Deferred Correction Scheme Research Articles

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.

Numerical Stabilization for Flutter Analysis Procedure

Severe mode switching is often observed when the PK-method is used in the flutter analysis of complex aircraft configurations, in particular when nearly 100 vibrational modes are considered. In the commonly used commercial software NASTRAN, the resulting eigenroots are sorted in an ascending order of frequency. Therefore, the appearance of massive mode-switching instances cannot be avoided in the PK-method flutter analyses, especially for engineering applications with real-world complex configurations. In this study, as a post-processing procedure, an extensive sorting capability was developed in order to compensate for NASTRAN’s lack of a mode-tracking procedure in between the airspeed steps. The capability was developed based on both the complex eigenvalues and their corresponding eigenvectors. In addition, numerical techniques commonly used in computational fluid dynamics (CFD) were introduced to improve the convergence of the traditional PK-method. A hybrid approach was applied to the initial guess of the reduced frequency, followed by a deferred correction scheme for the PK-iteration process. Additionally, mode matching was specifically addressed when locking eigenroots onto the aerodynamics within the PK iterations. In addition to the PK iterations, a damping iteration or modified g-method was implemented by extending the PK-method solver. The combination of these special techniques effectively improved the numerical stability of the iterations in the stability eigensolution process and significantly reduced the appearance of the misleading mode switching, minimizing risks in aircraft flight.

Deferred defect‐correction finite element method for the Darcy‐Brinkman model

AbstractThis paper proposes a deferred defect‐correction method for the time‐dependent nonlinear Darcy‐Brinkman model based on the mixed finite element method. The presented method combines the deferred correction strategy with the defect‐correction method and involves two steps. In the first step, a defect Darcy‐Brinkman model is solved. Then in the second step, a correction Darcy‐Brinkman model based on the deferred correction approach is solved. Moreover, unconditional stability and convergence of the presented method are deduced. Finally, several numerical examples are given to support the theoretical analysis.

A Novel Arbitrary Higher-Order Discontinuous Galerkin-Spectral Deferred Correction Scheme for the Variable Coefficients Advection Equation

A Novel Arbitrary Higher-Order Discontinuous Galerkin-Spectral Deferred Correction Scheme for the Variable Coefficients Advection Equation

High order methods for acoustic scattering: Coupling farfield expansions ABC with deferred-correction methods

Arbitrary high order numerical methods for time-harmonic acoustic scattering problems originally defined on unbounded domains are constructed. This is done by coupling recently developed high order local absorbing boundary conditions (ABCs) with finite difference methods for the Helmholtz equation. These ABCs are based on exact representations of the outgoing waves by means of farfield expansions. The finite difference methods, which are constructed from a deferred-correction (DC) technique, approximate the Helmholtz equation and the ABCs, with the appropriate number of terms, to any desired order. As a result, high order numerical methods with an overall order of convergence equal to the order of the DC schemes are obtained. A detailed construction of these DC finite difference schemes is presented. Additionally, a rigorous proof of the consistency of the DC schemes with the Helmholtz equation and the ABCs in polar coordinates is also given. The results of several numerical experiments corroborate the high order convergence of the novel method.

Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes

Production-destruction systems (PDS) of ordinary differential equations (ODEs) are used to describe physical and biological reactions in nature. The considered quantities are subject to natural laws. Therefore, they preserve positivity and conservation of mass at the analytical level. In order to maintain these properties at the discrete level, the so-called modified Patankar-Runge-Kutta (MPRK) schemes are often used in this context. However, up to our knowledge, the family of MPRK has been only developed up to third order of accuracy. In this work, we propose a method to solve PDS problems, but using the Deferred Correction (DeC) process as a time integration method. Applying the modified Patankar approach to the DeC scheme results in provable conservative and positivity preserving methods. Furthermore, we demonstrate that these modified Patankar DeC schemes can be constructed up to arbitrarily high order. Finally, we validate our theoretical analysis through numerical simulations.

A Class of Exponential Integrators Based on Spectral Deferred Correction

This paper introduces a new class of exponential integrators based on spectral deferred correction. These new methods are simple to implement at any order of accuracy and can be used to efficiently solve initial value problems when high precision is desired. We begin by deriving exponential spectral deferred correction (ESDC) methods for solving both partitioned and unpartitioned initial value problems. We then analyze the linear stability properties of these new integrators and show that they are comparable to those of existing semi-implicit spectral deferred correction schemes. Finally, we present five numerical experiments to demonstrate the improved efficiency of our new exponential integrator compared to semi-implicit spectral deferred correction schemes and existing fourth-order exponential Runge--Kutta methods.

