2,192 research outputs found
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
Stability of explicit one-step methods for P1-finite element approximation of linear diffusion equations on anisotropic meshes
We study the stability of explicit one-step integration schemes for the
linear finite element approximation of linear parabolic equations. The derived
bound on the largest permissible time step is tight for any mesh and any
diffusion matrix within a factor of , where is the spatial
dimension. Both full mass matrix and mass lumping are considered. The bound
reveals that the stability condition is affected by two factors. The first one
depends on the number of mesh elements and corresponds to the classic bound for
the Laplace operator on a uniform mesh. The other factor reflects the effects
of the interplay of the mesh geometry and the diffusion matrix. It is shown
that it is not the mesh geometry itself but the mesh geometry in relation to
the diffusion matrix that is crucial to the stability of explicit methods. When
the mesh is uniform in the metric specified by the inverse of the diffusion
matrix, the stability condition is comparable to the situation with the Laplace
operator on a uniform mesh. Numerical results are presented to verify the
theoretical findings.Comment: Revised WIAS Preprin
Dispersive properties of high order nedelec/edge element approximation of the time-harmonic Maxwell equations
The dispersive behaviour of high-order Næ#169;dæ#169;lec element approximation of the time harmonic Maxwell equations at a prescribed temporal frequency ω on tensor-product meshes of size h is analysed. A simple argument is presented, showing that the discrete dispersion relation may be expressed in terms of that for the approximation of the scalar Helmholtz equation in one dimension. An explicit form for the one-dimensional dispersion relation is given, valid for arbitrary order of approximation. Explicit expressions for the leading term in the error in the regimes where ωh is small, showing that the dispersion relation is accurate to order 2p for a pth-order method; and in the high-wavenumber limit where 1«ωh, showing that in this case the error reduces at a super-exponential rate once the order of approximation exceeds a certain threshold, which is given explicitly
Numerical wave propagation for the triangular - finite element pair
Inertia-gravity mode and Rossby mode dispersion properties are examined for
discretisations of the linearized rotating shallow-water equations using the
- finite element pair on arbitrary triangulations in planar
geometry. A discrete Helmholtz decomposition of the functions in the velocity
space based on potentials taken from the pressure space is used to provide a
complete description of the numerical wave propagation for the discretised
equations. In the -plane case, this decomposition is used to obtain
decoupled equations for the geostrophic modes, the inertia-gravity modes, and
the inertial oscillations. As has been noticed previously, the geostrophic
modes are steady. The Helmholtz decomposition is used to show that the
resulting inertia-gravity wave equation is third-order accurate in space. In
general the \pdgp finite element pair is second-order accurate, so this leads
to very accurate wave propagation. It is further shown that the only spurious
modes supported by this discretisation are spurious inertial oscillations which
have frequency , and which do not propagate. The Helmholtz decomposition
also allows a simple derivation of the quasi-geostrophic limit of the
discretised - equations in the -plane case, resulting in a
Rossby wave equation which is also third-order accurate.Comment: Revised version prior to final journal submissio
Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization
In this work, we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping. The method relies on an artificial diffusion method, based on a graph-Laplacian operator. It is nonlinear, since it depends on a shock detector. Further, the resulting method is linearity preserving. The same shock detector is used to gradually lump the mass matrix. The resulting method is LED, positivity preserving, and also satisfies a global DMP. Lipschitz continuity has also been proved. However, the resulting scheme is highly nonlinear, leading to very poor nonlinear convergence rates. We propose a smooth version of the scheme, which leads to twice differentiable nonlinear stabilization schemes. It allows one to straightforwardly use Newton’s method and obtain quadratic convergence. In the numerical experiments, steady and transient linear transport, and transient Burgers’ equation have been considered in 2D. Using the Newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of Anderson acceleration with the original non-smooth scheme. In any case, these properties are only true for the converged solution, but not for iterates. In this sense, we have also proposed the concept of projected nonlinear solvers, where a projection step is performed at the end of every nonlinear iterations onto a FE space of admissible solutions. The space of admissible solutions is the one that satisfies the desired monotonic properties (maximum principle or positivity).Peer ReviewedPostprint (author's final draft
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