In the recent paper by Dutt, Greengard and Rokhlin, a variant of deferred or defect correction methods is presented which couples Gaussian quadrature with the Picard integral equation formulation of the initial value ordinary differential equation. The resulting spectral deferred correction methods (SDC) have been shown to possess favorable accuracy and stability properties even for versions with very high order of accuracy. In this paper, we show that for linear problems, the iterations in the SDC algorithm are equivalent to constructing a preconditioned Neumann series expansion for the solution of the standard collocation discretization of the ODE. This observation is used to accelerate the convergence of SDC using the GMRES Krylov subspace method. For nonlinear problems, the GMRES acceleration is coupled with a linear implicit approach. Stability and accuracy analyses show the accelerated scheme provides an improvement in the accuracy, efficiency, and stability of the original SDC approach. Furthermore, preliminary numerical experiments show that accelerating the convergence of SDC methods can effectively eliminate the order reduction previously observed for stiff ODE systems. Key words: spectral deferred correction methods, stiff equations, Krylov subspace methods, GMRES.

High-order semi-implicit Picard integral deferred correction (SIPIDC) methods have previously been proposed for the time-integration of partial differential equations with two or more disparate time scales. The SIPIDC methods studied to date compute a high-order approximation by first computing a provisional solution with a first-order semi-implicit method and then using a similar semi-implicit method to solve a series of correction equations, each of which raises the order of accuracy of the solution by one. This study assesses the efficiency of SIPIDC methods that instead use standard semi-implicit methods with orders two through four to compute the provisional solution. Numerical results indicate that using a method with more than first-order accuracy in the computation of the provisional solution increases the efficiency of SIPIDC methods in some cases. First-order PIDC corrections can improve the efficiency of semi-implicit integration methods based on backward difference formulae (BDF) or Runge–Kutta methods while maintaining desirable stability properties. Finally, the phenomenon of order reduction, which may be encountered in the integration of stiff problems, can be partially alleviated by the use of BDF methods in the computation of the provisional solution.

In this paper, a new framework for the construction of accurate and efficient numerical methods for differential algebraic equation (DAE) initial value problems is presented. The methods are based on applying spectral deferred correction techniques as preconditioners to a Picard integral collocation formulation for the solution. The resulting preconditioned nonlinear system is solved using Newton-Krylov schemes such as the Newton-GMRES method. Least squares based orthogonal polynomial approximations are computed using Gaussian type quadratures, and spectral integration is used to avoid the numerically unstable differentiation operator. The resulting Krylov deferred correction (KDC) methods are of arbitrary order of accuracy and very stable. Preliminary results show that these new methods are very competitive with existing DAE solvers, particularly when high precision is desired. Key words: spectral deferred correction methods, differential algebraic equations, spectral integration, preconditioners, Krylov subspace methods, GMRES.

Abstract: In the recently developed Krylov deferred correction (KDC) methods for ordinary differential equation initial value problems [11], a Picard-type collocation formulation is preconditioned using low-order time integration schemes based on spectral deferred correction (SDC), and the resulting system is solved efficiently using a Newton-Krylov method. Existing analyses show that these KDC methods are super convergent, A-stable, B-stable, symplectic, and symmetric. In this paper, we investigate the efficiency of semi-implicit KDC (SI-KDC) methods for problems which can be decomposed into stiff and non-stiff components. Preliminary analysis and numerical results show that SI-KDC methods display very similar convergence of Newton-Krylov iterations compared with fully-implicit (FI-KDC) methods but can significantly reduce the computational cost in each SDC iteration for the same accuracy requirement for certain problems.

A fourth-order numerical method for the zero-Mach-number limit of the equations for compressible flow is presented. The method is formed by discretizing a new auxiliary variable formulation of the conservation equations, which is a variable density analog to the impulse or gauge formulation of the incompressible Euler equations. An auxiliary variable projection method is applied to this formulation, and accuracy is achieved by combining a fourth-order finite-volume spatial discretization with a fourth-order temporal scheme based on spectral deferred corrections. Numerical results are included which demonstrate fourth-order spatial and temporal accuracy for non-trivial flows in simple geometries.

The method of regularized Stokeslets is a numerical approach to approximating solutions of fluid-structure interaction problems in the Stokes regime. Regularized Stokeslets are fundamental solutions to the Stokes equations with a regularized point-force term that are used to represent forces generated by rigid or elastic object interacting with the fluid. Due to the linearity of the Stokes equations, the velocity at any point in the fluid can be computed by summing the contributions of regularized Stokeslets, and the time evolution of positions can be computed using standard methods for ordinary differential equations. Rigid or elastic objects in the flow are usually treated as immersed boundaries represented by a collection of regularized Stokeslets coupled together by virtual springs which determine the forces exerted by the boundary in the fluid. For problems with boundaries modeled by springs with large spring constants, the resulting ordinary differential equations become stiff, and hence the time step for explicit time integration methods is severely constrained. Unfortunately, the

The parareal algorithm introduced in 2001 by Lions, Maday, and Turinici is an iterative method for the parallelization of the numerical solution of ODEs or discretized PDEs in the temporal direction. The temporal interval of interest is partitioned into successive domains which are assigned to separate processor units. Each iteration of the parareal algorithm consists of a high accuracy solution procedure performed in parallel on each domain using approximate initial conditions and a serial step which propagates a correction to the initial conditions through the entire time interval. The original method is designed to use classical single-step numerical methods for both of these steps. This paper investigates a variant of the parareal algorithm first outlined by Minion and Williams in 2008 that utilizes a deferred correction strategy within the parareal iterations. Here, the connections between parareal, parallel deferred corrections, and a hybrid parareal-spectral deferred correction method are further explored. The parallel speedup and efficiency of the hybrid methods are analyzed, and numerical results for ODEs and discretized PDEs are presented to demonstrate the performance of the hybrid approach.

We introduce a solver for stiff ordinary differential equations (ODEs) that is based on the deferred correction scheme for the corresponding Picard integral equation. Our solver relies on the assumption that the solution can be accurately represented by a combination of carefully selected complex exponentials. The solver’s accuracy and stability rely on the computation of highly accurate quadrature weights for the integration of the selected exponentials on equidistant nodes. We analyze our solver stability and accuracy regions, and demonstrate its fast convergence on stiff problems. The solver is combined with an adaptive step-size scheme employing interpolation formulas for the exponentially fitted solution.

Abstract. In the recently developed Krylov deferred correction (KDC) methods for differential algebraic equation initial value problems [31], a Picard-type collocation formulation is preconditioned using low-order time integration schemes based on spectral deferred correction (SDC), and the resulting system is solved efficiently using Newton-Krylov methods. In this paper, we further improve the efficiency of these KDC methods by introducing the semi-implicit KDC (SI-KDC) methods, in which the stiff component of the preconditioner is solved by implicit schemes and the non-stiff parts by explicit methods. Compared with fully implicit KDC (FI-KDC) methods, preliminary analyses show that the convergence of Newton-Krylov iterations in the SI-KDC methods is similar to that in FI-KDC, while for systems with a nonlinear non-stiff component and a linear stiff part, the SI-KDC can greatly reduce the computational cost in each spectral deferred correction iteration for the same accuracy requirement, as only linear solves are required in each SI-KDC iteration. The analyses are validated by preliminary numerical results.