Second-order Backward Difference Formula Research Articles

A space-time second-order method based on modified two-grid algorithm with second-order backward difference formula for the extended Fisher–Kolmogorov equation

In this paper, a modified two-grid algorithm based on block-centred finite difference method is developed for the fourth-order nonlinear extended Fisher–Kolmogorov equation. To further improve the computational efficiency, an effective second-order accurate backward difference formula is considered. The modified two-grid method based on Newton iteration is constructed to linearize the nonlinear system. The method solves a miniature nonlinear system on a coarse grid accompanying a larger time step to get the numerical solution, then computes a linear system constructed by the previous result with the Taylor expansion on a fine grid accompanying a smaller time step to get the correct numerical solution. Theoretical analysis shows that the modified two-grid algorithm can achieve second-order convergence accuracy both in time and space domain. Several numerical experiments are provided to verify the theoretical result and the high efficiency of this approach. The practical problem illustrates the actual applicable value of the algorithm.

Optimal time two-mesh mixed finite element method for a nonlinear fractional hyperbolic wave model

<abstract><p>In this article, a second-order time discrete algorithm with a shifted parameter $ \theta $ combined with a fast time two-mesh (TT-M) mixed finite element (MFE) scheme was considered to look for the numerical solution of the nonlinear fractional hyperbolic wave model. The second-order backward difference formula including a shifted parameter $ \theta $ (BDF2-$ \theta $) with the weighted and shifted Grünwald difference (WSGD) approximation for fractional derivative was used to discretize time direction at time $ t_{n-\theta} $, the $ H^1 $-Galerkin MFE method was applied to approximate the spatial direction, and the fast TT-M method was used to save computing time of the developed MFE system. A priori error estimates for the fully discrete TT-M MFE system were analyzed and proved in detail, where the second-order space-time convergence rate in both $ L^2 $-norm and $ H^1 $-norm were derived. Detailed numerical algorithms with smooth and weakly regular solutions were provided. Finally, some numerical examples were provided to illustrate the feasibility and effectiveness for our scheme.</p></abstract>

Second-order efficient algorithm for coupled nonlinear model of groundwater transport system

In this paper, an efficient modified two-grid block-centered finite difference algorithm with second-order backward difference formula is presented for coupled nonlinear reaction-diffusion equations of the solutes in the groundwater flow. The main steps of the modified two-grid algorithm include dealing with a small-scale nonlinear problem on the coarse grid with a large time step size and then dealing with a large-scale linear problem on the fine grid with a smaller time step size base on the second-order backward difference formula. Moreover, the theoretical analysis illustrates that the modified two-grid algorithm is able to maintain second convergence order in space-time domain. Several numerical experiments including a practical example are carried out to support the theoretical results and verify that the modified two-grid block-centered finite difference with second-order backward difference formula algorithm can achieve high efficiency while maintaining the second-order accuracy, meanwhile, has a certain applicable value in actual problem.

Thermodynamically consistent hydrodynamic phase-field computational modeling for fluid-structure interaction with moving contact lines

This paper proposes a novel computational modeling approach to investigate the fluid-structure interactions with moving contact lines. By embracing the generalized Onsager principle, a coupled hydrodynamics and phase field system is introduced that can describe the fluid-structure interactions with moving contact lines in a thermodynamically consistent manner. The resulting partial differential equation (PDE) model consists of the Navier-Stokes equation and a nonlinear Allen-Cahn type equation. Volume conservation is enforced through an additional penalty term. A fully-discrete structure-preserving numerical scheme is proposed by combining several techniques to solve this coupled PDE system effectively and accurately. For the temporal discretization, we utilize the supplementary variable method for preserving the thermodynamic structure and the projection approach for reducing the problem size. Then, we use the finite difference method on the staggered grid for spatial discretization. Furthermore, we have rigorously proved that the proposed numerical scheme based on the second-order backward difference formula respects the original energy stability, i.e., the scheme is energy stable. Additionally, with the aid of the supplementary variable method, the resultant scheme can be transformed into a constrained optimization problem, where the solutions of the supplementary variables are the arguments of the objective function that reaches the optimality. Then the augmented Lagrangian method is introduced in part to bring robustness and efficiency to solving such a constrained optimization problem. Finally, various numerical simulations verify the model's capability and demonstrate the scheme's effectiveness, accuracy, and stability.

