223 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
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
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