Spline Deferred Correction Methods for Ordinary Differential Equations
We introduce a new variant of Picard integral deferred correction, a class of methods aimed at solving initial value problems with arbitrarily high order of accuracy, called spline deferred correction (SplineDC). Similar to spectral deferred correction (SpectralDC), SplineDC applies a series of correction sweeps, derived from the Picard integral form, that are driven by either first-order explicit or implicit Euler integration schemes to produce a highly accurate numerical solution. However, as the name implies, SplineDC makes use of spline interpolation as opposed to Lagrange interpolation in the case of SpectralDC. This idea is motivated by our interest in circumventing the inherent restrictions of SpectralDC methods which are enforced to avoid the instabilities associated with high-order and/or equispaced polynomial interpolation, i.e. Runge's phenomenon. In doing so, we develop a class of methods with favorable convergence behavior and excellent stability properties, all of which we demonstrate through various numerical experiments. Additionally, because of the freedom SplineDC has with respect to time step selection, we demonstrate the advantages of implementing adaptive time-stepping in SplineDC over SpectralDC. Finally, we present an alternate method of spline construction that facilitates the ease by which one can construct splines with arbitrary continuity and interpolation conditions.