Deferred Correction Method Research Articles

An explicit two-grid spectral deferred correction method for nonlinear fractional pantograph differential equations

An explicit two-grid spectral deferred correction method for nonlinear fractional pantograph differential equations

An exponential spectral deferred correction method for multidimensional parabolic problems

We present some efficient algorithms based on an exponential time differencing spectral deferred correction (ETDSDC) method for multidimensional second and fourth-order parabolic problems with non-periodic boundary conditions including Dirichlet, Neumann, Robin boundary conditions. Similar to the Fourier spectral method for periodic problems, the key to the efficiency of our algorithms is to construct diagonal discrete linear operators via Legendre–Galerkin methods with Fourier-like basis functions. In combination with the ETDSDC scheme, the proposed methods are spectrally accurate in space and up to 10th-order accurate in time (as shown in this work). We demonstrate the high-order of convergence and efficiency of our algorithms in solving parabolic equations through a series of two-dimensional and three-dimensional examples including Ginzburg–Landau and Allen–Cahn equations.

Stability of implicit deferred correction methods based on BDF methods

The Dahlquist barrier states that the highest attainable order for an A-stable linear multistep method is limited to 2. In this paper, we adopt the deferred correction approach with the BDF methods to develop A-stable third and fourth-order multistep methods with low stages. The stability of the methods is investigated to show how A-stability can be achieved. Numerical experiments are conducted to validate the accuracy and stability of the proposed methods when applied to stiff problems.

Spectral deferred correction method for fractional initial value problem with Caputo–Hadamard derivative

This paper considers an efficient and accurate spectral deferred correction (SDC) method for the initial value problem (IVP) with Caputo–Hadamard derivative. We first apply the basic idea of the SDC method to derive the numerical scheme. Then the iteration matrix which is the key to convergence of the proposed scheme can be obtained for the linear problem. Detailed computation of history term is presented using the spectral collocation method based on mapped Jacobi log orthogonal functions (MJLOFs). Finally, numerical simulations for both linear and nonlinear cases are shown to verify the feasibility and efficiency of the proposed method.

Spectral Deferred Correction Methods for Second-Order Problems

Spectral Deferred Correction Methods for Second-Order Problems

A-stable spectral deferred correction method for nonlinear Allen-Cahn model

This paper presents a type of A-stable spectral deferred correction (SDC) method. The scheme is initiated by the first-order backward Euler method. We adopt the linear stabilization approach for the Allen-Cahn model to get the linear semi-implicit SDC scheme. This is done by the addition and subtraction of the linear stabilization operators that have been provided for the Allen-Cahn problem. The ranges of the stabilization factors are then given. The aim is to achieve A-stability and error estimates for the implicit part of the scheme. Finally, the semi-implicit SDC scheme coupled with the Fourier spectral method is used to simulate the Allen-Cahn problem, which is also suitable for the highly nonlinear problem. Numerical experiments are given to numerically demonstrate the high-order accuracy and the energy decay property of the scheme with A-stability parameters.

A stable numerical framework for long-time dynamic crack analysis

This paper introduces a stable numerical framework designed to address dynamic crack problems over long-time intervals. The initial step involves the temporal discretization of the governing dynamic equilibrium equations using the arbitrary order Krylov deferred correction method. To ensure precise boundary condition matching, a novel numerical implementation is incorporated into the Krylov deferred correction technique. Subsequently, the resulting system of spatial partial differential equations at each time node is solved using the meshless generalized finite difference method with 4th-order expansions in regions close to the crack-tips and 2nd-order expansions in areas far from the crack-tips. This combined approach capitalizes on the strengths of both the Krylov deferred correction technique and the generalized finite difference method, enabling stable simulations of dynamic cracks with the large time step and without the need for mesh generation. Notably, we refine the collocation nodes near crack tips to attain accurate numerical results for displacement and stress field. Several numerical experiments involving diverse impact loadings are conducted to validate the developed framework. Furthermore, a comparison is made between the dynamic stress intensity factors obtained using our approach and those from existing methods.

