Deferred Correction Method Research Articles

Error estimates for deferred correction methods in time

We consider the deferred correction principle for high order accurate time discretization of ordinary differential equations (ODEs) and partial differential equations (PDEs) using the method of lines. We derive error estimates for the deferred correction method based on the implicit midpoint rule and the implicit BDF2 scheme and it is shown that the methods have no stability restraint on the size of the timestep. We are interested in the application of deferred correction schemes to initial boundary value problems. The treatment of the boundary conditions has shown to be nontrivial for many high order time discretization methods. Straightforward implementation of the boundary conditions leads to a reduction of the order of accuracy in time. We present a way to treat the boundary conditions to retain high order approximations.

Accelerating the convergence of spectral deferred correction methods

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 (SDC) methods 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.

Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations

This paper concerns a class of deferred correction methods recently developed for initial value ordinary differential equations; such methods are based on a Picard integral form of the correction equation. These methods divide a given timestep [t n ,tn+1] into substeps, and use function values computed at these substeps to approximate the Picard integral by means of a numerical quadrature. The main purpose of this paper is to present a detailed analysis of the implications of the location of quadrature nodes on the accuracy and stability of the overall method. Comparisons between Gauss-Legendre, Gauss-Lobatto, Gauss-Radau, and uniformly spaced points are presented. Also, for a given set of quadrature nodes, quadrature rules may be formulated that include or exclude function values computed at the left-hand endpoint t n . Quadrature rules that do not depend on the left-hand endpoint (which are referred to as right-hand quadrature rules) are shown to lead to L(α)-stable implicit methods with α≈π/2. The semi-implicit analog of this property is also discussed. Numerical results suggest that the use of uniform quadrature nodes, as opposed to nodes based on Gaussian quadratures, does not significantly affect the stability or accuracy of these methods for orders less than ten. In contrast, a study of the reduction of order for stiff equations shows that when uniform quadrature nodes are used in conjunction with a right-hand quadrature rule, the form and extent of order-reduction changes considerably. Specifically, a reduction of order to $\mathcal{O}(\epsilon^2)$ is observed for uniform nodes as opposed to $\mathcal{O}(\epsilon\Delta{t})$ for non-uniform nodes, where Δt denotes the time step and e a stiffness parameter such that e→0 corresponds to the problem becoming increasingly stiff.

Semi-implicit projection methods for incompressible flow based on spectral deferred corrections

A semi-implicit form of spectral deferred corrections is used in a method of lines approach to create a projection method that is sixth-order accurate in both time and space for simple domains. The derivation of spectral deferred corrections is presented and compared to traditional deferred correction methods. An auxiliary variable form of the equations for Boussinesq flow is presented which facilitates a method of lines approach. A discussion of how these methods can be extended to complex spatial domains through the use of integral equation methods is also presented.

Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics

In most models of reacting gas dynamics, the characteristic time scales of chemical reactions are much shorter than the hydrodynamic and diffusive time scales, rendering the reaction part of the model equations stiff. Moreover, nonlinear forcings may introduce into the solutions sharp gradients or shocks, the robust behavior and correct propagation of which require the use of specialized spatial discretization procedures. This study presents high-order conservative methods for the temporal integration of model equations of reacting flows. By means of a method of lines discretization on the flux difference form of the equations, these methods compute approximations to the cell-averaged or finite-volume solution. The temporal discretization is based on a multi-implicit generalization of spectral deferred correction methods. The advection term is integrated explicitly, and the diffusion and reaction terms are treated implicitly but independently, with the splitting errors reduced via the spectral deferred correction procedure. To reduce computational cost, different time steps may be used to integrate processes with widely-differing time scales. Numerical results show that the conservative nature of the methods allows a robust representation of discontinuities and sharp gradients; the results also demonstrate the expected convergence rates for the methods of orders three, four, and five for smooth problems.

High-order multi-implicit spectral deferred correction methods for problems of reactive flow

Models for reacting flow are typically based on advection–diffusion–reaction (A–D–R) partial differential equations. Many practical cases correspond to situations where the relevant time scales associated with each of the three sub-processes can be widely different, leading to disparate time-step requirements for robust and accurate time-integration. In particular, interesting regimes in combustion correspond to systems in which diffusion and reaction are much faster processes than advection. The numerical strategy introduced in this paper is a general procedure to account for this time-scale disparity. The proposed methods are high-order multi-implicit generalizations of spectral deferred correction methods (MISDC methods), constructed for the temporal integration of A–D–R equations. Spectral deferred correction methods compute a high-order approximation to the solution of a differential equation by using a simple, low-order numerical method to solve a series of correction equations, each of which increases the order of accuracy of the approximation. The key feature of MISDC methods is their flexibility in handling several sub-processes implicitly but independently, while avoiding the splitting errors present in traditional operator-splitting methods and also allowing for different time steps for each process. The stability, accuracy, and efficiency of MISDC methods are first analyzed using a linear model problem and the results are compared to semi-implicit spectral deferred correction methods. Furthermore, numerical tests on simplified reacting flows demonstrate the expected convergence rates for MISDC methods of orders three, four, and five. The gain in efficiency by independently controlling the sub-process time steps is illustrated for nonlinear problems, where reaction and diffusion are much stiffer than advection. Although the paper focuses on this specific time-scales ordering, the generalization to any ordering combination is straightforward.

