Two-step Backward Differentiation Formula Research Articles

Temporal error analysis of an unconditionally energy stable second-order BDF scheme for the square phase-field crystal model

In this paper, we first propose and study the second-order time-discrete numerical scheme for the sixth-order nonlinear parabolic problem of the square phase-field crystal model. Then, we demonstrate the two-step backward differentiation formula (BDF-2) scheme with mass conservation and energy dissipation, where the higher order nonlinear term is treated implicitly. Moreover, a rigorous error analysis is presented and we prove the optimal second-order convergence rate O(τ2) in H1- norm, where τ is the time step. Finally, some numerical results are provided to confirm our theoretical analysis.

Modified multi-phase diffuse-interface model for compound droplets in contact with solid

In this study, a novel diffuse-interface (phase-field) model is developed to efficiently describe the dynamics of compound droplets in contact with a solid object. Based on a classical four-component Cahn–Hilliard-type system, we propose modified governing equations, in which the solid is represented by an initially fixed phase. By considering Young's equality between surface tensions and microscale contact angles, equilibrium profiles of diffuse interfaces, and horizontal force balance between contact and interfacial angles, a correction term is derived and added into the phase-field equations to reflect the accurate contact line property for each component. The proposed model can be implemented on Eulerian grids in the absence of complicated treatment on the liquid-solid boundary. The standard finite difference method (FDM) is adopted to perform discretization in space. The linear second-order time-accurate method based on the two-step backward differentiation formula (BDF2) and a stabilization technique are adopted to update the phase-field variables. To accelerate convergence in solving the resulting fully discrete system, we use the linear multigrid method. At each time step, the calculations are completely decoupled. The numerical experiments not only indicate the desired accuracy but also show superior capability in complex geometries. Furthermore, the numerical and analytical results for the compound droplets on a flat solid are in good agreement with each other.

Uniform convergence analysis of the BDF2 scheme on Bakhvalov-type meshes for a singularly perturbed Volterra integro-differential equation

A singularly perturbed Volterra integro-differential problem is considered. At first, this problem is discretized by using the variable two-step backward differentiation formula (BDF2) and the trapezoidal formula on a Bakhvalov-type mesh to approximate the first-order derivative term and the integral term, respectively. Then, the stability and convergence analysis of the proposed numerical method are carried out. It is shown that the proposed numerical method is second-order uniformly convergent with respect to perturbation parameter ɛ in the discrete maximum norm. Finally, the theoretical find is illustrated by numerical experiments.

Error estimate of BDF2 scheme on a Bakhvalov-type mesh for a singularly perturbed Volterra integro-differential equation

<abstract><p>A singularly perturbed Volterra integro-differential problem is considered. The variable two-step backward differentiation formula is used to approximate the first-order derivative term and the trapezoidal formula is used to discretize the integral term. Then, the stability and convergence analysis of the proposed numerical method are proved. It is shown that the proposed scheme is second-order uniformly convergent with respect to perturbation parameter $ \varepsilon $ in the discrete maximum norm. Finally, a numerical experiment verifies the theoretical results.</p></abstract>

On the two-step BDF finite element methods for the incompressible Navier–Stokes problem under boundary conditions of friction type

This article concerns the numerical approximation of the two-dimensional nonstationary Navier–Stokes equations with slip boundary conditions of friction type. It studies the well-posedness and error analysis properties of the numerical scheme in L2-norm for all positive time using the two-step backward differentiation formula (BDF) in time and the finite element approximation in space. Error estimates are proved under feasible assumptions on the regularity of the solution and with the aid of different versions of discrete Grownwall lemmas. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

Analysis of continuous data assimilation scheme for the Navier–Stokes equations using variational multiscale method

In this study, we analyse a continuous data assimilation (CDA) scheme which enables us to combine an observable data with a numerical method to obtain better solutions in which these solutions are also closely similar to the current state of the system. The scheme is applied on a Navier–Stokes system which is discretized with two-step Backward Differentiation Formula (BDF2) in time and finite element in space. In order to improve the accuracy and prevent some non-physical oscillations due to the effect of small viscosity and the dominance of convection, a projection based variational multiscale method (VMS) has also been applied to the system. We present the long-time stability and long-time convergence analyses of the scheme in details and several numerical tests in order to support theoretical findings and demonstrate the promise of the method.

The BDF2-Maruyama method for the stochastic Allen–Cahn equation with multiplicative noise

We investigate the numerical approximation of the stochastic Allen–Cahn equation on a bounded domain D under Dirichlet boundary conditions and with multiplicative noise. The considered numerical method combines the two-step backward differentiation formula (BDF2) for the temporal discretization in conjunction with an abstract Galerkin scheme for the spatial approximation. In dependence on the regularity of the exact solution we derive a rate of convergence for the BDF2-Maruyama method with respect to the root-mean-square error in discrete analogues of the spaces L∞([0,T];L2(D)) and L2([0,T];H01(D)). Our error analysis is based on the variational approach for stochastic evolution equations. Finally, several numerical experiments illustrate our theoretical results, where a finite element method is used as an example for a Galerkin scheme.

