Backward Differentiation Formula Research Articles

Robust Error Estimates for Second‐Order Stabilization Finite Element Method for Navier–Stokes Equations With Small Viscosity and Nonsmooth Initial Data

ABSTRACTIn this article, we study the fully discrete finite element approximation to the two‐dimensional Navier–Stokes equations with a small viscosity coefficient and initial data by using semi‐implicit two‐step backward differentiation formulae (BDF2) method with variable step‐sizes and grad‐div stabilization. The variable step sizes allow us to take special viscosity‐dependent, locally refined time step sizes and thus prove that the variable step‐size BDF2 method can achieve second‐order convergence in time. By adding the grad‐div stabilization, we obtain a robust, fully discrete error bound in which the constants reduce dependence on inverse powers of the viscosity. Numerical results illustrate the effectiveness of the proposed method for such types of Navier–Stokes equations.

Formulation of Variable Step Block Backward Differentiation Formula for the Computation of Nonlinear Fuzzy Differential Equations

The research paper formulates a variable step block backward differentiation formula (VSBBDF) for solving nonlinear fuzzy differential equations (FDEs). Developed to address uncertainties within differential equations by using fuzzy environments, VSBBDF offers a flexible approach to solve equations with triangular fuzzy numbers. This method incorporates a dynamic step-size selection, allowing it to adapt to changes and reduce computational costs while maintaining accuracy. The paper further discusses the stability and convergence properties of VSBBDF, demonstrating that it is both zero-stable and of high accuracy. The results obtained highlight the method's computational efficiency and reliability, particularly when compared with other existing methods for solving nonlinear FDEs.

An Enhanced and Highly Efficient Semi‐Implicit Combined Lagrange Multiplier Approach Preserving Original Energy Law for Dissipative Systems

ABSTRACTRecently, a new Lagrange multiplier approach was introduced by Cheng, Liu, and Shen, which has been broadly used to solve various challenging phase field problems. To design original energy‐stable schemes, they have to solve a nonlinear algebraic equation to determine the introduced Lagrange multiplier, which can be computationally expensive, especially for large‐scale and long‐time simulations involving complex nonlinear terms. In this article, we propose an essential improved technique to modify this issue, which can be seen as a semi‐implicit combined Lagrange multiplier approach. In general, the newly constructed schemes keep all the advantages of the Lagrange multiplier method and significantly reduce the computation costs. Besides, the new proposed second‐order backward difference formula (BDF2) scheme dissipates the original energy, as opposed to a modified energy for the classic Lagrange multiplier approach. In addition, we establish a general framework for extending our constructed method to dissipative systems. Finally, several examples have been presented to demonstrate the effectiveness of the proposed approach.

Optimizing of selective catalytic reduction urea injection and NOx conversion analysis of diesel engine based on higher-order physical model and sequential quadratic programming algorithm

In order to maximize the NOx conversion rate and minimize NH3 slip in the diesel selective catalytic reduction (SCR) system, first, the high-order SCR model was simplified and solved using the trapezoidal method and second-order backward difference formula, while the chemical reaction parameters were optimized using the Gauss-Newton method. The results of the engine bench test show that the mean absolute errors of NH3 and NOx concentrations under steady-state operating conditions were 2.67 ppm and 1.72 ppm, respectively, and 5.46 ppm and 42.50 ppm under severe operating conditions. Second, urea injection rate is optimized based on a high-order model combined with the sequential quadratic programming (SQP) algorithm. The change in NOx conversion rate at different temperatures and air speeds is analyzed, and the results show that the NOx conversion is best at 650 K and 29,000 h⁻1. Third, the experiment validation show that the optimal NH3/NOx ratio is 0.69 at 554 K and 54,200 h⁻1; 0.8; and 1.37 at 746 K and 116,700 h⁻1. Finally, this paper simulates and analyzes the SCR emission characteristics at different temperatures and air speeds when the upstream NOx concentration is 1,000 ppm.

