308 research outputs found
An almost symmetric Strang splitting scheme for the construction of high order composition methods
In this paper we consider splitting methods for nonlinear ordinary
differential equations in which one of the (partial) flows that results from
the splitting procedure can not be computed exactly. Instead, we insert a
well-chosen state into the corresponding nonlinearity ,
which results in a linear term whose exact flow can be
determined efficiently. Therefore, in the spirit of splitting methods, it is
still possible for the numerical simulation to satisfy certain properties of
the exact flow. However, Strang splitting is no longer symmetric (even though
it is still a second order method) and thus high order composition methods are
not easily attainable. We will show that an iterated Strang splitting scheme
can be constructed which yields a method that is symmetric up to a given order.
This method can then be used to attain high order composition schemes. We will
illustrate our theoretical results, up to order six, by conducting numerical
experiments for a charged particle in an inhomogeneous electric field, a
post-Newtonian computation in celestial mechanics, and a nonlinear population
model and show that the methods constructed yield superior efficiency as
compared to Strang splitting. For the first example we also perform a
comparison with the standard fourth order Runge--Kutta methods and find
significant gains in efficiency as well better conservation properties
Exponential Integrators on Graphic Processing Units
In this paper we revisit stencil methods on GPUs in the context of
exponential integrators. We further discuss boundary conditions, in the same
context, and show that simple boundary conditions (for example, homogeneous
Dirichlet or homogeneous Neumann boundary conditions) do not affect the
performance if implemented directly into the CUDA kernel. In addition, we show
that stencil methods with position-dependent coefficients can be implemented
efficiently as well.
As an application, we discuss the implementation of exponential integrators
for different classes of problems in a single and multi GPU setup (up to 4
GPUs). We further show that for stencil based methods such parallelization can
be done very efficiently, while for some unstructured matrices the
parallelization to multiple GPUs is severely limited by the throughput of the
PCIe bus.Comment: To appear in: Proceedings of the 2013 International Conference on
High Performance Computing Simulation (HPCS 2013), IEEE (2013
On the error propagation of semi-Lagrange and Fourier methods for advection problems
In this paper we study the error propagation of numerical schemes for the
advection equation in the case where high precision is desired. The numerical
methods considered are based on the fast Fourier transform, polynomial
interpolation (semi-Lagrangian methods using a Lagrange or spline
interpolation), and a discontinuous Galerkin semi-Lagrangian approach (which is
conservative and has to store more than a single value per cell).
We demonstrate, by carrying out numerical experiments, that the worst case
error estimates given in the literature provide a good explanation for the
error propagation of the interpolation-based semi-Lagrangian methods. For the
discontinuous Galerkin semi-Lagrangian method, however, we find that the
characteristic property of semi-Lagrangian error estimates (namely the fact
that the error increases proportionally to the number of time steps) is not
observed. We provide an explanation for this behavior and conduct numerical
simulations that corroborate the different qualitative features of the error in
the two respective types of semi-Lagrangian methods.
The method based on the fast Fourier transform is exact but, due to round-off
errors, susceptible to a linear increase of the error in the number of time
steps. We show how to modify the Cooley--Tukey algorithm in order to obtain an
error growth that is proportional to the square root of the number of time
steps.
Finally, we show, for a simple model, that our conclusions hold true if the
advection solver is used as part of a splitting scheme.Comment: submitted to Computers & Mathematics with Application
Splitting methods for constrained diffusion-reaction systems
We consider Lie and Strang splitting for the time integration of constrained
partial differential equations with a nonlinear reaction term. Since such
systems are known to be sensitive with respect to perturbations, the splitting
procedure seems promising as we can treat the nonlinearity separately. This has
some computational advantages, since we only have to solve a linear constrained
system and a nonlinear ODE. However, Strang splitting suffers from order
reduction which limits its efficiency. This is caused by the fact that the
nonlinear subsystem produces inconsistent initial values for the constrained
subsystem. The incorporation of an additional correction term resolves this
problem without increasing the computational cost. Numerical examples including
a coupled mechanical system illustrate the proven convergence results
On the convergence of Lawson methods for semilinear stiff problems
Since their introduction in 1967, Lawson methods have achieved constant
interest in the time discretization of evolution equations. The methods were
originally devised for the numerical solution of stiff differential equations.
Meanwhile, they constitute a well-established class of exponential integrators.
The popularity of Lawson methods is in some contrast to the fact that they may
have a bad convergence behaviour, since they do not satisfy any of the stiff
order conditions. The aim of this paper is to explain this discrepancy. It is
shown that non-stiff order conditions together with appropriate regularity
assumptions imply high-order convergence of Lawson methods. Note, however, that
the term regularity here includes the behaviour of the solution at the
boundary. For instance, Lawson methods will behave well in the case of periodic
boundary conditions, but they will show a dramatic order reduction for, e.g.,
Dirichlet boundary conditions. The precise regularity assumptions required for
high-order convergence are worked out in this paper and related to the
corresponding assumptions for splitting schemes. In contrast to previous work,
the analysis is based on expansions of the exact and the numerical solution
along the flow of the homogeneous problem. Numerical examples for the
Schr\"odinger equation are included
A strategy to suppress recurrence in grid-based Vlasov solvers
In this paper we propose a strategy to suppress the recurrence effect present
in grid-based Vlasov solvers. This method is formulated by introducing a cutoff
frequency in Fourier space. Since this cutoff only has to be performed after a
number of time steps, the scheme can be implemented efficiently and can
relatively easily be incorporated into existing Vlasov solvers. Furthermore,
the scheme proposed retains the advantage of grid-based methods in that high
accuracy can be achieved. This is due to the fact that in contrast to the
scheme proposed by Abbasi et al. no statistical noise is introduced into the
simulation. We will illustrate the utility of the method proposed by performing
a number of numerical simulations, including the plasma echo phenomenon, using
a discontinuous Galerkin approximation in space and a Strang splitting based
time integration
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