6,461 research outputs found
AMR, stability and higher accuracy
Efforts to achieve better accuracy in numerical relativity have so far
focused either on implementing second order accurate adaptive mesh refinement
or on defining higher order accurate differences and update schemes. Here, we
argue for the combination, that is a higher order accurate adaptive scheme.
This combines the power that adaptive gridding techniques provide to resolve
fine scales (in addition to a more efficient use of resources) together with
the higher accuracy furnished by higher order schemes when the solution is
adequately resolved. To define a convenient higher order adaptive mesh
refinement scheme, we discuss a few different modifications of the standard,
second order accurate approach of Berger and Oliger. Applying each of these
methods to a simple model problem, we find these options have unstable modes.
However, a novel approach to dealing with the grid boundaries introduced by the
adaptivity appears stable and quite promising for the use of high order
operators within an adaptive framework
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
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