379,079 research outputs found
Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods
In this paper, we develop a new tensor-product based preconditioner for
discontinuous Galerkin methods with polynomial degrees higher than those
typically employed. This preconditioner uses an automatic, purely algebraic
method to approximate the exact block Jacobi preconditioner by Kronecker
products of several small, one-dimensional matrices. Traditional matrix-based
preconditioners require storage and
computational work, where is the degree of basis polynomials used, and
is the spatial dimension. Our SVD-based tensor-product preconditioner requires
storage, work in two spatial
dimensions, and work in three spatial dimensions.
Combined with a matrix-free Newton-Krylov solver, these preconditioners allow
for the solution of DG systems in linear time in per degree of freedom in
2D, and reduce the computational complexity from to
in 3D. Numerical results are shown in 2D and 3D for the
advection and Euler equations, using polynomials of degree up to . For
many test cases, the preconditioner results in similar iteration counts when
compared with the exact block Jacobi preconditioner, and performance is
significantly improved for high polynomial degrees .Comment: 40 pages, 15 figure
Tensor Network Space-Time Spectral Collocation Method for Time Dependent Convection-Diffusion-Reaction Equations
Emerging tensor network techniques for solutions of Partial Differential
Equations (PDEs), known for their ability to break the curse of dimensionality,
deliver new mathematical methods for ultrafast numerical solutions of
high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev
spectral collocation method, in both space and time, for solution of the time
dependent convection-diffusion-reaction (CDR) equation with inhomogeneous
boundary conditions, in Cartesian geometry. Previous methods for numerical
solution of time dependent PDEs often use finite difference for time, and a
spectral scheme for the spatial dimensions, which leads to slow linear
convergence. Spectral collocation space-time methods show exponential
convergence, however, for realistic problems they need to solve large
four-dimensional systems. We overcome this difficulty by using a TT approach as
its complexity only grows linearly with the number of dimensions. We show that
our TT space-time Chebyshev spectral collocation method converges
exponentially, when the solution of the CDR is smooth, and demonstrate that it
leads to very high compression of linear operators from terabytes to kilobytes
in TT-format, and tens of thousands times speedup when compared to full grid
space-time spectral method. These advantages allow us to obtain the solutions
at much higher resolutions
Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations
Computational costs of numerically solving multidimensional partial
differential equations (PDEs) increase significantly when the spatial
dimensions of the PDEs are high, due to large number of spatial grid points.
For multidimensional reaction-diffusion equations, stiffness of the system
provides additional challenges for achieving efficient numerical simulations.
In this paper, we propose a class of Krylov implicit integration factor (IIF)
discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion
equations on high spatial dimensions. The key ingredient of spatial DG
discretization is the multiwavelet bases on nested sparse grids, which can
significantly reduce the numbers of degrees of freedom. To deal with the
stiffness of the DG spatial operator in discretizing reaction-diffusion
equations, we apply the efficient IIF time discretization methods, which are a
class of exponential integrators. Krylov subspace approximations are used to
evaluate the large size matrix exponentials resulting from IIF schemes for
solving PDEs on high spatial dimensions. Stability and error analysis for the
semi-discrete scheme are performed. Numerical examples of both scalar equations
and systems in two and three spatial dimensions are provided to demonstrate the
accuracy and efficiency of the methods. The stiffness of the reaction-diffusion
equations is resolved well and large time step size computations are obtained
Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution
We present a new solver for nonlinear parabolic problems that is L-stable and
achieves high order accuracy in space and time. The solver is built by first
constructing a single-dimensional heat equation solver that uses fast O(N)
convolution. This fundamental solver has arbitrary order of accuracy in space,
and is based on the use of the Green's function to invert a modified Helmholtz
equation. Higher orders of accuracy in time are then constructed through a
novel technique known as successive convolution (or resolvent expansions).
These resolvent expansions facilitate our proofs of stability and convergence,
and permit us to construct schemes that have provable stiff decay. The
multi-dimensional solver is built by repeated application of dimensionally
split independent fundamental solvers. Finally, we solve nonlinear parabolic
problems by using the integrating factor method, where we apply the basic
scheme to invert linear terms (that look like a heat equation), and make use of
Hermite-Birkhoff interpolants to integrate the remaining nonlinear terms. Our
solver is applied to several linear and nonlinear equations including heat,
Allen-Cahn, and the Fitzhugh-Nagumo system of equations in one and two
dimensions
Nonlinear competition between asters and stripes in filament-motor-systems
A model for polar filaments interacting via molecular motor complexes is
investigated which exhibits bifurcations to spatial patterns. It is shown that
the homogeneous distribution of filaments, such as actin or microtubules, may
become either unstable with respect to an orientational instability of a finite
wave number or with respect to modulations of the filament density, where long
wavelength modes are amplified as well. Above threshold nonlinear interactions
select either stripe patterns or periodic asters. The existence and stability
ranges of each pattern close to threshold are predicted in terms of a weakly
nonlinear perturbation analysis, which is confirmed by numerical simulations of
the basic model equations. The two relevant parameters determining the
bifurcation scenario of the model can be related to the concentrations of the
active molecular motors and of the filaments respectively, which both could be
easily regulated by the cell.Comment: 13 pages, 7 figure
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