70,063 research outputs found
Adaptive multiscale methods for 3D streamer discharges in air
We discuss spatially and temporally adaptive implicit-explicit (IMEX) methods
for parallel simulations of three-dimensional fluid streamer discharges in
atmospheric air. We examine strategies for advancing the fluid equations and
elliptic transport equations (e.g. Poisson) with different time steps,
synchronizing them on a global physical time scale which is taken to be
proportional to the dielectric relaxation time. The use of a longer time step
for the electric field leads to numerical errors that can be diagnosed, and we
quantify the conditions where this simplification is valid. Likewise, using a
three-term Helmholtz model for radiative transport, the same error diagnostics
show that the radiative transport equations do not need to be resolved on time
scales finer than the dielectric relaxation time. Elliptic equations are
bottlenecks for most streamer simulation codes, and the results presented here
potentially provide computational savings. Finally, a computational example of
3D branching streamers in a needle-plane geometry that uses up to 700 million
grid cells is presented.Comment: 17 pages, 5 figure
Adaptive computational methods for aerothermal heating analysis
The development of adaptive gridding techniques for finite-element analysis of fluid dynamics equations is described. The developmental work was done with the Euler equations with concentration on shock and inviscid flow field capturing. Ultimately this methodology is to be applied to a viscous analysis for the purpose of predicting accurate aerothermal loads on complex shapes subjected to high speed flow environments. The development of local error estimate strategies as a basis for refinement strategies is discussed, as well as the refinement strategies themselves. The application of the strategies to triangular elements and a finite-element flux-corrected-transport numerical scheme are presented. The implementation of these strategies in the GIM/PAGE code for 2-D and 3-D applications is documented and demonstrated
On Asymptotic Global Error Estimation and Control of Finite Difference Solutions for Semilinear Parabolic Equations
The aim of this paper is to extend the global error estimation and control
addressed in Lang and Verwer [SIAM J. Sci. Comput. 29, 2007] for initial value
problems to finite difference solutions of semilinear parabolic partial
differential equations. The approach presented there is combined with an
estimation of the PDE spatial truncation error by Richardson extrapolation to
estimate the overall error in the computed solution. Approximations of the
error transport equations for spatial and temporal global errors are derived by
using asymptotic estimates that neglect higher order error terms for
sufficiently small step sizes in space and time. Asymptotic control in a
discrete -norm is achieved through tolerance proportionality and uniform
or adaptive mesh refinement. Numerical examples are used to illustrate the
reliability of the estimation and control strategies
Efficient Resolution of Anisotropic Structures
We highlight some recent new delevelopments concerning the sparse
representation of possibly high-dimensional functions exhibiting strong
anisotropic features and low regularity in isotropic Sobolev or Besov scales.
Specifically, we focus on the solution of transport equations which exhibit
propagation of singularities where, additionally, high-dimensionality enters
when the convection field, and hence the solutions, depend on parameters
varying over some compact set. Important constituents of our approach are
directionally adaptive discretization concepts motivated by compactly supported
shearlet systems, and well-conditioned stable variational formulations that
support trial spaces with anisotropic refinements with arbitrary
directionalities. We prove that they provide tight error-residual relations
which are used to contrive rigorously founded adaptive refinement schemes which
converge in . Moreover, in the context of parameter dependent problems we
discuss two approaches serving different purposes and working under different
regularity assumptions. For frequent query problems, making essential use of
the novel well-conditioned variational formulations, a new Reduced Basis Method
is outlined which exhibits a certain rate-optimal performance for indefinite,
unsymmetric or singularly perturbed problems. For the radiative transfer
problem with scattering a sparse tensor method is presented which mitigates or
even overcomes the curse of dimensionality under suitable (so far still
isotropic) regularity assumptions. Numerical examples for both methods
illustrate the theoretical findings
Optimal predictive control of water transport systems: Arrêt-Darré/Arros case study
This paper proposes the use of predictive optimal control as a suitable methodology to manage efficiently transport water networks. The predictive optimal controller is implemented using MPC control techniques. The Arrêt-Darré/Arros dam-river system located in the Southwest region of France is proposed as case study. A high-fidelity dynamic simulator based on the full Saint-Venant equations and able to reproduce this system is developed in MATLAB/SIMULINK to validate the performance of the developed predictive optimal control system. The control objective in the Arrêt-Darré/Arros dam-river system is to guarantee an ecological flow rate at a control point downstream of the Arrêt-Darré dam by controlling the outflow of this dam in spite of the unmeasured disturbances introduced by rainfalls incomings and farmer withdrawals
Divergence-Free Adaptive Mesh Refinement for Magnetohydrodynamics
In this paper we present a full-fledged scheme for the second order accurate,
divergence-free evolution of vector fields on an adaptive mesh refinement (AMR)
hierarchy. We focus here on adaptive mesh MHD. The scheme is based on making a
significant advance in the divergence-free reconstruction of vector fields. In
that sense, it complements the earlier work of Balsara and Spicer (1999) where
we discussed the divergence-free time-update of vector fields which satisfy
Stoke's law type evolution equations. Our advance in divergence-free
reconstruction of vector fields is such that it reduces to the total variation
diminishing (TVD) property for one-dimensional evolution and yet goes beyond it
in multiple dimensions. Divergence-free restriction is also discussed. An
electric field correction strategy is presented for use on AMR meshes. The
electric field correction strategy helps preserve the divergence-free evolution
of the magnetic field even when the time steps are sub-cycled on refined
meshes. The above-mentioned innovations have been implemented in Balsara's
RIEMANN framework for parallel, self-adaptive computational astrophysics which
supports both non-relativistic and relativistic MHD. Several rigorous, three
dimensional AMR-MHD test problems with strong discontinuities have been run
with the RIEMANN framework showing that the strategy works very well.Comment: J.C.P., figures of reduced qualit
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