Computational Modeling of Individual Red Blood Cell Dynamics Using Discrete Flow Composition and Adaptive Time-Stepping Strategies

In this article, we present a finite element method for studying the dynamic behavior of deformable vesicles, which mimic red blood cells, in a non-Newtonian Casson fluid. The fluid membrane, represented by an implicit level-set function, adheres to the Canham–Helfrich model and maintains surface inextensibility constraint through penalty. We propose a two-step time integration scheme that incorporates higher-order accuracy by using an asymmetric composition of discrete flow based on the second-order backward difference formula, followed by a projection onto the real axis. Our framework incorporates variable time steps generated by an appropriate adaptation criterion. We validate our model through numerical simulations against existing experimental and numerical results in the case of purely Newtonian flow. Furthermore, we provide preliminary results demonstrating the influence of the non-Newtonian fluid model on membrane regimes.

A high-order convergence analysis for semi-Lagrangian scheme of the Burgers' equation

<abstract><p>In this article, we provide a comprehensive convergence and stability analysis of a semi-Lagrangian scheme for solving nonlinear Burgers' equations with a high-order spatial discretization. The analysis is for the iteration-free semi-Lagrangian scheme comprising the second-order backward finite difference formula (BDF2) for total derivative and the fourth-order central finite difference for diffusion term along the trajectory. The main highlight of the study is to thoroughly analyze the order of convergence of the discrete $ \ell_2 $-norm error $ \mathcal{O}(h^2+\triangle x^4+ \triangle x^{p+1}/h) $ by managing the relationship between the local truncation errors from each discretization procedure and the interpolation properties with a symmetric high-order discretization of the diffusion term. Furthermore, stability is established by the uniform boundedness of the numerical solution using the discrete Grönwall's Lemma. We provide numerical examples to support the validity of the theoretical convergence and stability analysis for the propounded backward semi-Lagrangian scheme.</p></abstract>

Optimal convergence analysis of a linearized second-order BDF-PPIFE method for semi-linear parabolic interface problems

The article proposes and analyzes the optimal error estimates of a second-order backward difference formula (BDF2) numerical scheme for the semi-linear parabolic interface problems. The partially penalized immersed finite element (PPIFE) methods are used for the spatial discretization to resolve discontinuity of the diffusion coefficient across the interface. The classical extrapolation method is adopted to treat the nonlinear term, which effectively avoids the complicated numerical calculation of the nonlinearity. Our error analysis is based on the corresponding time-discrete system, which neatly splits the error into two parts: the temporal discretization error and the spatial discretization error. Since the spatial discretization error is independent of time step size τ, we can unconditionally derive the optimal error estimates in both L2 norm and semi-H1 norm, while previous works always require the coupling condition of time step and space size. Numerical experiments are given to confirm the theoretical analysis.

The Construction of High-Order Robust Theta Methods with Applications in Subdiffusion Models

An exponential-type function was discovered to transform known difference formulas by involving a shifted parameter θ to approximate fractional calculus operators. In contrast to the known θ methods obtained by polynomial-type transformations, our exponential-type θ methods take the advantage of the fact that they have no restrictions in theory on the range of θ such that the resultant scheme is asymptotically stable. As an application to investigate the subdiffusion problem, the second-order fractional backward difference formula is transformed, and correction terms are designed to maintain the optimal second-order accuracy in time. The obtained exponential-type scheme is robust in that it is accurate even for very small α and can naturally resolve the initial singularity provided θ=−12, both of which are demonstrated rigorously. All theoretical results are confirmed by extensive numerical tests.

Energy-stable method for the Cahn–Hilliard equation in arbitrary domains

Phase transition in an irregular domain is a common phenomenon in the natural world. In this study, we develop novel linear, temporally first- and second-order accurate, and unconditionally energy-stable methods for the Cahn–Hilliard (CH) equation in arbitrary domains. We consider a three-component CH system and fix one component as the complex domain. The other two components are updated to simulate the CH dynamics. The contact angle boundary conditions at the interface among three components are achieved by solving the modified version of the CH equation. The scalar auxiliary variable approach transforms the governing equations into equivalent forms. The implicit Euler method and second-order backward difference formula (BDF2) are used to construct time-marching schemes. The main merits of this approach are as follows: (i) The contact angle condition is implicitly achieved by solving a modified ternary CH model. (ii) The proposed schemes are highly efficient because the nonlinear terms are explicitly treated. (iii) The time-discretized energy dissipation laws can be proven analytically. (iv) The implementation in each time step is easy to follow. Numerical experiments indicate that the proposed schemes demonstrate accuracy, energy stability, and superior performance on the CH dynamics in various domains with complex shapes.

