1,287 research outputs found
High-order implicit palindromic discontinuous Galerkin method for kinetic-relaxation approximation
We construct a high order discontinuous Galerkin method for solving general
hyperbolic systems of conservation laws. The method is CFL-less, matrix-free,
has the complexity of an explicit scheme and can be of arbitrary order in space
and time. The construction is based on: (a) the representation of the system of
conservation laws by a kinetic vectorial representation with a stiff relaxation
term; (b) a matrix-free, CFL-less implicit discontinuous Galerkin transport
solver; and (c) a stiffly accurate composition method for time integration. The
method is validated on several one-dimensional test cases. It is then applied
on two-dimensional and three-dimensional test cases: flow past a cylinder,
magnetohydrodynamics and multifluid sedimentation
Adaptive discontinuous Galerkin approximations to fourth order parabolic problems
An adaptive algorithm, based on residual type a posteriori indicators of
errors measured in and norms, for a numerical
scheme consisting of implicit Euler method in time and discontinuous Galerkin
method in space for linear parabolic fourth order problems is presented. The a
posteriori analysis is performed for convex domains in two and three space
dimensions for local spatial polynomial degrees . The a posteriori
estimates are then used within an adaptive algorithm, highlighting their
relevance in practical computations, which results into substantial reduction
of computational effort
Explicit local time-stepping methods for time-dependent wave propagation
Semi-discrete Galerkin formulations of transient wave equations, either with
conforming or discontinuous Galerkin finite element discretizations, typically
lead to large systems of ordinary differential equations. When explicit time
integration is used, the time-step is constrained by the smallest elements in
the mesh for numerical stability, possibly a high price to pay. To overcome
that overly restrictive stability constraint on the time-step, yet without
resorting to implicit methods, explicit local time-stepping schemes (LTS) are
presented here for transient wave equations either with or without damping. In
the undamped case, leap-frog based LTS methods lead to high-order explicit LTS
schemes, which conserve the energy. In the damped case, when energy is no
longer conserved, Adams-Bashforth based LTS methods also lead to explicit LTS
schemes of arbitrarily high accuracy. When combined with a finite element
discretization in space with an essentially diagonal mass matrix, the resulting
time-marching schemes are fully explicit and thus inherently parallel.
Numerical experiments with continuous and discontinuous Galerkin finite element
discretizations validate the theory and illustrate the usefulness of these
local time-stepping methods.Comment: overview paper, typos added, references updated. arXiv admin note:
substantial text overlap with arXiv:1109.448
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