15,569 research outputs found
Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: explicit time-stepping and efficient mass matrix inversion
We present a class of spline finite element methods for time-domain wave
propagation which are particularly amenable to explicit time-stepping. The
proposed methods utilize a discontinuous Galerkin discretization to enforce
continuity of the solution field across geometric patches in a multi-patch
setting, which yields a mass matrix with convenient block diagonal structure.
Over each patch, we show how to accurately and efficiently invert mass matrices
in the presence of curved geometries by using a weight-adjusted approximation
of the mass matrix inverse. This approximation restores a tensor product
structure while retaining provable high order accuracy and semi-discrete energy
stability. We also estimate the maximum stable timestep for spline-based finite
elements and show that the use of spline spaces result in less stringent CFL
restrictions than equivalent piecewise continuous or discontinuous finite
element spaces. Finally, we explore the use of optimal knot vectors based on L2
n-widths. We show how the use of optimal knot vectors can improve both
approximation properties and the maximum stable timestep, and present a simple
heuristic method for approximating optimal knot positions. Numerical
experiments confirm the accuracy and stability of the proposed methods
von Neumann Stability Analysis of Globally Constraint-Preserving DGTD and PNPM Schemes for the Maxwell Equations using Multidimensional Riemann Solvers
The time-dependent equations of computational electrodynamics (CED) are
evolved consistent with the divergence constraints. As a result, there has been
a recent effort to design finite volume time domain (FVTD) and discontinuous
Galerkin time domain (DGTD) schemes that satisfy the same constraints and,
nevertheless, draw on recent advances in higher order Godunov methods. This
paper catalogues the first step in the design of globally constraint-preserving
DGTD schemes. The algorithms presented here are based on a novel DG-like method
that is applied to a Yee-type staggering of the electromagnetic field variables
in the faces of the mesh. The other two novel building blocks of the method
include constraint-preserving reconstruction of the electromagnetic fields and
multidimensional Riemann solvers; both of which have been developed in recent
years by the first author. We carry out a von Neumann stability analysis of the
entire suite of DGTD schemes for CED at orders of accuracy ranging from second
to fourth. A von Neumann stability analysis gives us the maximal CFL numbers
that can be sustained by the DGTD schemes presented here at all orders. It also
enables us to understand the wave propagation characteristics of the schemes in
various directions on a Cartesian mesh. We find that the CFL of DGTD schemes
decreases with increasing order. To counteract that, we also present
constraint-preserving PNPM schemes for CED. We find that the third and fourth
order constraint-preserving DGTD and P1PM schemes have some extremely
attractive properties when it comes to low-dispersion, low-dissipation
propagation of electromagnetic waves in multidimensions. Numerical accuracy
tests are also provided to support the von Neumann stability analysis
On discretely entropy conservative and entropy stable discontinuous Galerkin methods
High order methods based on diagonal-norm summation by parts operators can be
shown to satisfy a discrete conservation or dissipation of entropy for
nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as
nodal discontinuous Galerkin methods with diagonal mass matrices. In this work,
we describe how use flux differencing, quadrature-based projections, and
SBP-like operators to construct discretely entropy conservative schemes for DG
methods under more arbitrary choices of volume and surface quadrature rules.
The resulting methods are semi-discretely entropy conservative or entropy
stable with respect to the volume quadrature rule used. Numerical experiments
confirm the stability and high order accuracy of the proposed methods for the
compressible Euler equations in one and two dimensions
An unconditionally stable algorithm for generalized thermoelasticity based on operator-splitting and time-discontinuous Galerkin finite element methods
An efficient time-stepping algorithm is proposed based on operator-splitting and the space–time discontinuous Galerkin finite element method for problems in the non-classical theory of thermoelasticity. The non-classical theory incorporates three models: the classical theory based on Fourier’s law of heat conduction resulting in a hyperbolic–parabolic coupled system, a non-classical theory of a fully-hyperbolic extension, and a combination of the two. The general problem is split into two contractive sub-problems, namely the mechanical phase and the thermal phase. Each sub-problem is discretized using the space–time discontinuous Galerkin finite element method. The sub-problems are stable which then leads to unconditional stability of the global product algorithm. A number of numerical examples are presented to demonstrate the performance and capability of the method
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