Transparent boundary condition and its high frequency approximation for the Schrödinger equation on a rectangular computational domain

This paper addresses the numerical implementation of the transparent boundary condition (TBC) and its various approximations for the free Schrödinger equation on a rectangular computational domain. In particular, we consider the exact TBC and its spatially local approximation under high frequency assumption along with an appropriate corner condition. For the spatial discretization, we use a Legendre–Galerkin spectral method where Lobatto polynomials serve as the basis. Within variational formalism, we first arrive at the time-continuous dynamical system using spatially discrete form of the initial boundary-value problem incorporating the boundary conditions. This dynamical system is then discretized using various time-stepping methods, namely, the backward-differentiation formula of order 1 and 2 (i.e., BDF1 and BDF2, respectively) and the trapezoidal rule (TR) to obtain a fully discrete system. Next, we extend this approach to the novel Padé based implementation of the TBC presented by Yadav and Vaibhav (2024). Finally, several numerical tests are presented to demonstrate the effectiveness of the boundary maps (incorporating the corner conditions) where we study the stability and convergence behaviour empirically.

New implicit time integration schemes for structural dynamics combining high frequency damping and high second order accuracy

The time integration schemes, GA-23 and GA-234, recently proposed by the authors for first order problems, are extended to solve second-order problems in structural dynamics. The resulting methods maintain unconditional stability and user-controlled high-frequency damping. They offer improved accuracy and exhibit less numerical damping in the low-frequency regime, outperforming the well-known generalised-α method. When the high-frequency damping is maximised the new schemes can be cast in the format of backward difference formulae, offering more accurate alternatives to the standard second order formula. The effectiveness of the new time integration schemes is validated through a number of numerical examples, including a linear elastic cantilever beam, a nonlinear spring pendulum, and wave propagation on a string.

Superconvergence analysis and extrapolation of a BDF2 fully discrete scheme for nonlinear reaction–diffusion equations

The main aim of this paper is to propose a 2-step backward differential formula (BDF2) fully discrete scheme with the bilinear Q11 finite element method (FEM) for the nonlinear reaction–diffusion equation. By use of the combination technique of the element’s interpolation and Ritz projection, and through the interpolation post-processing approach, the superclose and global superconvergence estimates with order O(h2+τ2) in H1-norm are deduced rigorously. Furthermore, with the help of the asymptotic error expansion of the Q11 element, a new suitable fully discrete scheme is developed, and the extrapolation result of order O(h3+τ2) in H1-norm is derived, which is one order higher than that of the above traditional superconvergence estimate with respect to h. Here h is the mesh size and τ is the time step. Finally, some numerical results are provided to verify the theoretical analysis. It seems that the extrapolation of the fully discrete finite element scheme has never been seen in the previous studies.

Higher-order iterative decoupling for poroelasticity

For the iterative decoupling of elliptic–parabolic problems such as poroelasticity, we introduce time discretization schemes up to order five based on the backward differentiation formulae. Its analysis combines techniques known from fixed-point iterations with the convergence analysis of the temporal discretization. As the main result, we show that the convergence depends on the interplay between the time step size and the parameters for the contraction of the iterative scheme. Moreover, this connection is quantified explicitly, which allows for balancing the single error components. Several numerical experiments illustrate and validate the theoretical results, including a three-dimensional example from biomechanics.

Improving the accuracy and consistency of the energy quadratization method with an energy-optimized technique

We propose an energy-optimized invariant energy quadratization method to solve the gradient flow models in this paper, which requires only one linear energy-optimized step to correct the auxiliary variables on each time step. In addition to inheriting the benefits of the baseline and relaxed invariant energy quadratization method, our approach has several other advantages. Firstly, in the process of correcting auxiliary variables, we can directly solve linear programming problem by the energy-optimized technique, which greatly simplifies the nonlinear optimization problem in the previous relaxed invariant energy quadratization method. Secondly, we construct new linear unconditionally energy stable schemes by applying backward differentiation formulas and Crank–Nicolson formula, so that the accuracy in time can reach the first- and second-order. Thirdly, comparing with relaxation technique, the modified energy obtained by energy-optimized technique is closer to the original energy, and the accuracy and consistency of the numerical solutions can be improved. Ample numerical examples have been presented to demonstrate the accuracy, efficiency and energy stability of the proposed schemes.

