multilayer shallow water models for the barotropic case can be
also applied to the variable density case with Boussinesq approximation.
Furthermore, also for the variable density equations, a variable number of
layers can be used, so as to achieve greater flexibility and efficiency of the
resulting multilayer approach. An analysis of the linearized system, which
allows to derive linear stability parameters in simple configurations, and the
resulting spatially semi-discretized equations are presented. A number of
numerical experiments demonstrate the effectiveness of the proposed approach
Multilayer shallow water models with locally variable number of layers and semi-implicit time discretization
We propose an extension of the discretization approaches for multilayer
shallow water models, aimed at making them more flexible and efficient for
realistic applications to coastal flows. A novel discretization approach is
proposed, in which the number of vertical layers and their distribution are
allowed to change in different regions of the computational domain.
Furthermore, semi-implicit schemes are employed for the time discretization,
leading to a significant efficiency improvement for subcritical regimes. We
show that, in the typical regimes in which the application of multilayer
shallow water models is justified, the resulting discretization does not
introduce any major spurious feature and allows again to reduce substantially
the computational cost in areas with complex bathymetry. As an example of the
potential of the proposed technique, an application to a sediment transport
problem is presented, showing a remarkable improvement with respect to standard
discretization approaches