5 research outputs found
A high order semi-Lagrangian discontinuous Galerkin method for the two-dimensional incompressible Euler equations and the guiding center Vlasov model without operator splitting
In this paper, we generalize a high order semi-Lagrangian (SL) discontinuous
Galerkin (DG) method for multi-dimensional linear transport equations without
operator splitting developed in Cai et al. (J. Sci. Comput. 73: 514-542, 2017)
to the 2D time dependent incompressible Euler equations in the vorticity-stream
function formulation and the guiding center Vlasov model. We adopt a local DG
method for Poisson's equation of these models. For tracing the characteristics,
we adopt a high order characteristics tracing mechanism based on a
prediction-correction technique. The SLDG with large time-stepping size might
be subject to extreme distortion of upstream cells. To avoid this problem, we
propose a novel adaptive time-stepping strategy by controlling the relative
deviation of areas of upstream cells.Comment: arXiv admin note: text overlap with arXiv:1709.0253
High Order Semi-Lagrangian Discontinuous Galerkin Method Coupled with Runge-Kutta Exponential Integrators for Nonlinear Vlasov Dynamics
In this paper, we propose a semi-Lagrangian discontinuous Galerkin method
coupled with Runge-Kutta exponential integrators (SLDG-RKEI) for nonlinear
Vlasov dynamics. The commutator-free Runge-Kutta (RK) exponential integrators
(EI) were proposed by Celledoni, et al. (FGCS, 2003). In the nonlinear
transport setting, the RKEI can be used to decompose the evolution of the
nonlinear transport into a composition of a sequence of linearized dynamics.
The resulting linearized transport equations can be solved by the
semi-Lagrangian (SL) discontinuous Galerkin (DG) method proposed in Cai, et al.
(JSC, 2017). The proposed method can achieve high order spatial accuracy via
the SLDG framework, and high order temporal accuracy via the RK EI. Due to the
SL nature, the proposed SLDG-RKEI method is not subject to the CFL condition,
thus they have the potential in using larger time-stepping sizes than those in
the Eulerian approach. Inheriting advantages from the SLDG method, the proposed
SLDG-RKEI schemes are mass conservative, positivity-preserving, have no
dimensional splitting error, perform well in resolving complex solution
structures, and can be evolved with adaptive time-stepping sizes. We show the
performance of the SLDG-RKEI algorithm by classical test problems for the
nonlinear Vlasov-Poisson system, as well as the Guiding center Vlasov model.
Though that it is not our focus of this paper to explore the SLDG-RKEI scheme
for nonlinear hyperbolic conservation laws that develop shocks, we show some
preliminary results on schemes' performance on the Burgers' equation
High-order Solution Transfer between Curved Triangular Meshes
The problem of solution transfer between meshes arises frequently in
computational physics, e.g. in Lagrangian methods where remeshing occurs. The
interpolation process must be conservative, i.e. it must conserve physical
properties, such as mass. We extend previous works --- which described the
solution transfer process for straight sided unstructured meshes --- by
considering high-order isoparametric meshes with curved elements. To facilitate
solution transfer, we numerically integrate the product of shape functions via
Green's theorem along the boundary of the intersection of two curved elements.
We perform a numerical experiment and confirm the expected accuracy by
transferring test fields across two families of meshes
Some continuous and discontinuous Galerkin methods and structure preservation for incompressible flows
In this paper, we present consistent and inconsistent discontinuous Galerkin
methods for incompressible Euler and Navier-Stokes equations with the kinematic
pressure, Bernoulli function and EMAC function. Semi- and fully discrete energy
stability of the proposed dG methods are proved in a unified fashion.
Conservation of total energy, linear and angular momentum is discussed with
both central and upwind fluxes. Numerical experiments are presented to
demonstrate our findings and compare our schemes with conventional schemes in
the literature in both unsteady and steady problems. Numerical results show
that global conservation of the physical quantities may not be enough to
demonstrate the performance of the schemes, and our schemes are competitive and
able to capture essential physical features in several benchmark problems.Comment: 32 pages, 24 figure
A unified framework of continuous and discontinuous Galerkin methods for solving the incompressible Navier--Stokes equation
In this paper, we propose a unified numerical framework for the
time-dependent incompressible Navier--Stokes equation which yields the -,
-conforming, and discontinuous Galerkin methods with the use of
different viscous stress tensors and penalty terms for pressure robustness.
Under minimum assumption on Galerkin spaces, the semi- and fully-discrete
stability is proved when a family of implicit Runge--Kutta methods are used for
time discretization. Furthermore, we present a unified discussion on the
penalty term. Numerical experiments are presented to compare our schemes with
classical schemes in the literature in both unsteady and steady situations. It
turns out that our scheme is competitive when applied to well-known benchmark
problems such as Taylor--Green vortex, Kovasznay flow, potential flow, lid
driven cavity flow, and the flow around a cylinder.Comment: 33 pages, 28 figure