Semi-implicit spectral deferred correction methods for ordinary differential equations

A semi-implicit formulation of the method of spectral deferred corrections (SISDC) for ordinary differential equations with both stiff and non-stiff terms is presented. Several modifications and variations to the original spectral deferred corrections method by Dutt, Greengard, and Rokhlin concerning the choice of integration points and the form of the correction iteration are presented. The stability and accuracy of the resulting ODE methods are explored analytically and numerically. The SISDC methods are intended to be combined with the method of lines approach to yield a flexible framework for creating higher-order semi-implicit methods for partial differential equations. A discussion and numerical examples of the SISDC method applied to advection-diffusion type equations are included. The results suggest that higher-order SISDC methods are more efficient than semi-implicit Runge-Kutta methods for moderately stiff problems in terms of accuracy per function evaluation.

Upwinding with deferred correction (UPDC): an effective implementation of higher-order convection schemes for implicit finite volume methods

A conceptually simple and computationally inexpensive implementation of higher-order convection interpolation schemes for stable and faster convergence of the implicit finite volume method (FVM) is evaluated for multi-dimensional viscoelastic flow calculations as well as convection-dominated Newtonian flow calculations. Based on the deferred-correction method, this effective formulation consistently satisfies the four rules that guarantee bounded numerical solutions. In addition, artificial numerical diffusion, commonly found in numerical solutions with the first-order upwinding convection scheme, can be effectively controlled by consistently using a third-order quadratic upstream interpolation over the entire computational domain. Extensive tests performed for the steady-state wall-driven square enclosure flow of a Newtonian fluid at Reynolds number ( Re) up to 5000 demonstrate that the formulation is stable, accurate, and converges well. The accuracy of the formulation for multi-dimensional viscoelastic flow calculations is evaluated by solving the Oldroyd-B equations for the problem of a fully two-dimensional steady frictionless plane flow, and comparing the numerical results with the exact solution available. With the first-order upwinding scheme, the results show that, for hyperbolic constitutive models, regardless of the grid size, there always exists artificial diffusion whenever the flow streamlines are not closely aligned with the grid lines. With the new implementation, artificial numerical diffusion is effectively controlled and the results demonstrate third-order accuracy.

Deferred Correction Methods for Initial Boundary Value Problems

In this paper, we consider the deferred correction principle for initial boundary value problems. The method will here be applied to the discretization in time. We obtain a method of even order p by applying the implicit midpoint rule p/2 times in each time step. For the space discretization we will use a compact implicit difference scheme. We derive error estimates for the case of time dependent coefficients and present numerical experiments confirming the theoretical analysis.

The effect of numerical diffusion on simulation of isolated bubbles in a gas–solid fluidized bed

A two-fluid hydrodynamic model of a two-dimensional gas–solids fluidized bed is used to calculate the shape of an isolated bubble. The calculations are done by applying four different discretization schemes to the convection term. The bubble shape calculated by using the first-order upwind (FOW) scheme is unphysical and pointed, where as the bubble shape calculated by the second-order schemes SMART, Superbee, and MINMOD is physically realistic and rounded. Grid independence of the solutions is established. Superbee and SMART are found to give the most accurate results for a given grid resolution. Higher order discretization schemes are implemented with a downwind factor (DWF) method and with a deferred correction (DC) method. The DC method is significantly faster than the DWF method.

Superconvergent Deferred Correction Methods For First Order Systems of Nonlinear Two-Point Boundary Value Problems

Iterated deferred correction is a widely used approach to the numerical solution of first order systems of nonlinear two-point boundary value problems. Normally the orders of accuracy of the various methods used in a deferred correction scheme differ by 2, and, as a direct result, each time a deferred correction is applied the order of the overall scheme is increased by a maximum of 2. In this paper we consider the construction of mono-implicit Runge--Kutta (MIRK) methods where an increase of four orders of accuracy is obtained for each deferred correction. We develop a very powerful yet rather straightforward theory which allows us to identify the appropriate Runge--Kutta formulae for inclusion in such schemes. In particular, we will focus on the construction of pairs of MIRK formulae of order 4 and 8 which will allow this superconvergence to be realized. We will further show that it is possible to derive formulae of this type for which high order interpolants and accurate error estimates are readily available.

Deferred correction methods for initial value problems

In this paper we derive high order implicit difference methods for large systems of ODE. The methods are based on the deferred correction principle, yielding accuracy of order p by applying the trapezoidal rule p/2 times in each timestep. Numerical experiments demonstrate the efficiency of the method.