Semi-Implicit Spectral Deferred Correction Methods Based on Second-Order Time Integration Schemes for Nonlinear PDEs

Semi-Implicit Spectral Deferred Correction Methods Based on Second-Order Time Integration Schemes for Nonlinear PDEs

Higher-order methods for the Stokes equations based on the coupling of discontinuous Galerkin method and spectral deferred correction method

In this paper, the spatial discontinuous Galerkin (DG) approximation coupled with the temporal spectral deferred correction (SDC) evolution for the Stokes equations is adopted to construct the higher-order discretization method. First, the artificial compressibility strategy method is used to convert the Stokes equations into the Cauchy–Kovalevskaja type equations. Second, the corresponding equations can be rewritten as a first-order system by introducing the new variable equal to the gradient of the velocity. Then, the DG and the SDC methods are properly combined to construct the expected higher-order method. Theoretically, the stability analysis of the second-order fully discrete method is proved. The numerical experiments are given to verify the effectiveness of the presented methods.

L-stable spectral deferred correction methods and applications to phase field models

This paper presents the L-stable spectral deferred correction (SDC) methods with low stages. These schemes are initiated by the Crank-Nicolson method. We adopt the linear stabilization approach for the phase field models to obtain the linear implicit SDC scheme. This is done by adding and subtracting the linear stabilization operators that are provided for the different phase field problems. Without loss of the low-stage property, the extrapolation technique is also used in the prediction step of the semi-implicit SDC method. Numerical experiments are given to validate the high-order accuracy and the energy decay property of the proposed semi-implicit SDC methods for the Allen-Cahn, Cahn-Hilliard, and molecular beam epitaxy equations.

Accurate numerical simulations for fractional diffusion equations using spectral deferred correction methods

This paper mainly studies the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for discretization in space and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. Therefore, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Finally, several numerical examples are given to illustrate the effectiveness and feasibility of the numerical scheme.

An arbitrary order numerical framework for transient heat conduction problems

This paper presents an arbitrary order numerical approach for achieving highly accurate solutions in transient heat conduction problems. In this framework, the arbitrary order Krylov deferred correction method (KDC) is first utilized to discretize the temporal direction of the problems, ensuring high-accuracy results by selecting an adequate number of Gaussian nodes. During each time iteration, a boundary value problem in terms of the inhomogeneous modified Helmholtz equation is generated by utilizing the KDC methodology. Subsequently, the arbitrary order GFDM is introduced to solve the boundary value problem, allowing for flexible selection of the Taylor series expansion order to improve spatial accuracy. Consequently, the present numerical framework is of arbitrary order of accuracy and very stable through integrating advantages of the arbitrary order KDC and the arbitrary order GFDM. The developed numerical framework is ultimately verified for its characteristic of high accuracy and stability through two numerical examples in two-dimensions and three numerical examples in three-dimensions. The results highlight the significant improvement in numerical accuracy achieved by the present arbitrary-order numerical approach for long-time simulations.

Deferred correction method for the continuous data assimilation model

This paper presents a numerical assessment of a continuous data assimilation (CDA) model for incompressible Navier–Stokes equations. The proposed model employs the deferred correction method, which is an approach to increase temporal accuracy. What makes this work valuable is the necessity of obtaining high temporal accuracy results, which is only possible with the deferred correction method for some problems, e.g., the decoupling of the atmosphere–ocean interaction problem (Temam et al., 1993 [1]; Erkmen et al., 2018). The scheme involves incorporating a nudging term into a traditional deferred correction scheme in order to calibrate a model to match coarse mesh or spatial observations at a larger scale. We present stability, long time stability, and convergence results of the scheme in detail. We also illustrate some numerical experiments to validate and reinforce the theoretical results.

Spectral deferred correction method for Landau–Brazovskii model with convex splitting technique

The Landau–Brazovskii model is a well-known Landau model for finding the complex phase structures in microphase-separating systems ranging from block copolymers to liquid crystals. It is critical to design efficient numerical schemes for the Landau–Brazovskii model with energy dissipation and mass conservation properties. Here, we propose a mass conservative and energy stable scheme by combining the spectral deferred correction (SDC) method with the convex splitting technique to solve the Landau–Brazovskii model efficiently. An adaptive correction strategy for the SDC method is implemented to reduce the cost time and preserve energy stability. Numerical experiments, including two- and three-dimensional periodic crystals in Landau–Brazovskii model, are presented to show the efficiency of the proposed numerical method.

A high order accurate numerical algorithm for the space-fractional Swift-Hohenberg equation

This paper presents a high order time discretization method by combining the time splitting scheme with semi-implicit spectral deferred correction method for the space-fractional Swift-Hohenberg equation (SFSH). Based on the operator splitting method, the original problem is split into linear and nonlinear subproblems, respectively. The Fourier spectral method is adopted for the linear part, and a first-order accurate finite difference scheme in time together with the Fourier spectral method in space is used for the nonlinear part. The stability and convergence of the obtained numerical scheme are analyzed theoretically. Moreover, the spectral deferred correction (SDC) method is then employed to improve the temporal accuracy. Various two and three dimensional numerical experiments are performed to validate the theoretical results and efficiency of the proposed method.

