88,072 research outputs found
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
Modelling of radionuclide migration through the geosphere with radial basis function method and geostatistics
The modelling of radionuclide transport through the geosphere is necessary in the safety assessment of repositories for radioactive waste. A number of key geosphere processes need to be considered when predicting the movement of radionuclides through the geosphere. The most important input data are obtained from field measurements, which are not available for all regions of interest. For example, the hydraulic conductivity, as input parameter, varies from place to place. In such cases geostatistical science offers a variety of spatial estimation procedures. To assess the a long term safety of a radioactive waste disposal system, mathematical models are used to describe the complicated groundwater flow, chemistry and potential radionuclide migration through geological formations. The numerical solution of partial differential equations (PDEs) has usually been obtained by finite difference methods (FDM), finite element methods (FEM), or finite volume methods (FVM). Kansa introduced the concept of solving PDEs using radial basis functions (RBFs) for hyperbolic, parabolic and elliptic PDEs. The aim of this study was to present a relatively new approach to the modelling of radionuclide migration through the geosphere using radial basis functions methods and to determine the average and sample variance of radionuclide concentration with regard to spatial variability of hydraulic conductivity modelled by a geostatistical approach. We will also explore residual errors and their influence on optimal shape parameters
The Road to Quantum Computational Supremacy
We present an idiosyncratic view of the race for quantum computational
supremacy. Google's approach and IBM challenge are examined. An unexpected
side-effect of the race is the significant progress in designing fast classical
algorithms. Quantum supremacy, if achieved, won't make classical computing
obsolete.Comment: 15 pages, 1 figur
Kernel-based stochastic collocation for the random two-phase Navier-Stokes equations
In this work, we apply stochastic collocation methods with radial kernel
basis functions for an uncertainty quantification of the random incompressible
two-phase Navier-Stokes equations. Our approach is non-intrusive and we use the
existing fluid dynamics solver NaSt3DGPF to solve the incompressible two-phase
Navier-Stokes equation for each given realization. We are able to empirically
show that the resulting kernel-based stochastic collocation is highly
competitive in this setting and even outperforms some other standard methods
High Order Cell-Centered Lagrangian-Type Finite Volume Schemes with Time-Accurate Local Time Stepping on Unstructured Triangular Meshes
We present a novel cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE)
finite volume scheme on unstructured triangular meshes that is high order
accurate in space and time and that also allows for time-accurate local time
stepping (LTS). The new scheme uses the following basic ingredients: a high
order WENO reconstruction in space on unstructured meshes, an element-local
high-order accurate space-time Galerkin predictor that performs the time
evolution of the reconstructed polynomials within each element, the computation
of numerical ALE fluxes at the moving element interfaces through approximate
Riemann solvers, and a one-step finite volume scheme for the time update which
is directly based on the integral form of the conservation equations in
space-time. The inclusion of the LTS algorithm requires a number of crucial
extensions, such as a proper scheduling criterion for the time update of each
element and for each node; a virtual projection of the elements contained in
the reconstruction stencils of the element that has to perform the WENO
reconstruction; and the proper computation of the fluxes through the space-time
boundary surfaces that will inevitably contain hanging nodes in time due to the
LTS algorithm. We have validated our new unstructured Lagrangian LTS approach
over a wide sample of test cases solving the Euler equations of compressible
gasdynamics in two space dimensions, including shock tube problems, cylindrical
explosion problems, as well as specific tests typically adopted in Lagrangian
calculations, such as the Kidder and the Saltzman problem. When compared to the
traditional global time stepping (GTS) method, the newly proposed LTS algorithm
allows to reduce the number of element updates in a given simulation by a
factor that may depend on the complexity of the dynamics, but which can be as
large as 4.7.Comment: 31 pages, 13 figure
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