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In this study, novel numerical methods are presented for solving nonlinear Fisher equations. These equations have a wide range of applications in various scientific and engineering fields, particularly in the biomedical sciences for determining the size of brain tumors. The challenges posed by the nonlinearity of the equations are effectively addressed through the development of numerical techniques. The nonlinearity is tackled using a combination of the method of lines and backward differentiation formulas of varied orders. This method is unconditionally stable, and its accuracy is evaluated using error norms. The methods are successfully validated against test problems with known solutions, demonstrating their superiority through comparative analyses with existing methodologies in the literature.

In this paper, we present a second order parameter-uniform numerical method for a singularly perturbed Volterra delay-integro-differential equation on a Bakhvalov-type mesh. The equation is discretized by using the variable two-step backward differentiation formula of the first derivative term and the trapezoidal formula of the integral term. The stability and convergence of the numerical method in the discrete maximum norm are proved. Finally, the theoretical results are verified by some numerical experiments. [ABSTRACT FROM AUTHOR]

This paper presents a novel numerical approach, which is based on the Method of Lines. This method semi-discretizes the problem and produces a system of ordinary differential equations (ODEs) in time. To solve this system, a stiff solver, BDF2, is used, which yields very precise results. The linearization is handled by the Taylor series method. To validate the numerical method, various test examples are considered. These formulas find extensive applications across various scientific and engineering domains. [ABSTRACT FROM AUTHOR]

We introduce a semi-explicit time-stepping scheme of second order for linear poroelasticity satisfying a weak coupling condition. Here, semi-explicit means that the system, which needs to be solved in each step, decouples and hence improves the computational efficiency. The construction and the convergence proof are based on the connection to a differential equation with two time delays, namely one and two times the step size. Numerical experiments confirm the theoretical results and indicate the applicability to higher-order schemes. [ABSTRACT FROM AUTHOR]

In this study, two numerical schemes with second-order accuracy in time for a modified Ericksen–Leslie model are constructed. The highlight is based on a novel convex splitting method for dealing with the nonlinear potentials, which is integrated with the second-order backward differentiation formula (BDF2) and leap frog method for temporal discretization and the finite element method for spatial discretization. The unconditional energy stability of both schemes is further demonstrated. Finally, several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this study, two numerical schemes with second-order accuracy in time for a modified Ericksen–Leslie model are constructed. The highlight is based on a novel convex splitting method for dealing with the nonlinear potentials, which is integrated with the second-order backward differentiation formula (BDF2) and leap frog method for temporal discretization and the finite element method for spatial discretization. The unconditional energy stability of both schemes is further demonstrated. Finally, several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed schemes.

Magnetization dynamics in magnetic materials is often modeled by the Landau-Lifshitz equation, which is solved numerically in general. In micromagnetics simulations, the computational cost relies heavily on the time-marching scheme and the evaluation of the stray field. In this work, we propose a new method, dubbed as GSPM-BDF2, by combining the advantages of the Gauss-Seidel projection method (GSPM) and the second-order backward differentiation formula scheme (BDF2). Like GSPM, this method is first-order accurate in time and second-order accurate in space, and it is unconditionally stable with respect to the damping parameter. Remarkably, GSPM-BDF2 updates the stray field only once per time step, leading to an efficiency improvement of about 60 % compared with the state-of-the-art of GSPM for micromagnetics simulations. For Standard Problems #4 and #5 from National Institute of Standards and Technology, GSPM-BDF2 reduces the computational time over the popular software OOMMF by 82 % and 96 % , respectively. Thus, the proposed method provides a more efficient choice for micromagnetics simulations. [ABSTRACT FROM AUTHOR]

Singular source terms in sub-diffusion equations may lead to the unboundedness of solutions, which will bring a severe reduction of convergence order of existing time-stepping schemes. In this work, we propose two efficient time-stepping schemes for solving sub-diffusion equations with a class of source terms mildly singular in time. One discretization is based on the Grünwald-Letnikov and backward Euler methods. First-order error estimate with respect to time is rigorously established for singular source terms and nonsmooth initial data. The other scheme derived from the second-order backward differentiation formula (BDF) is proved to possess second-order accuracy in time. Further, piecewise linear finite element and lumped mass finite element discretizations in space are applied and analyzed rigorously. Numerical investigations confirm our theoretical results. [ABSTRACT FROM AUTHOR]