Polydisperse vapour-droplet mixture dynamics model with phase transitions, fragmentation, and coagulation of droplets

This work presents mathematical model and numerical method for polydisperse vapour-droplet mixture current description. A system of Euler equations for the carrier medium and disperse phase motion is solved via Mac-Cormack’s explicit method based on operator dimensional splitting and deferred correction scheme. The results of vapour-droplet flow calculations in cylindrical conduits are shown.

A spectral deferred correction strategy for low Mach number reacting flows subject to electric fields

We propose an algorithm for low Mach number reacting flows subjected to electric field that includes the chemical production and transport of charged species. This work is an extension of a multi-implicit spectral deferred correction (MISDC) algorithm designed to advance the conservation equations in time at scales associated with advective transport. The fast and nontrivial interactions of electrons with the electric field are treated implicitly using a Jacobian-Free Newton Krylov approach for which a preconditioning strategy is developed. Within the MISDC framework, this enables a close and stable coupling of diffusion, reactions and dielectric relaxation terms with advective transport and is shown to exhibit second-order convergence in space and time. The algorithm is then applied to a series of steady and unsteady problems to demonstrate its capability and stability. Although developed in a one-dimensional case, the algorithmic ingredients are carefully designed to be amenable to multi-dimensional applications.

Explicit Deferred Correction Methods for Second-Order Forward Backward Stochastic Differential Equations

This is the second part of our series papers on the deferred correction method for forward backward stochastic differential equations. In this work, we extend our previous work in Tang et al. (Numer Math Theory Methods Appl 10(2):222–242, 2017) to solve second-order forward backward stochastic differential equations (2FBSDEs). More precisely, we propose a class of explicit deferred correction schemes for 2FBSDEs. The key feature is that the simple Euler scheme is used as an initialization. Then, by a simple deferred correction iteration scheme, one can obtain an approximated solution with very high accuracy. Yet in each iteration, the computational complexity is always comparable to the Euler solver. Numerical examples are presented to show the effectiveness of the proposed scheme. We believe that the scheme proposed in this work is promising when dealing with 2FBSDEs in moderate dimensions.

Concurrent implicit spectral deferred correction scheme for low-Mach number combustion with detailed chemistry

We present a parallel multi-implicit time integration scheme for the advection-diffusion-reaction systems arising from the equations governing low-Mach number combustion with complex chemistry. Our strategy employs parallelisation across the method to accelerate the serial Multi-Implicit Spectral Deferred Correction (MISDC) scheme used to couple the advection, diffusion, and reaction processes. In our approach, the diffusion solves and the reaction solves are performed concurrently by different processors. Our analysis shows that the proposed parallel scheme is stable for stiff problems and that the sweeps converge to the fixed-point solution at a faster rate than with serial MISDC. We present numerical examples to demonstrate that the new algorithm is high-order accurate in time, and achieves a parallel speedup compared to serial MISDC.

High-Order Accurate Adaptive Kernel Compression Time-Stepping Schemes for Fractional Differential Equations

High-order adaptive methods for fractional differential equations are proposed. The methods rely on a kernel compression scheme for the approximation and localization of the history term. To avoid complications typical to multistep methods, we focus our study on 1-step methods and approximate the local part of the fractional integral by integral deferred correction to enable high order accuracy. We study the local truncation error of integral deferred correction schemes for Volterra equations and present numerical results obtained with both implicit and the explicit methods applied to different problems.

A high-order spectral deferred correction strategy for low Mach number flow with complex chemistry

We present a fourth-order finite-volume algorithm in space and time for low Mach number reacting flow with detailed kinetics and transport. Our temporal integration scheme is based on a Multi-Implicit Spectral Deferred Correction (MISDC) strategy that iteratively couples advection, diffusion, and reactions evolving subject to a constraint. Our new approach overcomes a stability limitation of our previous second-order method encountered when trying to incorporate higher-order polynomial representations of the solution in time to increase accuracy. We have developed a new iterative scheme that naturally fits within our MISDC framework and allows us to conserve mass and energy while simultaneously satisfying the equation of state. We analyse the conditions for which the iterative schemes are guaranteed to converge to the fixed point solution. We present numerical examples illustrating the performance of the new method on premixed hydrogen, methane, and dimethyl ether flames.