FINITE ELEMENT ANALYSIS OF LINEARLY EXTRAPOLATED BLENDED BACKWARD DIFFERENCE FORMULA (BLEBDF) FOR THE NATURAL CONVECTION FLOWS

In this paper, we study the stability and convergence of fully discrete finite element method with grad-div stabilization for the incompressible non-isothermal fluid flows. The proposed scheme uses finite element discretization in space and linearly extrapolated blended Backward Differentiation Formula (BLEBDF) in time. We prove the unconditional stability over finite time interval and optimally convergence of the scheme. We also present numerical experiments to verify our theoretical convergence rates and show the reliability of the scheme.

Local discontinuous Galerkin methods with implicit–explicit BDF time marching for Newell–Whitehead–Segel equations

The Newell–Whitehead–Segel type equations with time-dependent Dirichlet boundary conditions are solved by the local discontinuous Galerkin (LDG) method coupled with the implicit–explicit backward difference formulas (IMEX-BDF). With a suitable setting of numerical fluxes and by the aid of the multiplier technique and the a priori error assumption technique, the optimal error estimate for the corresponding fully discrete LDG-IMEX-BDF schemes is obtained by energy analysis, under the condition τ ≤ C h 1 / s , where h and τ are mesh size and time step, respectively, the positive constant C is independent of h, and s = 1 , … , 5 is the order of the IMEX-BDF method. Numerical experiments are also presented to verify the accuracy of the considered schemes.

Dynamically regularized Lagrange multiplier schemes with energy dissipation for the incompressible Navier-Stokes equations

In this paper, we present efficient numerical schemes based on the Lagrange multiplier approach for the Navier-Stokes equations. By introducing a dynamic equation (involving the kinetic energy, the Lagrange multiplier, and a regularization parameter), we form a new system which incorporates the energy evolution process but is still equivalent to the original equations. Such nonlinear system is then discretized in time based on the backward differentiation formulas, resulting in a dynamically regularized Lagrange multiplier (DRLM) method. First- and second-order DRLM schemes are derived and shown to be unconditionally energy stable with respect to the original variables. The proposed schemes require only the solutions of two linear Stokes systems and a scalar quadratic equation at each time step. Moreover, with the introduction of the regularization parameter, the Lagrange multiplier can be uniquely determined from the quadratic equation, even with large time step sizes, without affecting accuracy and stability of the numerical solutions. Fully discrete energy stability is also proved with the Marker-and-Cell (MAC) discretization in space. Various numerical experiments in two and three dimensions verify the convergence and energy dissipation as well as demonstrate the accuracy and robustness of the proposed DRLM schemes.

Enhancing Accuracy and Efficiency in Stiff ODE Integration Using Variable Step Diagonal BBDF Approaches

Recent advancements in mathematical modelling have uncovered a growing number of systems exhibiting stiffness, a phenomenon that challenges the effectiveness of traditional numerical methods. Motivated by the need for more robust numerical techniques to address this issue, this paper presents an enhanced version of the Diagonally Block Backward Differentiation Formula (BBDF) that incorporates intermediate points, known as off-step points, to improve the accuracy and efficiency of solutions for stiff ordinary differential equations (ODEs). The new scheme leverages an adaptive step-size strategy to refine accuracy and efficiency between regular and off-grid integration steps. Theoretical analysis confirms that the proposed scheme is an A-stable and convergent method, as it satisfies the fundamental criteria of consistency, zero-stability, and A-stability. Numerical experiments on single and multivariable systems across varying time scales demonstrate significant improvements in solving stiff ODEs compared to existing techniques. Therefore, the new proposed method is an effective solver for stiff ODEs.

