165 research outputs found
Multirate explicit Adams methods for time integration of conservation laws
This paper constructs multirate linear multistep time discretizations based on Adams-Bashforth methods. These methods are aimed at solving conservation laws and allow different timesteps to be used in different parts of the spatial domain. The proposed family of discretizations is second order accurate in time and has conservation and linear and nonlinear stability properties under local CFL conditions. Multirate timestepping avoids the necessity to take small global timesteps - restricted by the largest value of the Courant number on the grid - and therefore results in more efficient computations. Numerical results obtained for the advection and Burgers' equations confirm the theoretical findings
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
Convergence analysis of energy conserving explicit local time-stepping methods for the wave equation
Local adaptivity and mesh refinement are key to the efficient simulation of
wave phenomena in heterogeneous media or complex geometry. Locally refined
meshes, however, dictate a small time-step everywhere with a crippling effect
on any explicit time-marching method. In [18] a leap-frog (LF) based explicit
local time-stepping (LTS) method was proposed, which overcomes the severe
bottleneck due to a few small elements by taking small time-steps in the
locally refined region and larger steps elsewhere. Here a rigorous convergence
proof is presented for the fully-discrete LTS-LF method when combined with a
standard conforming finite element method (FEM) in space. Numerical results
further illustrate the usefulness of the LTS-LF Galerkin FEM in the presence of
corner singularities
A high-order, conservative integrator with local time-stepping
We present a family of multistep integrators based on the Adams-Bashforth
methods. These schemes can be constructed for arbitrary convergence order with
arbitrary step size variation. The step size can differ between different
subdomains of the system. It can also change with time within a given
subdomain. The methods are linearly conservative, preserving a wide class of
analytically constant quantities to numerical roundoff, even when numerical
truncation error is significantly higher. These methods are intended for use in
solving conservative PDEs in discontinuous Galerkin formulations or in
finite-difference methods with compact stencils. A numerical test demonstrates
these properties and shows that significant speed improvements over the
standard Adams-Bashforth schemes can be obtained.Comment: 29 page
Discontinuous Galerkin Discretizations of the Boltzmann Equations in 2D: semi-analytic time stepping and absorbing boundary layers
We present an efficient nodal discontinuous Galerkin method for approximating
nearly incompressible flows using the Boltzmann equations. The equations are
discretized with Hermite polynomials in velocity space yielding a first order
conservation law. A stabilized unsplit perfectly matching layer (PML)
formulation is introduced for the resulting nonlinear flow equations. The
proposed PML equations exponentially absorb the difference between the
nonlinear fluctuation and the prescribed mean flow. We introduce semi-analytic
time discretization methods to improve the time step restrictions in small
relaxation times. We also introduce a multirate semi-analytic Adams-Bashforth
method which preserves efficiency in stiff regimes. Accuracy and performance of
the method are tested using distinct cases including isothermal vortex, flow
around square cylinder, and wall mounted square cylinder test cases.Comment: 37 pages, 11 figure
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