A Numerical Framework for Integrating Deferred Correction Methods to Solve High Order Collocation Formulations of ODEs

Recent analysis and numerical experiments show that the deferred correction methods are competitive numerical schemes for time dependent differential equations. These methods differ in the mathematical formulations, choices of collocation points, and numerical integration or differentiation strategies. Existing analyses of these methods usually follow traditional ODE theory and study each algorithm's convergence and stability properties as the step size $$\varDelta t$$Δt varies. In this paper, we study the deferred correction methods from a different perspective by separating two different concepts in the algorithm: (1) the properties of the converged solution to the collocation formulation, and (2) the convergence procedure utilizing the deferred correction schemes to iteratively and efficiently reduce the error in the provisional solution. This new viewpoint allows the construction of a numerical framework to integrate existing techniques, by (1) selecting an appropriate collocation discretization based on the physical properties of the solution to balance the time step size and accuracy of the initial approximate solution; and by (2) applying different deferred correction strategies for reducing different components in the error of the provisional solution. This paper discusses properties of different components in the numerical framework, and presents preliminary results on the effective integration of these components for ODE initial value problems. Our results provide useful guidelines for implementing "optimal" time integration schemes for general time dependent differential equations.

Algorithm 927

In this article we describe the code bvptwp.m, a MATLAB code for the solution of two point boundary value problems. This code is based on the well-known Fortran codes, twpbvp.f, twpbvpl.f and acdc.f, that employ a mesh selection strategy based on the estimation of the local error, and on revisions of these codes, called twpbvpc.f, twpbvplc.f and acdcc.f, that employ a mesh selection strategy based on the estimation of the local error and the estimation of two parameters which characterize the conditioning of the problem. The codes twpbvp.f/tpbvpc.f use a deferred correction scheme based on Mono-Implicit Runge-Kutta methods (MIRK); the other codes use a deferred correction scheme based on Lobatto formulas. The acdc.f/acdcc.f codes implement an automatic continuation strategy. The performance and features of the new solver are checked by performing some numerical tests to show that the new code is robust and able to solve very difficult singularly perturbed problems. The results obtained show that bvptwp.m is often able to solve problems requiring stringent accuracies and problems with very sharp changes in the solution. This code, coupled with the existing boundary value codes such as bvp4c.m, makes the MATLAB BVP section an extremely powerful one for a very wide range of problems.

Fourth-Order Deferred Correction Scheme for Solving Heat Conduction Problem

A deferred correction method is utilized to increase the order of spatial accuracy of the Crank-Nicolson scheme for the numerical solution of the one-dimensional heat equation. The fourth-order methods proposed are the easier development and can be solved by using Thomas algorithms. The stability analysis and numerical experiments have been limited to one-dimensional heat-conducting problems with Dirichlet boundary conditions and initial data.

A deferred correction coupling strategy for low Mach number flow with complex chemistry

We present a new thermodynamic coupling strategy for complex reacting flow in a low Mach number framework. In such flows, the advection, diffusion and reaction processes span a broad range of time scales. In order to reduce splitting errors inherent in Strang splitting approaches, we couple the processes with a multi-implicit spectral deferred correction strategy. Our iterative scheme uses a series of relatively simple correction equations to reduce the error in the solution. The new method retains the efficiencies of Strang splitting compared to a traditional method-of-lines approach in that each process is discretised sequentially using a numerical method well suited for its particular time scale. We demonstrate that the overall scheme is second-order accurate and provides increased accuracy with less computational work compared to Strang splitting for terrestrial and astrophysical flames. The overall framework also sets the stage for higher-order coupling strategies.

A Highly Accurate Solver for Stiff Ordinary Differential Equations

We introduce a solver for stiff ordinary differential equations 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.

A hybrid parareal spectral deferred corrections method

The parareal algorithm introduced in 2001 by Lions, Maday, and Turinici is an iterative method for the parallelization of the numerical solution of ordinary differential equations or partial differential equations discretized 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.

A high-order upwind control-volume method based on integrated RBFs for fluid-flow problems

This paper is concerned with the development of a high-order upwind conservative discretization method for the simulation of flows of a Newtonian fluid in two dimensions. The fluid-flow domain is discretized using a Cartesian grid from which non-overlapping rectangular control volumes are formed. Line integrals arising from the integration of the diffusion and convection terms over control volumes are evaluated using the middle-point rule. One-dimensional integrated radial basis function schemes using the multiquadric basis function are employed to represent the variations of the field variables along the grid lines. The convection term is effectively treated using an upwind scheme with the deferred-correction strategy. Several highly non-linear test problems governed by the Burgers and the Navier–Stokes equations are simulated, which show that the proposed technique is stable, accurate and converges well. Copyright © 2010 John Wiley & Sons, Ltd.