16,962 research outputs found
Space-time discontinuous Galerkin discretization of rotating shallow water equations on moving grids
A space-time discontinuous Galerkin (DG) discretization is presented for the (rotating) shallow water equations over varying topography. We formulate the space-time DG finite element discretization in an efficient and conservative discretization. The HLLC flux is used as numerical flux through the finite element boundaries. When discontinuities are present, we locally apply dissipation around these discontinuities with the help of Krivodonova's discontinuity indicator such that spurious oscillations are suppressed. The non-linear algebraic system resulting from the discretization is solved using a pseudo-time integration with a second-order five-stage Runge-Kutta method. A thorough verification of the space-time DG finite element method is undertaken by comparing numerical and exact solutions. We also carry out a discrete Fourier analysis of the one dimensional linear rotating shallow water equations to show that the method is unconditionally stable with minimal dispersion and dissipation error. The numerical scheme is validated in a novel way by considering various simulations of bore-vortex interactions in combination with a qualitative analysis of PV generation by non-uniform bores. Finally, the space-time DG method is particularly suited for problems where dynamic grid motion is required. To demonstrate this we simulate waves generated by a wave maker and verify these for low amplitude waves where linear theory is approximately valid
Diagnosing numerical Cherenkov instabilities in relativistic plasma simulations based on general meshes
Numerical Cherenkov radiation (NCR) or instability is a detrimental effect
frequently found in electromagnetic particle-in-cell (EM-PIC) simulations
involving relativistic plasma beams. NCR is caused by spurious coupling between
electromagnetic-field modes and multiple beam resonances. This coupling may
result from the slow down of poorly-resolved waves due to numerical (grid)
dispersion and from aliasing mechanisms. NCR has been studied in the past for
finite-difference-based EM-PIC algorithms on regular (structured) meshes with
rectangular elements. In this work, we extend the analysis of NCR to
finite-element-based EM-PIC algorithms implemented on unstructured meshes. The
influence of different mesh element shapes and mesh layouts on NCR is studied.
Analytic predictions are compared against results from finite-element-based
EM-PIC simulations of relativistic plasma beams on various mesh types.Comment: 31 pages, 20 figure
The GeoClaw software for depth-averaged flows with adaptive refinement
Many geophysical flow or wave propagation problems can be modeled with
two-dimensional depth-averaged equations, of which the shallow water equations
are the simplest example. We describe the GeoClaw software that has been
designed to solve problems of this nature, consisting of open source Fortran
programs together with Python tools for the user interface and flow
visualization. This software uses high-resolution shock-capturing finite volume
methods on logically rectangular grids, including latitude--longitude grids on
the sphere. Dry states are handled automatically to model inundation. The code
incorporates adaptive mesh refinement to allow the efficient solution of
large-scale geophysical problems. Examples are given illustrating its use for
modeling tsunamis, dam break problems, and storm surge. Documentation and
download information is available at www.clawpack.org/geoclawComment: 18 pages, 11 figures, Animations and source code for some examples at
http://www.clawpack.org/links/awr10 Significantly modified from original
posting to incorporate suggestions of referee
Statistical and systematic uncertainties in pixel-based source reconstruction algorithms for gravitational lensing
Gravitational lens modeling of spatially resolved sources is a challenging
inverse problem with many observational constraints and model parameters. We
examine established pixel-based source reconstruction algorithms for de-lensing
the source and constraining lens model parameters. Using test data for four
canonical lens configurations, we explore statistical and systematic
uncertainties associated with gridding, source regularisation, interpolation
errors, noise, and telescope pointing. Specifically, we compare two gridding
schemes in the source plane: a fully adaptive grid that follows the lens
mapping but is irregular, and an adaptive Cartesian grid. We also consider
regularisation schemes that minimise derivatives of the source (using two
finite difference methods) and introduce a scheme that minimises deviations
from an analytic source profile. Careful choice of gridding and regularisation
can reduce "discreteness noise" in the surface that is inherent in the
pixel-based methodology. With a gridded source, some degree of interpolation is
unavoidable, and errors due to interpolation need to be taken into account
(especially for high signal-to-noise data). Different realisations of the noise
and telescope pointing lead to slightly different values for lens model
parameters, and the scatter between different "observations" can be comparable
to or larger than the model uncertainties themselves. The same effects create
scatter in the lensing magnification at the level of a few percent for a peak
signal-to-noise ratio of 10, which decreases as the data quality improves.Comment: 20 pages, 18 figures, accepted to MNRAS, see
http://physics.rutgers.edu/~tagoreas/papers/ for high resolution image
High resolution finite volume methods on arbitrary grids via wave propagation
A generalization of Godunov's method for systems of conservation laws has been developed and analyzed that can be applied with arbitrary time steps on arbitrary grids in one space dimension. Stability for arbitrary time steps is achieved by allowing waves to propagate through more than one mesh cell in a time step. The method is extended here to second order accuracy and to a finite volume method in two space dimensions. This latter method is based on solving one dimensional normal and tangential Riemann problems at cell interfaces and again propagating waves through one or more mesh cells. By avoiding the usual time step restriction of explicit methods, it is possible to use reasonable time steps on irregular grids where the minimum cell area is much smaller than the average cell. Boundary conditions for the Euler equations are discussed and special attention is given to the case of a Cartesian grid cut by an irregular boundary. In this case small grid cells arise only near the boundary, and it is desirable to use a time step appropriate for the regular interior cells. Numerical results in two dimensions show that this can be achieved
- …