Local discontinuous Galerkin method based on a family of second-order time approximation schemes for fractional mobile/immobile convection-diffusion equations

In this article, we introduce a local discontinuous Galerkin (LDG) method combined with the generalized second-order backward difference formula with a shifted parameter θ (BDF2-θ) to solve the fractional mobile/immobile convection-diffusion equations, where the temporal direction is approximated by the generalized BDF2-θ and the spatial direction is discretized by the LDG method. We prove the stability for the numerical scheme and derive the rigorous error results that are related to the regularity of the solution. In order to investigate the correctness of the theoretical results and the effectiveness of the algorithm, we provide some numerical tests with Pk(k=1,3) elements for periodic boundary conditions and Pk(k=0,1,2,3) elements for compactly supported boundary conditions. Especially, with a comparison to the numerical scheme without the starting part, the corrected scheme yielded by adding the starting part can restore the second-order convergence rate for nonsmooth problems.

Second-Order Time Stepping Scheme Combined with a Mixed Element Method for a 2D Nonlinear Fourth-Order Fractional Integro-Differential Equations

In this article, we study a class of two-dimensional nonlinear fourth-order partial differential equation models with the Riemann–Liouville fractional integral term by using a mixed element method in space and the second-order backward difference formula (BDF2) with the weighted and shifted Grünwald integral (WSGI) formula in time. We introduce an auxiliary variable to transform the nonlinear fourth-order model into a low-order coupled system including two second-order equations and then discretize the resulting equations by the combined method between the BDF2 with the WSGI formula and the mixed finite element method. Further, we derive stability and error results for the fully discrete scheme. Finally, we develop two numerical examples to verify the theoretical results.

Original variables based energy-stable time-dependent auxiliary variable method for the incompressible Navier–Stokes equation

In this study, we develop an efficiently linear and energy-stable method for the incompressible Navier–Stokes equation. A time-dependent Lagrange multiplier is introduced to change the original equation into an equivalent form. Using the equivalent equation, we design a second-order time-accurate scheme based on the second-order backward difference formula (BDF2). The proposed scheme explicitly treats the advection term. In each time iteration, some linear elliptic type equations need to be solved. Therefore, the calculation is highly efficient. Moreover, the time-discretized energy stability with respect to original variables can be easily proved. Various benchmark tests, such as lid-driven cavity flow, Kelvin–Helmholtz instability, and Taylor–Green vortices, are performed to validate the performance of the proposed method.

A time two-grid algorithm for the two dimensional nonlinear fractional PIDE with a weakly singular kernel

The main aim of this paper is to solve the two-dimensional nonlinear fractional partial integro-differential equation (PIDE) with a weakly singular kernel by using the time two-grid finite difference (FD) algorithm. The second-order backward difference formula (BDF) and L1 scheme are used in time. The time two-grid algorithm is constructed to improve the solving efficiency of nonlinear systems. The Newton iteration is used to solve nonlinear discrete system on the coarse grid, and then we apply Lagrangian linear interpolation to attain the function value used in constructing the difference scheme on the fine grid. The second-order finite difference method (FDM) is used in space. The unconditional stability and convergence are attained for the two-grid fully discrete system. Numerical experiments show that the used CPU time for the presented two-grid numerical algorithm is lower than the general finite difference method for solving the nonlinear system.

High-order time-accurate, efficient, and structure-preserving numerical methods for the conservative Swift–Hohenberg model

In this study, we develop high-order time-accurate, efficient, and energy stable schemes for solving the conservative Swift–Hohenberg equation that can be used to describe the L2-gradient flow based phase-field crystal dynamics. By adopting a modified exponential scalar auxiliary variable approach, we first transform the original equations into an expanded system. Based on the expanded system, the first-, second-, and third-order time-accurate schemes are constructed using the backward Euler formula, second-order backward difference formula (BDF2), and third-order backward difference formula (BDF3), respectively. The energy dissipation law can be easily proved with respect to a modified energy. In each time step, the local variable is updated by solving one elliptic type equation and the non-local variables are explicitly computed. The whole algorithm is totally decoupled and easy to implement. Extensive numerical experiments in two- and three-dimensional spaces are performed to show the accuracy, energy stability, and practicability of the proposed schemes.

Error analysis of finite difference/collocation method for the nonlinear coupled parabolic free boundary problem modeling plaque growth in the artery

The main target of this paper is to present a new and efficient method to solve a nonlinear free boundary mathematical model of atherosclerosis. This model consists of three parabolic, one elliptic and one ordinary differential equations that are coupled together and describes the growth of a plaque in the artery. We start our discussion by using the front fixing method to fix the free domain and simplify the model by changing the mixed boundary condition to a Neumann one by applying suitable changes of variables. Then, after employing a finite difference using the second-order backward difference formula (BDF2) and the collocation method on this model, we prove the stability and convergence of methods. Finally, some numerical results are considered to show the efficiency of the method.

