Integral Deferred Correction Methods Research Articles

Improvements to a class of hybrid methods for radiation transport: Nyström reconstruction and defect correction methods

In this paper, two modifications are introduced for improving the accuracy, versatility, and robustness of a class of hybrid methods for radiation transport. In general, such methods are constructed by splitting the radiative flux into collided and uncollided components to which low- and high-resolution angular approximations are applied, respectively. In this work we focus on discrete ordinates discretizations of high and low order. The first modification we introduce changes the way in which the collided component is mapped into the uncollided component at the end of each time step in a simulation. The new mapping is a Nyström-type reconstruction that is applicable to arbitrary discrete ordinates quadratures, is guaranteed to preserve positivity of the solution provided that all ordinate weights are positive, is significantly more accurate than previous methods, and can be readily extended to other discretizations such as moment methods, finite element methods, and diffusion approximations. The second modification leverages integral deferred correction (IDC) to iteratively correct for the splitting error introduced by the inconsistency in angular discretization between the collided and uncollided components, in addition to improving the accuracy of the low-order temporal error that is treated by traditional IDC methods. Numerical tests in one- and two-dimensional geometries are used to demonstrate the increased accuracy and efficiency of the proposed modifications. It is found that the two techniques combined yield methods with solution accuracy and memory requirements comparable to that of monolithic discrete ordinates methods while reducing runtime by as much as a factor of between two and ten, depending on the problem.

Simulation of advection\u2013diffusion\u2013dispersion equations based on a composite time discretization scheme

In this work, we develop a high-order composite time discretization scheme based on classical collocation and integral deferred correction methods in a backward semi-Lagrangian framework (BSL) to simulate nonlinear advection–diffusion–dispersion problems. The third-order backward differentiation formula and fourth-order finite difference schemes are used in temporal and spatial discretizations, respectively. Additionally, to evaluate function values at non-grid points in BSL, the constrained interpolation profile method is used. Several numerical experiments demonstrate the efficiency of the proposed techniques in terms of accuracy and computation costs, compare with existing departure traceback schemes.

Numerical analysis of a 4th-order time parallel algorithm for the time-dependent Navier-Stokes equations

This work studies the stability and convergency properties for a time parallel algorithm: RIDC4 with reduced stencils, namely reduced 4th revisionist integral deferred correction method. For the spatial approximation of the velocity and the pressure, a finite element space discretization is applied. We proved that the scheme is almost unconditionally stable and is of 4th-order accuracy in time. Finally, numerical experiments are carried out to verify its time and space convergence orders and the formidable parallel efficiency demonstrated by the time speed-up compared with the classical Backward Euler (BE) methods. In conclusion, within almost same time to BE, the (reduced) RIDC4 scheme could achieve 4th order accuracy in time direction, provided four cores are available.

An arbitrary-order, fully implicit, hybrid kinetic solver for linear radiative transport using integral deferred correction

In this work, we describe the implementation of an arbitrarily high-order hybrid solver for linear, kinetic, radiative transport equations. The hybrid method is derived from a splitting of the radiative flux into free-streaming and collisional components to which high- and low-resolution discrete ordinates methods are applied, respectively. Arbitrarily high orders of accuracy with respect to time and space are attained by combining an integral deferred correction (IDC) time integration scheme constructed with implicit Euler substepping on Radau II (right biased) nodes with an upwind discontinuous Galerkin (DG) spatial discretization on uniform Cartesian meshes. Numerical experiments are used to demonstrate that the aforementioned IDC methods can be constructed such that they are unconditionally stable (L-stable) to within machine precision. Asymptotic analysis is used to show that such IDC methods also preserve the diffusion limit of the underlying transport equation on the semi-discrete level, in the sense that the semi-discrete transport system under an implicit IDC discretization recovers the same IDC discretization of the limiting diffusion equation. Convergence results in one spatial dimension are provided, and it is found that while the hybrid method exhibits convergence stagnation and order reduction in certain scenarios, the overall accuracy of the hybrid approximation is comparable to a standard discrete ordinates approximation in many cases. Numerical results for two test problems in two spatial dimensions are given to compare the computational efficiency of the hybrid method against a standard discrete ordinates method, and to compare the efficiency of space–time discretizations of different orders of accuracy. The results indicate that a properly chosen hybrid discrete ordinates method can be more efficient than a standard discrete ordinates method by a factor of between 2 and 8 when IDC timestepping methods are used. Further, among the first-, third-, and fifth-order space–time discretizations considered here, the first-order method is the least efficient and the third-order method is the most efficient.

Algorithm 965

Revisionist integral deferred correction methods are a family of parallel-in-time methods to solve systems of initial values problems. The approach is able to bootstrap lower-order time integrators to provide high-order approximations in approximately the same wall-clock time, hence providing a multiplicative increase in the number of compute cores utilized. Here we provide a library that automatically produces a parallel-in-time solution of a system of initial value problems given user-supplied code for the right-hand side of the system and a sequential code for a first-order timestep. The user-supplied timestep routine may be explicit or implicit and may make use of any auxiliary libraries that take care of the solution of any nonlinear algebraic systems that may arise or the numerical linear algebra required.

