Spectral Deferred Correction Method Research Articles

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

Spectral Deferred Correction Methods for Second-Order Problems

Spectral Deferred Correction Methods for Second-Order Problems

Sensitivity of Simulations of Double-detonation Type Ia Supernovae to Integration Methodology

We study the coupling of hydrodynamics and reactions in simulations of the double-detonation model for Type Ia supernovae. When assessing the convergence of simulations, the focus is usually on spatial resolution; however, the method of coupling the physics together as well as the tolerances used in integrating a reaction network also play an important role. In this paper, we explore how the choices made in both coupling and integrating the reaction portion of a simulation (operator/Strang splitting versus the simplified spectral deferred corrections method we introduced previously) influences the accuracy, efficiency, and nucleosynthesis of simulations of double detonations. We find no need to limit reaction rates or reduce the simulation time step to the reaction timescale. The entire simulation methodology used here is GPU-accelerated and made freely available as part of the Castro simulation code.

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.

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.

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.

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.

Assessment of high‐order IMEX methods for incompressible flow

SummaryThis paper investigates the competitiveness of semi‐implicit Runge‐Kutta (RK) and spectral deferred correction (SDC) time‐integration methods up to order six for incompressible Navier‐Stokes problems in conjunction with a high‐order discontinuous Galerkin method for space discretization. It is proposed to harness the implicit and explicit RK parts as a partitioned scheme, which provides a natural basis for the underlying projection scheme and yields a straight‐forward approach for accommodating nonlinear viscosity. Numerical experiments on laminar flow, variable viscosity and transition to turbulence are carried out to assess accuracy, convergence and computational efficiency. Although the methods of order 3 or higher are susceptible to order reduction due to time‐dependent boundary conditions, two third‐order RK methods are identified that perform well in all test cases and clearly surpass all second‐order schemes including the popular extrapolated backward difference method. The considered SDC methods are more accurate than the RK methods, but become competitive only for relative errors smaller than ca .

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.

Stability and Error Estimates of a Novel Spectral Deferred Correction Time-Marching with Local Discontinuous Galerkin Methods for Parabolic Equations

Abstract In this paper, we discuss the stability and error estimates of the fully discrete schemes for parabolic equations, in which local discontinuous Galerkin methods with generalized alternating numerical fluxes and a novel spectral deferred correction method based on second-order time integration methods are adopted. With the energy techniques, we obtain both the second- and fourth-order spectral deferred correction time-marching with local discontinuous Galerkin spatial discretization are unconditional stable. The optimal error estimates for the corresponding fully discrete scheme are derived by the aid of the generalized Gauss–Radau projection. We extend the analysis to problems with higher even-order derivatives. Numerical examples are displayed to verify our theoretical results.

A Two-Grid Spectral Deferred Correction Method for the Multi-Order Fractional Differential Equations

A Two-Grid Spectral Deferred Correction Method for the Multi-Order Fractional Differential Equations

Spectral deferred correction methods for high‐order accuracy in poroelastic problems

AbstractWe investigate high‐order accuracy in time integration by examining two operator splitting methods for poroelastic problems: the two‐pass and the spectral deferred correction (SDC) methods. To enhance the order of accuracy, the two‐pass method partitions a coupled operator symmetrically, whereas the SDC method corrects truncation errors by establishing an error equation. These high‐order methods are applied to underlying solution strategies, that is, monolithic, fixed‐stress sequential, and undrained sequential methods. We observe that semi‐discretized systems from spatial discretization have forms similar to those of index‐1 differential algebraic equations (DAEs), causing order reduction against the two‐pass method when it is used in conjunction with either the monolithic or sequential method. On the other hand, the SDC in conjunction with the monolithic method exhibits the desired second‐order accuracy in poroelastic problems while increasing the order of accuracy for index‐1 DAEs. However, the SDC in conjunction with either of the two sequential methods does not achieve the desired order of accuracy, and maintains first order because the flow equation for poroelasticity has an additional approximation associated with the volumetric strain rate term, which does not yield exactly the same forms as those of conventional DAEs. Thus, the monolithic SDC method can achieve higher‐order accuracy, but may require higher computational costs because it involves solving matrix systems larger than those for the sequential methods.

