383 research outputs found
A discontinuous finite element baroclinic marine model on unstructured prismatic meshes: I. Space discretization
We describe the space discretization of a three-dimensional baroclinic finite element model, based upon a discontinuous Galerkin method, while the companion paper (Comblen et al. 2010a) describes the discretization in time. We solve the hydrostatic Boussinesq equations governing marine flows on a mesh made up of triangles extruded from the surface toward the seabed to obtain prismatic three-dimensional elements. Diffusion is implemented using the symmetric interior penalty method. The tracer equation is consistent with the continuity equation. A Lax–Friedrichs flux is used to take into account internal wave propagation. By way of illustration, a flow exhibiting internal waves in the lee of an isolated seamount on the sphere is simulated. This enables us to show the advantages of using an unstructured mesh, where the resolution is higher in areas where the flow varies rapidly in space, the mesh being coarser far from the region of interest. The solution exhibits the expected wave structure. Linear and quadratic shape functions are used, and the extension to higher-order discretization is straightforward
Discontinuous Galerkin Methods for Solving Acoustic Problems
Parciální diferenciální rovnice hrají důležitou v inženýrských aplikacích. Často je možné tyto rovnice řešit pouze přibližně, tj. numericky. Z toho důvodu vzniklo množství diskretizačních metod pro řešení těchto rovnic. Uvedená nespojitá Galerkinova metoda se zdá jako velmi obecná metoda pro řešení těchto rovnic, především pak pro hyperbolické systémy. Naším cílem je řešit úlohy aeroakustiky, přičemž šíření akustických vln je popsáno pomocí linearizovaných Eulerových rovnic. A jelikož se jedná o hyperbolický systém, byla vybrána právě nespojitá Galerkinova metoda. Mezi nejdůležitější aspekty této metody patří schopnost pracovat s geometricky složitými oblastmi, možnost dosáhnout metody vysokého řádu a dále lokální charakter toho schématu umožnuje efektivní paralelizaci výpočtu. Nejprve uvedeme nespojitou Galerkinovu metodu v obecném pojetí pro jedno- a dvoudimenzionalní úlohy. Algoritmus následně otestujeme pro řešení rovnice advekce, která byla zvolena jako modelový případ hyperbolické rovnice. Metoda nakonec bude testována na řadě verifikačních úloh, které byly formulovány pro testování metod pro výpočetní aeroakustiku, včetně oveření okrajových podmínek, které, stejně jako v případě teorie proudění tekutin, jsou nedílnou součástí výpočetní aeroakustiky.Partial differential equations play an important role in engineering applications. It is often possible to solve these equations only approximately, i.e. numerically. Therefore number of successful discretization techniques has been developed to solve these equations. The presented discontinuous Galerkin method seems to be very general method to solve this type of equations, especially useful for hyperbolic systems. Our aim is to solve aeroacoustic problems, where propagation of acoustic waves is described using linearized Euler equations. This system of equations is indeed hyperbolic and therefore the discontinuous Galerkin method was chosen. The most important aspects of this method is ability to deal with complex geometries, possibility of high-order method and its local character enabling efficient computation parallelization. We first introduce the discontinuous Galerkin method in general for one- and two-dimensional problems. We then test the algorithm to solve advection equation, which was chosen as a model case of hyperbolic equation. The method will be finally tested using number of verification problems, which were formulated to test methods for computational equations, including verification of boundary conditions, which, similarly to computational fluid dynamics, are important part of computational aeroacoustics.
