Second-order Backward Differentiation Formula Research Articles

Analysis and application of a compact multistep ADI solver for a class of nonlinear viscous wave equations

In this article, a compact multistep alternating direction implicit (ADI) method is derived for solving a class of two-dimensional (2D) nonlinear viscous wave equations. This ADI method uses the combination of second-order backward differentiation formula (BDF2) solver with approximation factorization for time integration, and fourth-order Padé approximations to the second spatial derivatives for spatial discretization. It is shown by the discrete energy method that the present ADI method has good stability, and can attain convergence rate of O(Δt2+hx4+hy4) in L2- and H1-norms. Besides, the application of a three-grid Richardson extrapolation algorithm to the ADI solution can make final solution fourth-order accurate in both time and space. Numerical results are given to demonstrate the usefulness and efficiency of the resulting algorithms.

Up to sixth-order accurate A-stable implicit schemes applied to the Discontinuous Galerkin discretized Navier–Stokes equations

In this paper a high-order implicit multi-step method, known in the literature as Two Implicit Advanced Step-point (TIAS) method, has been implemented in a high-order Discontinuous Galerkin (DG) solver for the unsteady Euler and Navier–Stokes equations.Application of the absolute stability condition to this class of multi-step multi-stage time discretization methods allowed to determine formulae coefficients which ensure A-stability up to order 6. The stability properties of such schemes have been verified by considering linear model problems. The dispersion and dissipation errors introduced by TIAS method have been investigated by looking at the analytical solution of the oscillation equation.The performance of the high-order accurate, both in space and time, TIAS-DG scheme has been evaluated by computing three test cases: an isentropic convecting vortex under two different testing conditions and a laminar vortex shedding behind a circular cylinder. To illustrate the effectiveness and the advantages of the proposed high-order time discretization, the results of the fourth- and sixth-order accurate TIAS schemes have been compared with the results obtained using the standard second-order accurate Backward Differentiation Formula, BDF2, and the five stage fourth-order accurate Strong Stability Preserving Runge–Kutta scheme, SSPRK4.

Adaptive discontinuous evolution Galerkin method for dry atmospheric flow

We present a new adaptive genuinely multidimensional method within the framework of the discontinuous Galerkin method. The discontinuous evolution Galerkin (DEG) method couples a discontinuous Galerkin formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of multidimensional hyperbolic systems, such that all of the infinitely many directions of wave propagation are considered explicitly. In order to take into account multiscale phenomena that typically appear in atmospheric flows, nonlinear fluxes are split into a linear part governing the acoustic and gravitational waves and a nonlinear part that models advection. Time integration is realized by the IMEX type approximation using the semi-implicit second-order backward differentiation formula (BDF2). Moreover in order to approximate efficiently small scale phenomena, adaptive mesh refinement using the space filling curves via the AMATOS function library is employed. Four standard meteorological test cases are used to validate the new discontinuous evolution Galerkin method for dry atmospheric convection. Comparisons with the Rusanov flux, a standard one-dimensional approximate Riemann solver used for the flux integration, demonstrate better stability and accuracy, as well as the reliability of the new multidimensional DEG method.

Accuracy improvement of a multistep splitting method for nonlinear viscous wave equations

This paper focuses on a multistep splitting method for a class of nonlinear viscous equations in two spaces, which uses second-order backward differentiation formula (BDF2) combined with approximation factorization for time integration, and second-order centred difference approximation to second derivatives for spatial discretization. By the discrete energy method, it is shown that this splitting method can attain second-order accuracy in both time and space with respect to the discrete L2- and H1-norms. Moreover, for improving computational efficiency, we introduce a Richardson extrapolation method and obtain extrapolation solution of order four in both time and space. Numerical experiments illustrate the accuracy and performance of our algorithms.

A Spectral Deferred Correction Method Applied to the Shallow Water Equations on a Sphere

Abstract Although significant gains have been made in achieving high-order spatial accuracy in global climate modeling, less attention has been given to the impact imposed by low-order temporal discretizations. For long-time simulations, the error accumulation can be significant, indicating a need for higher-order temporal accuracy. A spectral deferred correction (SDC) method is demonstrated of even order, with second- to eighth-order accuracy and A-stability for the temporal discretization of the shallow water equations within the spectral-element High-Order Methods Modeling Environment (HOMME). Because this method is stable and of high order, larger time-step sizes can be taken while still yielding accurate long-time simulations. The spectral deferred correction method has been tested on a suite of popular benchmark problems for the shallow water equations, and when compared to the explicit leapfrog, five-stage Runge–Kutta, and fully implicit (FI) second-order backward differentiation formula (BDF2) time-integration methods, it achieves higher accuracy for the same or larger time-step sizes. One of the benchmark problems, the linear advection of a Gaussian bell height anomaly, is extended to run for longer time periods to mimic climate-length simulations, and the leapfrog integration method exhibited visible degradation for climate length simulations whereas the second-order and higher methods did not. When integrated with higher-order SDC methods, a suite of shallow water test problems is able to replicate the test with better accuracy.

