In this work we analyze the convergence properties of the spectral deferred correction
(SDC) method originally proposed by Dutt et al. (BIT
40 (2000), no. 2, 241–266).
The framework for this high-order ordinary differential equation (ODE)
solver is typically described as a low-order approximation (such as forward or
backward Euler) 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 the difference between the current and previous iterates 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 ODE “solver” is inconsistent.
From this vantage, we solidify the prospects of extending spectral deferred
correction methods to a larger class of solvers, of which we present some
examples.
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