Towards exponentially-convergent simulations of extreme-mass-ratio inspirals: A time-domain solver for the scalar Teukolsky equation with singular source terms
Gravitational wave signals from extreme mass ratio inspirals are a key target
for space-based gravitational wave detectors. These systems are typically
modeled as a distributionally-forced Teukolsky equation, where the smaller
black hole is treated as a Dirac delta distribution. Time-domain solvers often
use regularization approaches that approximate the Dirac distribution that
often introduce small length scales and are a source of systematic error,
especially near the smaller black hole. We describe a multi-domain
discontinuous Galerkin method for solving the distributionally-forced Teukolsky
equation that describes scalar fields evolving on a Kerr spacetime. To handle
the Dirac delta, we expand the solution in spherical harmonics and recast the
sourced Teukolsky equation as a first-order, one-dimensional symmetric
hyperbolic system. This allows us to derive the method's numerical flux to
correctly account for the Dirac delta. As a result, our method achieves global
spectral accuracy even at the source's location. To connect the near field to
future null infinity, we use the hyperboloidal layer method, allowing us to
supply outer boundary conditions and providing direct access to the far-field
waveform. We document several numerical experiments where we test our method,
including convergence tests against exact solutions, energy luminosities for
circular orbits, the scheme's superconvergence properties at future null
infinity, and the late-time tail behavior of the scalar field. We also compare
two systems that arise from different choices of the first-order reduction
variables, finding that certain choices are numerically problematic in
practice. The methods developed here may be beneficial when computing
gravitational self-force effects, where the regularization procedure has been
developed for the spherical harmonic modes and high accuracy is needed at the
Dirac delta's location.Comment: 20 pages, 7 figures and 2 table
A mixed discontinuous/continuous finite element pair for shallow-water ocean modelling
We introduce a mixed discontinuous/continuous finite element pair for ocean
modelling, with continuous quadratic pressure/layer depth and discontinuous
velocity. We investigate the finite element pair applied to the linear
shallow-water equations on an f-plane. The element pair has the property that
all geostrophically balanced states which strongly satisfy the boundary
conditions have discrete divergence equal to exactly zero and hence are exactly
steady states of the discretised equations. This means that the finite element
pair has excellent geostrophic balance properties. We illustrate these
properties using numerical tests and provide convergence calculations which
show that the discretisation has quadratic errors, indicating that the element
pair is stable
High Order Fluctuation Splitting Schemes for Hyperbolic Conservation Laws
This thesis presents the construction, the analysis and the verification of a new form of higher than second order fluctuation splitting discretisation for the solution of steady conservation laws on unstructured meshes. This is an alternative approach to the two existing higher than second order fluctuation splitting schemes, which use submesh reconstruction (developed by Abgrall and Roe) and gradient recovery (developed by Caraemi) to obtain the loacl higher degree polynomials used to evaluate the fluctuation. The new higher than second order approach constructs the polynomial interpolant of the values of the dependent variables at an appropriate number of carefully chosen mesh nodes.
As they stand, none of the higher than second order methods can guarantee the absence of spurious oscillations from the flow without the application of an additional smoothing stage. The implementation of a technique that removes unphysical oscillations (devised by Hubbard) as part of a new higher than second order approach will be outlined. The design steps and theoretical bases are discussed in depth.
The new higher than second order approach is examined and analysed through application to a series of linear and nonlinear scalar problems, using a pseudo-time-stepping technique to reach steady state solution on two-dimensional structured and unstructured meshes. The results demonstrate its effectiveness in approximating the linear and nolinear scalar problems.
This thesis also addresses the development and examination of a multistage high order (in space and time) fluctuation splitting scheme for two-dimensional unsteady scalar advection on triangular unstructured meshes. the method is similar in philosophy to that of multistep high order (in space and time) fluctuation splitting scheme for the approximation of time-dependent hyperbolic conservation laws. The construction and implementation of the high order multistage time-dependent method are discussed in detail and its performance is illustrated using several standard test problems. The multistage high order time-dependent method is evaluated in the context of existing fluctuation splitting approaches to modelling time-dependent problems and some suggestions for their future development are made. Results presented indicate that the multistage high orer method can produce a slightly more accurate solution than the multistep high order method
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
Efficient numerical solution of the Fokker-Planck equation using physics-conforming finite element methods
We consider the Fokker-Planck equation (FPE) for the orientation probability density of fiber suspensions. Using the continuous Galerkin method, we express the numerical solution in terms of Lagrange basis functions that are associated with N nodes of a computational mesh for a domain in the 3D physical space and M nodes of a mesh for the surface of a unit sphere representing the configuration space. The NM time-dependent unknowns of our finite element approximations are probabilities corresponding to discrete space locations and orientation angles. The framework of alternatingdirection methods enables us to update the numerical solution in parallel by solving N evolution equations on the sphere and M three-dimensional advection equations in each (pseudo-)time step. To ensure positivity preservation as well as the normalization property of the probability density, we perform algebraic flux correction for each equation and synchronize the correction factors corresponding to different orientation angles. The velocity field for the spatial advection step is obtained using a Schur complement method to solve a generalized system of the incompressible Navier-Stokes equations (NSE). Fiber-induced subgrid-scale effects are taken into account using an effective stress tensor that depends on the second- and fourth-order moments of the orientation density function. Numerical studies are performed for individual subproblems and for the coupled FPE-NSE system
Geometric Error of Finite Volume Schemes for Conservation Laws on Evolving Surfaces
This paper studies finite volume schemes for scalar hyperbolic conservation
laws on evolving hypersurfaces of . We compare theoretical
schemes assuming knowledge of all geometric quantities to (practical) schemes
defined on moving polyhedra approximating the surface. For the former schemes
error estimates have already been proven, but the implementation of such
schemes is not feasible for complex geometries. The latter schemes, in
contrast, only require (easily) computable geometric quantities and are thus
more useful for actual computations. We prove that the difference between
approximate solutions defined by the respective families of schemes is of the
order of the mesh width. In particular, the practical scheme converges to the
entropy solution with the same rate as the theoretical one. Numerical
experiments show that the proven order of convergence is optimal.Comment: 23 pages, 5 figures, to appear in Numerische Mathemati
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