Efficient and Long-Time Accurate Second-Order Methods for the Stokes--Darcy System

We propose and study two second-order in time implicit-explicit methods for the coupled Stokes--Darcy system that governs flows in karst aquifers and other subsurface flow systems. The first method is a combination of a second-order backward differentiation formula and the second-order Gear's extrapolation approach. The second is a combination of the second-order Adams--Moulton and second-order Adams--Bashforth methods. Both algorithms only require the solution of decoupled Stokes and Darcy problems at each time-step. Hence, these schemes are very efficient and can be easily implemented using legacy codes. We establish the unconditional and uniform in time stability for both schemes. The uniform in time stability leads to uniform in time control of the error which is highly desirable for modeling physical processes, e.g., contaminant sequestration and release, that occur over very long time scales. Error estimates for fully discretized schemes using finite element spatial discretizations are derived. Nume...

Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation

AbstractWe present the finite difference/element method for a two-dimensional modified fractional diffusion equation. The analysis is carried out first for the time semi-discrete scheme, and then for the full discrete scheme. The time discretization is based on theL1-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term. We use finite element method for the spatial approximation in full discrete scheme. We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent. Moreover, the optimal convergence rate is obtained. Finally, some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.

Unconditionally stable difference methods for delay partial differential equations

This paper is concerned with the numerical solution of parabolic partial differential equations with time-delay. We focus in particular on the delay dependent stability analysis of difference methods that use a non-constrained mesh, i.e., the time step-size is not required to be a submultiple of the delay. We prove that the fully discrete system unconditionally preserves the delay dependent asymptotic stability of the linear test problem under consideration, when the following discretization is used: a variant of the classical second-order central differences to approximate the diffusion operator, a linear interpolation to approximate the delay argument, and, finally, the trapezoidal rule or the second-order backward differentiation formula to discretize the time derivative. We end the paper with some numerical experiments that confirm the theoretical results.

Consistent SUPG-method for transient transport problems: Stability and convergence

We consider the time/space discretization of the transient advection equation. Discretization in space is performed by the streamline upwind Petrov–Galerkin method and in time we use an A -stable finite difference operator. The formulation is strongly consistent in the sense that the time derivative is included in the stabilization term. Uniform stability of the general formulation is proved under a regularity condition on data, or a moderate inverse CFL-condition that allows for optimal choices of the discretization parameters. Both the backward Euler method (BDF1), the Crank–Nicolson scheme and the second-order backward differentiation formula (BDF2) enter the framework and quasi-optimal convergence is proved for these schemes.

ADI orthogonal spline collocation methods for parabolic partial integro-differential equations

Alternating direction implicit (ADI) orthogonal spline collocation schemes are formulated and analysed for a class of partial integro-differential equations of parabolic type. These techniques are based on the θ-method, for θ ∈ [1/2, 1], (where θ = 1 yields the backward Euler method and θ = 1/2 yields the Crank-Nicolson method) and the second-order backward differentiation formula (BDF) method. For each method, optimal estimates in various norms at each time step are derived and confirmed by results of numerical experiments.

Efficient Fully Implicit Time Integration Methods for Modeling Cardiac Dynamics

Implicit methods are well known to have greater stability than explicit methods for stiff systems, but they often are not used in practice due to perceived computational complexity. This paper applies the backward Euler (BE) method and a second-order one-step two-stage composite backward differentiation formula (C-BDF2) for the monodomain equations arising from mathematically modeling the electrical activity of the heart. The C-BDF2 scheme is an L-stable implicit time integration method and easily implementable. It uses the simplest forward Euler and BE methods as fundamental building blocks. The nonlinear system resulting from application of the BE method for the monodomain equations is solved for the first time by a nonlinear elimination method, which eliminates local and nonsymmetric components by using a Jacobian-free Newton solver, called Newton--Krylov solver. Unlike other fully implicit methods proposed for the monodomain equations in the literature, the Jacobian of the global system after the nonlinear elimination has much smaller size, is symmetric and possibly positive definite, which can be solved efficiently by standard optimal solvers. Numerical results are presented demonstrating that the C-BDF2 scheme can yield accurate results with less CPU times than explicit methods for both a single patch and spatially extended domains.