Spectral Deferred Correction Methods for Ordinary Differential Equations

We introduce a new class of methods for the Cauchy problem for ordinary differential equations (ODEs). We begin by converting the original ODE into the corresponding Picard equation and apply a deferred correction procedure in the integral formulation, driven by either the explicit or the implicit Euler marching scheme. The approach results in algorithms of essentially arbitrary order accuracy for both non-stiff and stiff problems; their performance is illustrated with several numerical examples. For non-stiff problems, the stability behavior of the obtained explicit schemes is very satisfactory and algorithms with orders between 8 and 20 should be competitive with the best existing ones. In our preliminary experiments with stiff problems, a simple adaptive implementation of the method demonstrates performance comparable to that of a state-of-the-art extrapolation code (at least, at moderate to high precision).

A Deferred Correction Method for Nonlinear Two-Point Boundary Value Problems: Implementation and Numerical Evaluation

A deferred correction method for the numerical solution of nonlinear two-point boundary value problems has been derived and analyzed in two recent papers by the first author. The method is based on mono-implicit Runge–Kutta formulas and is specially designed to deal efficiently with problems whose solutions contain nonsmooth parts—in particular, singular perturbation problems of boundary layer or turning point type. This paper briefly describes an implementation of the method and gives the results of extensive numerical testing on a set of nonlinear problems that includes both smooth and increasingly stiff (and difficult) problems. Results on the test set are also given using the available codes COLSYS and COLNEW. Although the intent is not to make a formal comparison, the code described appears to be competitive in speed and storage requirements on these problems.

A numerical study of the planar contraction flow of a viscoelastic fluid using the SIMPLER algorithm

A finite volume technique has been introduced in an attempt to simulate the planar 4:1 contraction flow of the Oldroyd B fluid on a non-uniform staggered grid system, which incorporates the SIMPLER algorithm in discretizing the momentum equations and the deferred correction method in discretizing the constitutive equations. For some combinations of viscoelastic parameters, the size and shape of the corner vortex growth are shown to be in good qualitative agreement with those observed experimentally by others for constant viscosity viscoelastic fluids. In general, however, they seem to be sensitive to the ratio of retardation time to relaxation time.

A comparison of some global methods for solving two-point boundary value problems

A numerical comparison is made between three codes which implement global methods for solving two-point boundary value problems. The codes compared are a collocation method, COLSYS, a deferred correction method, PASVA 3, and a deferred correction method based on mono-implicit Runge-Kutta formulae, HAGRON. Although the difficulty of conducting a fair comparison of complex codes is widely recognised it is nevertheless of great practical importance to try to gauge the relative performance of different codes and the three codes considered in this paper are particularly appropriate for numerical comparison. By choosing a test set which includes both smooth and increasingly stiff problems some interesting characteristics of the codes are noted and some recommendations regarding their use are made.

On the Numerical Integration of Nonlinear Two-Point Boundary Value Problems Using Iterated Deferred Corrections. Part 2: The Development and Analysis of Highly Stable Deferred Correction Formulae

Iterated deferred correction methods have been very widely used for the numerical solution of general nonlinear two-point boundary value problems in ordinary differential equations. However, there may be loss of stability when this procedure is applied, since the process of deferred correction is normally explicit. This phenomenon is well known in the theory of initial value problems, where a classic stability theorem of Dahlquist is central in the design of A-stable methods. As a direct result of the deterioration in stability, deferred correction procedures are sometimes ineffective for problems with “rough” solutions, such as singular perturbation or turning point problems. (Such problems are often referred to as “stiff.”) In this paper we consider the idea of using implicit deferred corrections, which allows the derivation of high-order methods that maintain the good stability properties of the underlying low-order formula. Using the ideas developed in this paper, a highly stable, variable-order deferred correction scheme is derived, and the performance of its implementation is compared with that of COLSYS and D02GAF on some linear test problems.

AnO(h 6) quintic spline collocation method for fourth order two-point boundary value problems

AnO(h6) collocation method based on quintic splines is developed and analyzed for general fourth-order linear two-point boundary value problems. The method determines a quintic spline approximation to the solution by forcing it to satisfy a high order perturbation of the original boundary value problem at the nodal points of the spline. A variation of this method is formulated as a deferred correction method. The error analysis of the new method and its numerical behavior is presented.

Some recent applications of asymptotic error expansions to finite-difference schemes

In this paper we will present some results on asymptotic error expansions of the approximate solutions of differential equations by finite-difference schemes. We will present some quite well-known material, in an effort to make the presentation self-contained, as well as discuss recent work by Engquist on linear multistep methods for initial-value problems, by Kreiss on extrapolation procedures for elliptic finite-difference schemes, by Pereyra and co-workers on iterated deferred-correction methods for elliptic equations and by the author and Hald on the finite-element method. Practical aspects of the subject as well as the use of error expansions as a technical device in theoretical numerical analysis are discussed.

On the Construction and Comparison of Difference Schemes

On the Construction and Comparison of Difference Schemes