Stability of Variable-Step BDF2 and BDF3 Methods

We prove that the two-step backward differentiation formula (BDF) method is stable on arbitrary time grids; while the variable-step three-step backward differentiation formula scheme is stable if almost all adjacent step ratios are less than 2.553. These results relax the severe mesh restrictions in the literature and provide a new understanding of variable-step BDF methods. Our main tools include the discrete orthogonal convolution kernels and an elliptic type matrix norm.

Optimal a posteriori estimators for the variable step-size BDF2 method for linear parabolic equations

Optimal a posteriori error estimates for time discretizations of linear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. Based on second-order BDF reconstructions of the piecewise linear approximate solutions, the optimality of residual-based a posteriori error estimators is proved by using a novel stability inequality when the starting value is computed by the trapezoidal method. With a reasonable choice for the starting step-size, the optimality of the estimators when the starting value is computed by the backward Euler scheme can be also ensured. The effectiveness of the a posteriori error estimators is illustrated by a numerical example.

3D full-time anisotropic TEM modelling using a mixed BDF2/SAI method

Abstract Transient electromagnetic (TEM) data are affected by resistivity anisotropy, which should be considered in 3D modelling. The influence of anisotropy on full-time response is the main focus of this research. For spatial discretisation of an anisotropic model, the mimetic finite volume approach was applied. The accuracy of the shift-and-invert (SAI) Krylov subspace approach and the two-step backward differentiation formula (BDF2) for modelling 3D full-time electromagnetic data has been demonstrated. However, both algorithms require time-consuming calculations. The SAI technique requires a number of projection subspace constructions, whereas the BDF2 algorithm necessitates numerous coefficient matrix decompositions. We proposed a novel mixed BDF2/SAI algorithm in this paper, which combines the advantages of the two algorithms. The on-time response is computed using BDF2, while the off-time response is computed using the SAI-Krylov subspace method. The forward results of a 1D model with a half-sine waveform demonstrated that the new algorithm is accurate and faster than both the BDF2 algorithm and the SAI algorithm. During the full-time period, the forward results of a 3D anisotropic model with half-sine waveform show that abnormal responses can be induced. It was shown that the relative abnormal of ${{{\bf b}}_{\boldsymbol{z}}}$ is higher during the on-time period, while the relative abnormal of $\partial {{{\bf b}}_{\boldsymbol{z}}}/\partial t$ is higher during the off-time period. Furthermore, the change in relative anomaly is more obvious as the anisotropic block rotates around the x-axis. And the larger the rotation angle, the larger the relative anomaly.

Fully discrete a posteriori error estimates for parabolic integro-differential equations using the two-step backward differentiation formula

In this article, finite element a posteriori error estimates for the linear parabolic integro-differential equation using the two-step backward time descretization formula are explored. For space discretization, we use piecewise linear finite element spaces. The Ritz–Volterra reconstruction operator is used as a raw ingredient to obtain the optimal convergence in space. Further, a novel quadratic space–time reconstruction operator, namely BDF2 operator, is introduced to achieve second-order accuracy in time. Numerical results demonstrate the theoretical findings.

Analysis of BDF2 finite difference method for fourth-order integro-differential equation

In this work, the stability and error analysis are presented for fully discrete solutions of the fourth-order differential equation with the multi-term Riemann–Liouville fractional integral. Our numerical scheme is obtained by the standard central difference method in space and the formally two-step backward differentiation formula method and second-order convolution quadrature in time. Optimal order of the numerical scheme in $$L^2$$ -norm is established using the discrete energy method. The analysis is supported by two numerical experiments.

Stability and error estimates for the variable step-size BDF2 method for linear and semilinear parabolic equations

In this paper, stability and error estimates for time discretizations of linear and semilinear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. An affirmative answer is provided to the question: whether the upper bound of step-size ratios for the $l^{\infty }(0,T;H)$ -stability of the BDF2 method for linear and semilinear parabolic equations is identical with the upper bound for the zero-stability. The $l^{\infty }(0,T;V)$ -stability of the variable step-size BDF2 method is also established under more relaxed condition on the ratios of consecutive step-sizes. Based on these stability results, error estimates in several different norms are derived. To utilize the BDF method, the trapezoidal method and the backward Euler scheme are employed to compute the starting value. For the latter choice, order reduction phenomenon of the constant step-size BDF2 method is observed theoretically and numerically in several norms. Numerical results also illustrate the effectiveness of the proposed method for linear and semilinear parabolic equations.

Analysis of adaptive BDF2 scheme for diffusion equations

The variable two-step backward differentiation formula (BDF2) is revisited via a new theoretical framework using the positive semi-definiteness of BDF2 convolution kernels and a class of orthogonal convolution kernels. We prove that, if the adjacent time-step ratios $r_k:=\tau_k/\tau_{k-1}\le(3+\sqrt{17})/2\approx3.561$, the adaptive BDF2 time-stepping scheme for linear reaction-diffusion equations is unconditionally stable and (maybe, first-order) convergent in the $L^2$ norm. The second-order temporal convergence can be recovered if almost all of time-step ratios $r_k\le 1+\sqrt{2}$ or some high-order starting scheme is used. Specially, for linear dissipative diffusion problems, the stable BDF2 method preserves both the energy dissipation law (in the $H^1$ seminorm) and the $L^2$ norm monotonicity at the discrete levels. An example is included to support our analysis.

