# On the convergence of spectral deferred correction methods

## TL;DR

This paper analyzes the convergence of spectral deferred correction (SDC) methods, revealing that the quadrature rule determines the order of accuracy, and establishes conditions for high-order convergence even with inconsistent solvers.

## Contribution

It provides a new convergence analysis of SDC methods, emphasizing the role of quadrature rules and error sources, and extends the applicability to a broader class of solvers.

## Key findings

- Quadrature rule determines the order of accuracy in SDC methods.
- High-order accuracy can be achieved with inconsistent solvers under certain conditions.
- Identifies three sources of errors in SDC methods: current error, previous error, and quadrature error.

## Abstract

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.