386 research outputs found
Analysis of a space--time hybridizable discontinuous Galerkin method for the advection--diffusion problem on time-dependent domains
This paper presents the first analysis of a space--time hybridizable
discontinuous Galerkin method for the advection--diffusion problem on
time-dependent domains. The analysis is based on non-standard local trace and
inverse inequalities that are anisotropic in the spatial and time steps. We
prove well-posedness of the discrete problem and provide a priori error
estimates in a mesh-dependent norm. Convergence theory is validated by a
numerical example solving the advection--diffusion problem on a time-dependent
domain for approximations of various polynomial degree
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
A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows
The potential of the hybridized discontinuous Galerkin (HDG) method has been
recognized for the computation of stationary flows. Extending the method to
time-dependent problems can, e.g., be done by backward difference formulae
(BDF) or diagonally implicit Runge-Kutta (DIRK) methods. In this work, we
investigate the use of embedded DIRK methods in an HDG solver, including the
use of adaptive time-step control. Numerical results demonstrate the
performance of the method for both linear and nonlinear (systems of)
time-dependent convection-diffusion equations
An exactly mass conserving space-time embedded-hybridized discontinuous Galerkin method for the Navier-Stokes equations on moving domains
This paper presents a space-time embedded-hybridized discontinuous Galerkin
(EHDG) method for the Navier--Stokes equations on moving domains. This method
uses a different hybridization compared to the space-time hybridized
discontinuous Galerkin (HDG) method we presented previously in (Int. J. Numer.
Meth. Fluids 89: 519--532, 2019). In the space-time EHDG method the velocity
trace unknown is continuous while the pressure trace unknown is discontinuous
across facets. In the space-time HDG method, all trace unknowns are
discontinuous across facets. Alternatively, we present also a space-time
embedded discontinuous Galerkin (EDG) method in which all trace unknowns are
continuous across facets. The advantage of continuous trace unknowns is that
the formulation has fewer global degrees-of-freedom for a given mesh than when
using discontinuous trace unknowns. Nevertheless, the discrete velocity field
obtained by the space-time EHDG and EDG methods, like the space-time HDG
method, is exactly divergence-free, even on moving domains. However, only the
space-time EHDG and HDG methods result in divergence-conforming velocity
fields. An immediate consequence of this is that the space-time EHDG and HDG
discretizations of the conservative form of the Navier--Stokes equations are
energy stable. The space-time EDG method, on the other hand, requires a
skew-symmetric formulation of the momentum advection term to be energy-stable.
Numerical examples will demonstrate the differences in solution obtained by the
space-time EHDG, EDG, and HDG methods
A locally conservative and energy-stable finite element for the Navier--Stokes problem on time-dependent domains
We present a finite element method for the incompressible Navier--Stokes
problem that is locally conservative, energy-stable and pressure-robust on
time-dependent domains. To achieve this, the space--time formulation of the
Navier--Stokes problem is considered. The space--time domain is partitioned
into space--time slabs which in turn are partitioned into space--time
simplices. A combined discontinuous Galerkin method across space--time slabs,
and space--time hybridized discontinuous Galerkin method within a space--time
slab, results in an approximate velocity field that is -conforming and exactly divergence-free, even on time-dependent domains.
Numerical examples demonstrate the convergence properties and performance of
the method
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