Analysis of the local and parallel space-time algorithm for the heat equation

In this paper, a second-order local and parallel space-time algorithm is proposed and analyzed for the heat equation. This scheme is based on the parareal with spectral deferred correction method in time and the expandable local and parallel finite element method in space. It realizes the parallelism both in the temporal as well as in the spatial direction. We prove its stability and the optimal error estimation in L2-norm. At last, several numerical experiments are presented to demonstrate the effectiveness of our parallel scheme.

A high order operator splitting method based on spectral deferred correction for the nonlocal viscous Cahn-Hilliard equation

Recently, the viscous Cahn-Hilliard (VCH) equation has been proposed as a phenomenological continuum model for phase separation in glass and polymer systems where intermolecular friction forces become important. Compared with the classical local VCH model, the nonlocal VCH model equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions of microstructures in materials. This paper presents a high order fast explicit method based on operator splitting and spectral deferred correction (SDC) for solving the nonlocal VCH equation. We start with a second-order operator splitting spectral scheme, which is based on the Fourier spectral method and the strong stability preserving Runge-Kutta (SSP-RK) method. The scheme takes advantage of applying the fast Fourier transform (FFT) and avoiding nonlinear iteration. The stability and convergence analysis of the obtained numerical scheme are analyzed. To improve the temporal accuracy, the semi-implicit SDC method is then introduced. Various numerical simulations are performed to validate the accuracy and efficiency of the proposed method.

DeC and ADER: Similarities, Differences and a Unified Framework

In this paper, we demonstrate that the explicit ADER approach as it is used inter alia in [1] can be seen as a special interpretation of the deferred correction (DeC) method as introduced in [2]. By using this fact, we are able to embed ADER in a theoretical background of time integration schemes and prove the relation between the accuracy order and the number of iterations which are needed to reach the desired order. Next, we extend our investigation to stiff ODEs, treating these source terms implicitly. Some differences in the interpretation and implementation can be found. Using DeC yields typically a much simpler implementation, while ADER benefits from a higher accuracy, at least for our numerical simulations. Then, we also focus on the PDE case and present common space-time discretizations using DeC and ADER in closed forms. Finally, in the numerical section we investigate A-stability for the ADER approach - this is done for the first time up to our knowledge - for different order using several basis functions and compare them with the DeC ansatz. Then, we compare the performance of ADER and DeC for stiff and non-stiff ODEs and verify our analysis focusing on two basic hyperbolic problems. [1] O. Zanotti, F. Fambri, M. Dumbser, and A. Hidalgo. Space-time adaptive ader discontinuous galerkin finite element schemes with a posteriori sub-cell finite volume limiting. Computers & Fluids, 118:204-224, 2015. [2] A. Dutt, L. Greengard, and V. Rokhlin. Spectral Deferred Correction Methods for Ordinary Differential Equations. BIT Numerical Mathematics, 40(2):241-266, 2000.

A second-order decoupled algorithm with different subdomain time steps for the non-stationary Stokes/Darcy model

In this paper, we propose and analyze a second-order decoupled algorithm with different subdomain time steps for the non-stationary Stokes/Darcy model. It is based on the second-order spectral deferred correction method in time and the finite element method in space. We provide the stability and convergence results of our decoupled scheme. Last, some numerical experiments are given to illustrate the accuracy and effectiveness of our decoupled scheme.

PFASST-ER: combining the parallel full approximation scheme in space and time with parallelization across the method

To extend prevailing scaling limits when solving time-dependent partial differential equations, the parallel full approximation scheme in space and time (PFASST) has been shown to be a promising parallel-in-time integrator. Similar to space–time multigrid, PFASST is able to compute multiple time-steps simultaneously and is therefore in particular suitable for large-scale applications on high performance computing systems. In this work we couple PFASST with a parallel spectral deferred correction (SDC) method, forming an unprecedented doubly time-parallel integrator. While PFASST provides global, large-scale “parallelization across the step”, the inner parallel SDC method allows integrating each individual time-step “parallel across the method” using a diagonalized local Quasi-Newton solver. This new method, which we call “PFASST with Enhanced concuRrency” (PFASST-ER), therefore exposes even more temporal concurrency. For two challenging nonlinear reaction-diffusion problems, we show that PFASST-ER works more efficiently than the classical variants of PFASST and can use more processors than time-steps.