Optimization control strategy for diesel urea-selective catalytic reduction (SCR) urea injection based on the interior point method (IP) + higher order SCR model

In order to ensure that the NOx and NH3 emissions of diesel engines comply with the Euro VII emission standards, this paper optimizes the urea injection using a SCR model combined with the interior point method. First, the equations of the higher-order SCR model were simplified by the variable substitution method and the line method and solved by the backward difference formula to improve computational accuracy. The Levenberg-Marquardt algorithm was used to calibrate the chemical reaction rate parameters to optimize model performance. Second, an optimization cost function was designed to optimize the upstream NH3 concentration in combination with the interior point method and the higher-order SCR model to meet the requirements of Euro VII emission standards. Third, the model effect was verified by performing WHTC hot-start testing on an engine complying with the National VI emission standard. The results show that the average error between the measured NOx concentration and the calculated is 1.49 ppm, which verifies the high accuracy of the model. Finally, the effects of the interior point method, NSGA-II, and MOPSO algorithms in optimizing the upstream NH3 concentration were compared. The results showed that the interior point method outperformed the other algorithms in terms of optimization time and effect, with an optimization time of 149 s, a NOx conversion rate of 97.36 %, and an average downstream NH3 concentration of 4.65 ppm. This achieved a high NOx conversion rate and low NH3 slip in accordance with the requirements of the Euro VII regulation.

Unconditional superconvergence analysis of a new energy stable nonconforming BDF2 mixed finite element method for BBM–Burgers equation

The focus of this paper is to establish a new energy stable 2-step backward differentiation formula (BDF2) fully-discrete mixed finite element method (MFEM) for the Benjamin–Bona–Mahony–Burgers (BBM–Burgers) equation with the nonconforming rectangular EQ1rot element and zero-order Raviart–Thomas (R–T) element (EQ1rot/Q10×Q01). Based on the energy stable property, the existence and uniqueness of the numerical solution are proved by the Brouwer fixed point theorem. Subsequently, with the assistance of some typical properties of this element pair and the interpolation post-processing approach, the unconditional superclose and superconvergence results with O(h2+τ2) are obtained directly without any constraints between the spatial partition parameter h and the time step τ. It is worthy to mention that the method and analysis presented herein are very different from the time–space splitting approach utilized in the previous studies and simplify the implement. Finally, two numerical experiments are executed to validate the theoretical analysis.

Unconditional error analysis of weighted implicit-explicit virtual element method for nonlinear neutral delay-reaction–diffusion equation

In this paper, we develop a virtual element method in space for the nonlinear neutral delay-reaction–diffusion equation, while a weighted implicit-explicit scheme is utilized in time. The nonlinear term is adopted by using the Newton linearized method. The calculation efficiency is improved using an implicit scheme to analyze linear terms and an explicit scheme to deal with nonlinear terms. Then, the time–space error splitting technique and G-stability are creatively combined to rigorously analyze the unconditionally optimal convergence results of the numerical scheme without any restrictions on the mesh ratio. Finally, numerical examples on a set of polygonal meshes are provided to confirm the theoretical results. In particular, when the weighted parameter is taken θ=12 (or θ=1), the method degenerates into the Crank–Nicolson (or second-order backward differential formula (BDF2)) scheme.

Implicit EXP-RBF techniques for modeling unsaturated flow through soils with water uptake by plant roots

Modeling unsaturated flow through soils with water uptake by plant root has many applications in agriculture and water resources management. In this study, our aim is to develop efficient numerical techniques for solving the Richards equation with a sink term due to plant root water uptake. The Feddes model is used for water absorption by plant roots, and the van-Genuchten model is employed for capillary pressure. We introduce a numerical approach that combines the localized exponential radial basis function (EXP-RBF) method for space and the second-order backward differentiation formula (BDF2) for temporal discretization. The localized RBF methods eliminate the need for mesh generation and avoid ill-conditioning problems. This approach yields a sparse matrix for the global system, optimizing memory usage and computational time. The proposed implicit EXP-RBF techniques have advantages in terms of accuracy and computational efficiency thanks to the use of BDF2 and the localized RBF method. Modified Picards iteration method for the mixed form of the Richards equation is employed to linearize the system. Various numerical experiments are conducted to validate the proposed numerical model of infiltration with plant root water absorption. The obtained results conclusively demonstrate the effectiveness of the proposed numerical model in accurately predicting soil moisture dynamics under water uptake by plant roots. The proposed numerical techniques can be incorporated in the numerical models where unsaturated flows and water uptake by plant roots are involved such as in hydrology, agriculture, and water management.