Convolution quadrature time discretization of fractional diffusion-wave equations

We propose and study a numerical method for time discretization of linear and semilinear integro-partial differential equations that are intermediate between diffusion and wave equations, or are subdiffusive. The method uses convolution quadrature based on the second-order backward differentiation formula. Second-order error bounds of the time discretization and regularity estimates for the solution are shown in a unified way under weak assumptions on the data in a Banach space framework. Numerical experiments illustrate the theoretical results.

Time-stepping schemes for moving grid finite elements applied to reaction–diffusion systems on fixed and growing domains

In this paper, we illustrate the application of time-stepping schemes to reaction–diffusion systems on fixed and continuously growing domains by use of finite element and moving grid finite element methods. We present two schemes for our studies, namely a first-order backward Euler finite differentiation formula coupled with a special form of linearisation of the nonlinear reaction terms (1-SBEM) and a second-order semi-implicit backward finite differentiation formula (2-SBDF) with no linearisation of the reaction terms. Our results conclude that for the type of reaction–diffusion systems considered in this paper, the 1-SBEM is more stable than the 2-SBDF scheme and that the 1-SBEM scheme has a larger region of stability (at least by a factor of 10) than that of the 2-SBDF scheme. As a result, the 1-SBEM scheme becomes a natural choice when solving reaction–diffusion problems on continuously deforming domains.

Finite-difference electrochemical kinetic simulations using the Rosenbrock time integration scheme

Abstract Temporal integration based on multistep backward differentiation formulae may not be the best choice for finite-difference electrochemical kinetic simulations. A one-step, highly accurate, unconditionally stable and non-oscillatory implicit time scheme that also provides temporal error estimates, would be more desirable. The Lawson–Morris–Gourlay extrapolation (LMGE) method, suggested by Strutwolf and Schoeller, partially satisfies these requirements, but it is computationally rather expensive, especially for nonlinear governing equations. The Rosenbrock method, thus far unused in electrochemical simulations, is shown here to be competitive. By combining a three-stage ROWDA3 Rosenbrock method for differential–algebraic equations (DAEs), with a finite-difference spatial discretization of the governing equations, an effective simulation algorithm can be obtained. This is demonstrated for three example kinetic models represented by a single reaction–diffusion equation in one space dimension: potential step experiment, linear sweep voltammetry for a first- and second-order irreversible heterogeneous charge transfer, and linear sweep voltammetry for an enzymatic catalytic reaction mechanism. Numerical properties of the algorithm are compared in calculations with those of the finite-difference methods based on first- and second-order backward differentiation formulae, the Crank–Nicolson (CN) method, and the second-order LMGE method. Fixed, uniform grids are used. Under conditions of a large temporal-to-spatial grid step ratio, important for electrochemical simulations, the third-order accurate ROWDA3 integration proves more efficient than the second-order extrapolation method. In one case it is also more efficient than the method based on the second-order (two-step) backward differentiation formula.

The second-order backward differentiation formula is unconditionally zero-stable

Previous studies of the stability of the second-order backward differentiation formula have concluded that stability is possible only if restrictions are placed on the stepsize ratios, for example, limiting the ratio to some value less than 1 + 2 . However, actual implementations of the BDFs differ from the usual theoretical models of such methods; in particular, practical codes use scaled derivatives (e.g. EPISODE) or backward differences to represent current information about the solution. The representation makes no difference to truncation errors, but it has an important effect of the propagation of roundoff errors and of errors in the solution of the implicit equations. In this paper it is shown that the divided difference implementation of the variable coefficient (variable stepsize extension of the) second-order BDF is zero-stable for unrestricted stepsize ratios.

Infinite subharmonic bifurcation in an Sier epidemic model : Ira B. Schwartz, Mathematics Research Branch, Building 31, Room 4B54A, Ntional Institutes of Health, Bethesda, Maryland 20205 USA; Hal L. Smith, Department of Mathematics, Arizona State University, Tempe, Arizona 85287 USA

In this work, we propose and analyze a new method for the solution of the three-dimensional evolutionary equation with a nonlocal term. Then the method combines Galerkin finite element methods (FEMs) for the spatial discretization with an alternating direction implicit (ADI) algorithm based on the second-order backward differentiation formula (BDF2), where the Riemann-Liouville (R-L) integral term is approximated via second-order convolution quadrature (CQ) rule. The L2-norm stability and convergence are proved. Numerical results confirm the predicted space-time convergence rates.