Linear, Second-Order Accurate, and Energy Stable Scheme for a Ternary Cahn–Hilliard Model by Using Lagrange Multiplier Approach

We develop a second-order accurate, energy stable, and linear numerical method for a ternary Cahn–Hilliard (CH) model. The proposed scheme is an extension of typical Lagrange multiplier approach for binary CH system. The second-order backward difference formula (BDF2) is applied to construct time discretization. We theoretically prove the mass conservation, unique solvability, and energy stability of the proposed scheme. We efficiently solve the resulting discrete linear system by using a multigrid algorithm. The numerical solutions demonstrate that the proposed scheme is practically stable and second-order accurate in time and space. Moreover, we can use the proposed scheme as an effective solver to calculate the ternary CH equations in ternary phase-field fluid systems.

Analysis of the second-order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection

In this work, we are concerned with the stability and convergence analysis of the second-order backward difference formula (BDF2) with variable steps for the molecular beam epitaxial model without slope selection. We first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step constraint. Then we show that it preserves an energy dissipation law if the adjacent time-step ratios satisfy rk:= τk/τk−1 < 3.561. Moreover, with a novel discrete orthogonal convolution kernels argument and some new estimates on the corresponding positive definite quadratic forms, the L2 norm stability and rigorous error estimates are established, under the same step-ratio constraint that ensures the energy stability, i.e., 0 < rk < 3.561. This is known to be the best result in the literature. We finally adopt an adaptive time-stepping strategy to accelerate the computations of the steady state solution and confirm our theoretical findings by numerical examples.

Unconditionally energy stable second-order numerical schemes for the Functionalized Cahn–Hilliard gradient flow equation based on the SAV approach

In this paper, we devise and analyse three highly efficient second-order accurate (in time) schemes for solving the Functionalized Cahn–Hilliard (FCH) gradient flow equation where an asymmetric double-well potential function is considered. Based on the Scalar Auxiliary Variable (SAV) approach, we construct these schemes by splitting the FCH free energy in a novel and ingenious way. Utilizing the Crank–Nicolson formula, we firstly construct two semi-discrete second-order numerical schemes, which we denote by CN-SAV and CN-SAV-A, respectively. To be more specific, the CN-SAV scheme is constructed based on the fixed time step, while the CN-SAV-A scheme is a variable time step scheme. The BDF2-SAV scheme is another second-order scheme in which the fixed time step should be used. It is designed by applying the second-order backward difference (BDF2) formula. All the constructed schemes are proved to be unconditionally energy stable and uniquely solvable in theory. To the best of our knowledge, the CN-SAV-A scheme is the first unconditionally energy stable, second-order scheme with variable time steps for the FCH gradient flow equation. In addition, an effective adaptive time selection strategy introduced in Christlieb et al., (2014) is slightly modified and then adopted to select the time step for the CN-SAV-A scheme. Finally, several numerical experiments based on the Fourier pseudo-spectral method are carried out in two and three dimensions, respectively, to confirm the numerical accuracy and efficiency of the constructed schemes.

Unsteady magnetohydrodynamic flow of generalized second grade fluid through porous medium with Hall effects on heat and mass transfer

This work investigates the unsteady magnetohydrodynamic flow of generalized second grade fluid through a porous medium with Hall effects on heat and mass transfer. The second grade fluid with a fractional derivative is used for the constitutive equation. A second-order fractional backward difference formula in the temporal direction and a spectral collocation method in the spatial direction are proposed to solve the model numerically. In the numerical implementation, a fast method is applied to decrease the memory requirement and computational cost. The velocity, temperature, and concentration profiles are discussed through graphs. The effects of various parameters on the velocity profiles, temperature field, and concentration field are shown. Results indicate that as the fractional derivative γ increases and the Hall parameter m decreases, the amplitudes of the velocity components decrease.

Unconditional optimal error estimates of linearized second-order BDF Galerkin FEMs for the Landau-Lifshitz equation

In this paper, we establish the unconditionally optimal error estimates of linearized second-order backward difference formula (BDF2) Galerkin finite element methods (FEMs) for the Landau-Lifshitz equation describing the magnetic behavior in ferromagnetic materials. By using the temporal-spatial error splitting techniques, we split the error between the exact solution and the numerical solution into two parts which are called the temporal error and the spatial error. First, we analyze the temporal error by introducing a time-discrete system and derive some regularity of the solution of the time-discrete system. Second, by the above achievements, we obtain the τ-independent spatial error and the boundedness of the numerical solution in L∞-norm. Then, the optimal L2 and H1 error estimates for r-th order FEMs (r=1,2) are derived without any restriction on the time step size. Numerical results in both two and three dimensional spaces are presented to confirm the theoretical predictions and demonstrate the efficiency of the methods.