A high-order deferred correction method for the solution of free boundary problems using penalty iteration, with an application to American option pricing

This paper presents a high-order deferred correction algorithm combined with penalty iteration for solving free and moving boundary problems, using a fourth-order finite difference method. Typically, when free boundary problems are solved on a fixed computational grid, the order of the solution is low due to the discontinuity in the solution at the free boundary, even if a high-order method is used. Using a detailed error analysis, we observe that the order of convergence of the solution can be increased to fourth-order by solving successively corrected finite difference systems, where the corrections are derived from the previously computed lower order solutions. The corrections are applied solely to the right-hand side, and leave the finite difference matrix unchanged. The penalty iterations converge quickly given a good initial guess. We demonstrate the accuracy and efficiency of our algorithm using several examples. Numerical results show that our algorithm gives fourth-order convergence for both the solution and the free boundary location. We also test our algorithm on the challenging American put option pricing problem. Our algorithm gives the expected high-order convergence.

High-Order Accurate Solutions of the Point Kinetics Equations with the Spectral Deferred Correction Method

Solving initial value problems with high-order methods receives considerable attention in many fields because these methods can potentially improve the accuracy of the simulation results with lower computational cost than low-order methods. Most methods, however, are either complicated to implement or unstable when the order of accuracy is high. The spectral deferred correction (SDC) method is a stable, robust, and efficient high-order time-integration scheme capable of an arbitrary order of accuracy. In this paper, we apply the SDC method to solve the initial value problem of the point kinetics equations (PKEs). For our implementation, we show that SDC is -stable for orders up to eight and the order of accuracy is verified for PKE problems with a range of different reactivities. A fifth-order SDC method was then implemented to solve the exact PKE in the transient multilevel method of MPACT. The error from solutions of the exact PKE with SDC is shown to be negligible. The investigations made here can provide the foundation for future investigations simulating the neutron transport problem using the high-order methods for both spatial discretization and time integration.

Improved spectral deferred correction methods for fractional differential equations

In this study, we improved the spectral deferred correction method for nonlinear fractional differential equations. First, problems were transformed into equivalent nonlinear Volterra integral equations with weakly singular kernels. We then employed the fractional Adams–Bashforth method in the prediction step and used the Gauss quadrature formula and fractional Adams–Moulton scheme in the correction step. Moreover, a vigorous error analysis for the numerical scheme was conducted. Finally, computational results for some experiments were reported to demonstrate the accuracy and ease of implementation of the improved numerical scheme.

Defect deferred correction method for two domain natural convection problem

This paper concerns defect deferred correction method used to resolve interactions between the atmosphere and ocean for applications like climate research, hurricane modelling and regional weather forecasting. The defect deferred correction method is obtained by combining the defect correction method and the deferred correction method. There are two advantages of using the defect deferred correction method to solve a two domain natural convection problem, namely, reaching the second-order time accuracy and solving small viscosity or diffusion coefficient problems. Finally, we will verify the conclusion from both theoretical analysis and numerical experiments.

Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes

In [1] is proposed a simplified DeC method, that, when combined with the residual distribution (RD) framework, allows to construct a high order, explicit FE scheme with continuous approximation avoiding the inversion of the mass matrix for hyperbolic problems. In this paper, we close some open gaps in the context of deferred correction (DeC) and their application within the RD framework. First, we demonstrate the connection between the DeC schemes and the RK methods. With this knowledge, DeC can be rewritten as a convex combination of explicit Euler steps, showing the connection to the strong stability preserving (SSP) framework. Then, we can apply the relaxation approach introduced in [2] and construct entropy conservative/dissipative DeC (RDeC) methods, using the entropy correction function proposed in [3]. [1] R. Abgrall. High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices. Journal of Scientific Computing, 73(2):461--494, Dec 2017. [2] D. Ketcheson. Relaxation Runge--Kutta methods: Conservation and stability for inner-product norms. SIAM Journal on Numerical Analysis, 57(6):2850--2870, 2019. [3] R. Abgrall. A general framework to construct schemes satisfying additional conservation relations. application to entropy conservative and entropy dissipative schemes. Journal of Computational Physics, 372:640--666, 2018.