On Energy Stable, Maximum-Principle Preserving, Second-Order BDF Scheme with Variable Steps for the Allen--Cahn Equation

In this work, we investigate the two-step backward differentiation formula (BDF2) with nonuniform grids for the Allen-Cahn equation. We show that the nonuniform BDF2 scheme is energy stable under the time-step ratio restriction $r_k:=\tau_k/\tau_{k-1}<(3+\sqrt{17})/2\approx3.561.$ Moreover, by developing a novel kernel recombination and complementary technique, we show, for the first time, the discrete maximum principle of BDF2 scheme under the time-step ratio restriction $r_k<1+\sqrt{2}\approx 2.414$ and a practical time step constraint. The second-order rate of convergence in the maximum norm is also presented. Numerical experiments are provided to support the theoretical findings.

On the Variable Two-Step IMEX BDF Method for Parabolic Integro-differential Equations with Nonsmooth Initial Data Arising in Finance

In this paper the implicit-explicit (IMEX) two-step backward differentiation formula (BDF2) method with variable step-size, due to the nonsmoothness of the initial data, is developed for solving parabolic partial integro-differential equations (PIDEs), which describe the jump-diffusion option pricing model in finance. It is shown that the variable step-size IMEX BDF2 method is stable for abstract PIDEs under suitable time step restrictions. Based on the time regularity analysis of abstract PIDEs, the consistency error and the global error bounds for the variable step-size IMEX BDF2 method are provided. After time semidiscretization, spatial differential operators are treated by using finite difference methods, and the jump integral is computed using the composite trapezoidal rule. A local mesh refinement strategy is also considered near the strike price because of the nonsmoothness of the payoff function. Numerical results illustrate the effectiveness of the proposed method for European and American options under jump-diffusion models.

A posteriori estimates for the two-step backward differentiation formula and discrete regularity for the time-dependent Stokes equations

A posteriori estimates for the two-step backward differentiation formula and discrete regularity for the time-dependent Stokes equations

A Review of the Recent Advances Made in the Black-Scholes Models and Respective Solutions Methods

The Black-Scholes model has been a major advance in finance over a period of time; this paper examines this model in some detail, in terms of the latest developments in both analytical and numerical solutions. The paper initially briefly examines historical aspects but quickly moves to a critical analysis of the works in the applications, solutions and implications for the efficiency and use of these. More specifically, this paper critically reviews the existing literature on the proposed exact as well as the numerical solutions to the Black-Scholes model. For this purpose, the exact solution literature on the Black-Scholes model includes speed based solutions and techniques based on valuation issues, time varying instruments and stochastic volatility. Similarly, the key numerical solutions include finite difference methods, the semi-discretization technique, Crank-Nicolson and the R3C scheme, the cubic spline wavelets and multi-wavelet bases method, the two-step backward differentiation formula in the temporal discretization and a High-Order Difference approximation with Identity Expansion (HODIE) scheme and fractional Black-Scholes model (TFBSM) along with Fourier analysis. This analysis reveals that transaction costs, high volatility, illiquid markets and large investor preferences are the key issues of today’s financial derivatives markets, especially after the Global Financial Crisis (GFC). These issues require non-linear solutions to the Black-Scholes models; therefore, Crank-Nicolson and the R3C scheme should be focused upon more by incorporating more and more real-life assumptions of current day trading.

Numerical solution of generalized Black–Scholes model

This paper presents a numerical scheme that approximates the option prices for different option styles, governed by the generalized Black–Scholes equation in its degenerate form. The proposed method uses the HODIE scheme in the spacial direction and the two-step backward differentiation formula in the temporal direction. It is proved that the method has second order convergence in space as well as in time. Numerical experiments validate the theoretical results.

Adaptive multistep time discretization and linearization based on a posteriori error estimates for the Richards equation

We derive some a posteriori error estimates for the Richards equation. This parabolic equation is nonlinear in space and in time, thus its resolution requires fixed-point iterations within each time step. We measure the approximation error with the dual norm of the residual. A computable upper bound of this error consists of several estimators involving adequate reconstructions based on the degrees of freedom of the scheme. The space and time reconstructions are specified for a two-step backward differentiation formula and a discrete duality finite volume scheme. Our strategy to decrease the computational cost relies on an aggregation of the estimators in three components: space discretization, time discretization, and linearization. We propose an algorithm to stop the fixed-point iterations after the linearization error becomes negligible, and to choose the time step in order to balance the time and space errors. We analyze the influence of the parameters of this algorithm on three test cases and quantify the gain obtained in comparison with a classical simulation.