Wall-Modeled Large-Eddy Simulation Method for Unstructured-Grid Navier–Stokes Solvers

This paper reports on the implementation and assessment of a wall-modeled large-eddy simulation (WMLES) methodology in an unstructured-grid, node-centered flow solver, FUN3D, that is developed and supported at the NASA Langley Research Center. Finite-volume (FV) and finite-element (FE) discretization schemes considered in the study provide formal second-order spatial accuracy. Large-eddy simulations (LES) resolve large-scale turbulent-flow features, and small-scale effects are modeled using the Vreman subgrid-scale model. At solid-wall boundaries, a shear-stress model is employed to provide a proper boundary-flux closure. The nonlinear equations are integrated in time using either an optimized backward difference formula or an implicit multistage Runge–Kutta temporal scheme. The implicit equations at each time step are solved by strong nonlinear iteration schemes. WMLES demonstrations are shown for two high-lift configurations, namely, the McDonnell Douglas 30P30N multi-element airfoil and a NASA High-Lift Common Research Model. Results show that the WMLES approaches implemented in the FV and FE discretization methods produce consistent solutions and are capable of capturing key aerodynamic characteristics and flow structures for high-lift configurations at a wide range of angles of attack, including maximum-lift conditions. In the 30P30N example, correct trends in the variations of integrated aerodynamic forces and moments, surface pressure distributions, and boundary-layer profiles are captured as the Reynolds number is increased.

Spatially and temporally high-order dynamic nonlinear variational multiscale methods for generalized Newtonian flows

Non-Newtonian fluids are of interest in industrial sectors, biological problems and other natural phenomena. This work proposes rheologically-dependent, spatially and temporally high-order non-residual stabilized finite element formulations. The accuracy of the methods is assessed by tackling highly-convective time-dependent power-law flows. The spatial approximation uses Lagrangian finite elements up to fourth order. The temporal integration is done via backward differentiation formulas of order one, two and three. A key aspect of our work is using a non-residual orthogonal variational multiscale (VMS) formulation to stabilize dominant convection and to allow equal-order interpolation of velocity and pressure. Our VMS method uses dynamic nonlinear subscales, which have not been tested so far for generalized Newtonian fluids. In this work, the use of high-order temporal discretizations for the subscale components is systematically evaluated. Numerical experiments consider the flow over a confined cylinder for Reynolds numbers between 40 and 400 and power-law indices between 0.4 and 1.8. Numerical testing demonstrates the method to be stable in all combinations of spatial and temporal orders. Our results show that using high-order spatial discretizations more accurately approximates boundary layers and viscosity fields. Moreover, higher temporal orders allow using larger time steps while still capturing highly dynamic behaviors with better resolution in frequency spectra.

A New Methodology for the Development of Efficient Multistep Methods for First–Order IVPs with Oscillating Solutions IV: The Case of the Backward Differentiation Formulae

A theory for the calculation of the phase–lag and amplification–factor for explicit and implicit multistep techniques for first–order differential equations was recently established by the author. His presentation also covered how the approaches’ efficacy is affected by the elimination of the phase–lag and amplification–factor derivatives. This paper will apply the theory for computing the phase–lag and amplification–factor, originally developed for implicit multistep methods, to a subset of implicit methods, called backward differentiation formulae (BDF), and will examine the impact of the phase–lag and amplification–factor derivatives on the efficiency of these strategies. Next, we will show you the stability zones of these brand-new approaches. Lastly, we will discuss the results of numerical experiments and draw some conclusions about the established approaches.