6,093 research outputs found
On the convergence of spectral deferred correction methods
In this work we analyze the convergence properties of the Spectral Deferred
Correction (SDC) method originally proposed by Dutt et al. [BIT, 40 (2000), pp.
241--266]. The framework for this high-order ordinary differential equation
(ODE) solver is typically described wherein a low-order approximation (such as
forward or backward Euler) is lifted to higher order accuracy by applying the
same low-order method to an error equation and then adding in the resulting
defect to correct the solution. Our focus is not on solving the error equation
to increase the order of accuracy, but on rewriting the solver as an iterative
Picard integral equation solver. In doing so, our chief finding is that it is
not the low-order solver that picks up the order of accuracy with each
correction, but it is the underlying quadrature rule of the right hand side
function that is solely responsible for picking up additional orders of
accuracy. Our proofs point to a total of three sources of errors that SDC
methods carry: the error at the current time point, the error from the previous
iterate, and the numerical integration error that comes from the total number
of quadrature nodes used for integration. The second of these two sources of
errors is what separates SDC methods from Picard integral equation methods; our
findings indicate that as long as difference between the current and previous
iterate always gets multiplied by at least a constant multiple of the time step
size, then high-order accuracy can be found even if the underlying "solver" is
inconsistent the underlying ODE. From this vantage, we solidify the prospects
of extending spectral deferred correction methods to a larger class of solvers
to which we present some examples.Comment: 29 page
Towards optimal explicit time-stepping schemes for the gyrokinetic equations
The nonlinear gyrokinetic equations describe plasma turbulence in laboratory
and astrophysical plasmas. To solve these equations, massively parallel codes
have been developed and run on present-day supercomputers. This paper describes
measures to improve the efficiency of such computations, thereby making them
more realistic. Explicit Runge-Kutta schemes are considered to be well suited
for time-stepping. Although the numerical algorithms are often highly
optimized, performance can still be improved by a suitable choice of the
time-stepping scheme, based on spectral analysis of the underlying operator.
Here, an operator splitting technique is introduced to combine first-order
Runge-Kutta-Chebychev schemes for the collision term with fourth-order schemes
for the remaining terms. In the nonlinear regime, based on the observation of
eigenvalue shifts due to the (generalized) advection term, an
accurate and robust estimate for the nonlinear timestep is developed. The
presented techniques can reduce simulation times by factors of up to three in
realistic cases. This substantial speedup encourages the use of similar
timestep optimized explicit schemes not only for the gyrokinetic equation, but
also for other applications with comparable properties.Comment: 11 pages, 5 figures, accepted for publication in Computer Physics
Communication
Extrapolation-Based Super-Convergent Implicit-Explicit Peer Methods with A-stable Implicit Part
In this paper, we extend the implicit-explicit (IMEX) methods of Peer type
recently developed in [Lang, Hundsdorfer, J. Comp. Phys., 337:203--215, 2017]
to a broader class of two-step methods that allow the construction of
super-convergent IMEX-Peer methods with A-stable implicit part. IMEX schemes
combine the necessary stability of implicit and low computational costs of
explicit methods to efficiently solve systems of ordinary differential
equations with both stiff and non-stiff parts included in the source term. To
construct super-convergent IMEX-Peer methods with favourable stability
properties, we derive necessary and sufficient conditions on the coefficient
matrices and apply an extrapolation approach based on already computed stage
values. Optimised super-convergent IMEX-Peer methods of order s+1 for s=2,3,4
stages are given as result of a search algorithm carefully designed to balance
the size of the stability regions and the extrapolation errors. Numerical
experiments and a comparison to other IMEX-Peer methods are included.Comment: 22 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1610.0051
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