Error estimates of the integral deferred correction method for stiff problems

In this paper, we present error estimates of the integral deferred correction method constructed with stiffly accurate implicit Runge–Kutta methods with a nonsingular matrix A in its Butcher table representation, when applied to stiff problems characterized by a small positive parameter e . In our error estimates, we expand the global error in powers of e and show that the coefficients are global errors of the integral deferred correction method applied to a sequence of differential algebraic systems. A study of these errors and of the remainder of the expansion yields sharp error bounds for the stiff problem. Numerical results for the van der Pol equation are presented to illustrate our theoretical findings. Finally, we study the linear stability properties of these methods.

High order operator splitting methods based on an integral deferred correction framework

Integral deferred correction (IDC) methods have been shown to be an efficient way to achieve arbitrary high order accuracy and possess good stability properties. In this paper, we construct high order operator splitting schemes using the IDC procedure to solve initial value problems (IVPs). We present analysis to show that the IDC methods can correct for both the splitting and numerical errors, lifting the order of accuracy by r with each correction, where r is the order of accuracy of the method used to solve the correction equation. We further apply this framework to solve partial differential equations (PDEs). Numerical examples in two dimensions of linear and nonlinear initial–boundary value problems are presented to demonstrate the performance of the proposed IDC approach.

A Parallel Space-Time Algorithm

With the continued evolution of computing architectures towards many-core computing, algorithms that can effectively and efficiently use many cores are crucial. In this paper, we propose, as a proof of principle, a parallel space-time algorithm that layers time parallelization together with a parallel elliptic solver to solve time dependent partial differential equations (PDEs). The parallel elliptic solver utilizes domain decomposition to divide a spatial grid into subdomains, and applies a parallel Schwarz iteration to find consistent solutions. The high-order parallel time integrator employed belongs to the family of revisionist integral deferred correction methods (RIDC) introduced by Christlieb, Macdonald, and Ong [SIAM J. Sci. Comput., 32 (2010), pp. 818--835], which allows for the small scale parallelization of solutions to initial value problems. The two established algorithms are combined in this proposed space-time algorithm to add parallel scalability. As a proof of concept, we utilize a framew...

Semi-implicit integral deferred correction constructed with additive Runge–Kutta methods

In this paper, we consider construct high order semi-implicit integrators using integral deferred correction (IDC) to solve stiff initial value problems. The general framework for the construction of these semi-implicit methods uses uniformly distributed nodes and additive RungeKutta (ARK) integrators as base schemes inside an IDC framework, which we refer to as IDC-ARK methods. We establish under mild assumptions that, when an r order ARK method is used to predict and correct the numerical solution, the order of accuracy of the IDC method increases by r for each IDC prediction and correction loop. Numerical experiments support the established theorems, and also indicate that higher order IDC-ARK methods present an efficiency advantage over existing implicit-explicit (IMEX) ARK schemes in some cases.

Integral deferred correction methods constructed with high order Runge–Kutta integrators

Spectral deferred correction (SDC) methods for solving ordinary differential equations (ODEs) were introduced by Dutt, Greengard and Rokhlin (2000). It was shown in that paper that SDC methods can achieve arbitrary high order accuracy and possess nice stability properties. Their SDC methods are constructed with low order integrators, such as forward Euler or backward Euler, and are able to handle stiff and non-stiff terms in the ODEs. In this paper, we use high order Runge-Kutta (RK) integrators to construct a family of related methods, which we refer to as integral deferred correction (IDC) methods. The distribution of quadrature nodes is assumed to be uniform, and the corresponding local error analysis is given. The smoothness of the error vector associated with an IDC method, measured by the discrete Sobolev norm, is a crucial tool in our analysis. The expected order of accuracy is demonstrated through several numerical examples. Superior numerical stability and accuracy regions are observed when high order RK integrators are used to construct IDC methods.

Comments on high-order integrators embedded within integral deferred correction methods

A class of novel deferred correction methods, integral deferred correction (IDC) methods, is studied. This class of methods is an extension of ideas introduced by Dutt, Greengard and Rokhlin on spectral deferred correction (SDC) methods for solving ordinary differential equations (ODEs). The novel nature of this class of defect correction methods is that the correction of the defect is carried out using an accurate integral form of the residual instead of the more familiar differential form. As a family of methods, these schemes are capable of matching the efficiency of popular high-order RK methods. The smoothness of the error vector associated with an IDC method is an important indicator of the order of convergence that can be expected from a scheme (Christlieb, Ong, and Qiu; Hansen and Strain; Skeel). It is demonstrated that embedding an r-th order integrator in the correction loop of an IDC method does not always result in an r-th order increase in accuracy. Examples include IDC methods constructed using non-self-starting multistep integrators, and IDC methods constructed using a nonuniform distribution of quadrature nodes. Additionally, the integral deferred correction concept is reposed as a framework to generate high-order Runge–Kutta (RK) methods; specifically, we explain how the prediction and correction loops can be incorporated as stages of a high-order RK method. This alternate point of view allows us to utilize standard methods for quantifying the performance (efficiency, accuracy and stability) of integral deferred correction schemes. It is found that IDC schemes constructed using uniformly distributed nodes and high-order integrators are competitive in terms of efficiency with IDC schemes constructed using Gauss–Lobatto nodes and forward Euler integrators. With respect to regions of absolute stability, however, IDC methods constructed with uniformly distributed nodes and high-order integrators are far superior. It is observed that as the order of the embedded integrator increases, the stability region of the IDC method increases.

Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods

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 byfirst 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.