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A second order multirate scheme for the evolutionary coupled Stokes–Darcy problem is proposed and analyzed. The scheme decouples the problem into two smaller subproblems at each time step by exploiting the mismatch in flow activity between the domains. Specifically, the slower Darcy subproblem is integrated with a larger time step compared to the more active Stokes subproblem. This avoids expending unnecessary computational effort on sampling the slow changing Darcy solution. The slow varying Darcy solution is extrapolated/interpolated onto the fine grid for the integration of the Stokes subproblem. The long-time stability and convergence of the method is proved under a time step restriction that depends on physical parameters and the ratio of the time steps in each subdomain. The scheme includes a stabilization term that relaxes the time step restriction. Numerical experiments confirm convergence of the method and the effectiveness of the stabilization technique at relaxing the stability time step restriction. [ABSTRACT FROM AUTHOR]

In this paper, we propose a parallel-in-time algorithm for approximately solving parabolic equations. In particular, we apply the k-step backward differentiation formula and then develop an iterative solver by using the waveform relaxation technique. Each resulting iteration represents a periodic-like system, which could be further solved in parallel by using the diagonalization technique. The convergence of the waveform relaxation iteration is theoretically examined by using the generating function method. The argument could be further applied to the time-fractional subdiffusion equation, whose discretization shares common properties of the standard BDF methods due to the nonlocality of the fractional differential operator. Illustrative numerical results are presented to complement the theoretical analysis. [ABSTRACT FROM AUTHOR]

• First, second, and third order BDF schemes are assessed for the numerical approximation of thermally coupled flows. • The BDF schemes are numerically tested for different Richardson numbers in the range 0

The numerical approximation for the Landau-Lifshitz equation, which models the dynamics of the magnetization in a ferromagnetic material, is taken into consideration. This highly nonlinear equation, with a non-convex constraint, has several equivalent forms, and involves solving an auxiliary problem in the infinite domain. All these features have posed interesting challenges in developing numerical methods. In this paper, we first present a fully discrete semi-implicit method for solving the Landau-Lifshitz equation based on the second-order backward differentiation formula and the one-sided extrapolation (using previous time-step numerical values). A projection step is further used to preserve the length of the magnetization. Subsequently, we provide a rigorous convergence analysis for the fully discrete numerical solution by the introduction of two sets of approximated solutions where one set of solutions solves the Landau-Lifshitz equation and the other is projected onto the unit sphere. Second-order accuracy in both time and space is obtained provided that the spatial step-size is the same order as the temporal step-size. And also, the unique solvability of the numerical solution without any assumption for the step-size in both time and space is theoretically justified, using a monotonicity analysis. All these theoretical properties are verified by numerical examples in both 1D and 3D spaces. [ABSTRACT FROM AUTHOR]

A numerical scheme based on backward differentiation formula (BDF) and generalized differential quadrature method (GDQM) has been developed. The proposed scheme has been employed to investigate three cases of Burgers' equation, one‐dimensional, two‐dimensional, and two‐dimensional coupled models. The results showed effectiveness accuracy in absolute error and error norms L2 and L∞ compared to other methods. The results were evaluated at different values of Reynold's number, Re, and viscosity, ν. [ABSTRACT FROM AUTHOR]

Many hyperbolic and kinetic equations contain a non-stiff convection/transport part and a stiff relaxation/collision part (characterized by the relaxation or mean free time ε). To solve this type of problems, implicit-explicit (IMEX) multistep methods have been widely used and their performance is understood well in the non-stiff regime (ε = O(1)) and limiting regime (ε → 0). However, in the intermediate regime (say, ε =O(Δ t)), uniform accuracy has been reported numerically without a complete theoretical justification (except some asymptotic or stability analysis). In this work, we prove the uniform accuracy - an optimal a priori error bound - of a class of IMEX multistep methods, IMEX backward differentiation formulas (IMEX-BDF), for linear hyperbolic systems with stiff relaxation. The proof is based on the energy estimate with a new multiplier technique. For nonlinear hyperbolic and kinetic equations, we numerically verify the same property using a series of examples. [ABSTRACT FROM AUTHOR]

We propose a stable scheme to solve numerically the Cahn–Hilliard–Hele–Shaw system in three-dimensional space. In the proposed scheme, we discretize the space and time derivative terms by combining with backward differentiation formula, which turns out to be both second-order accurate in space and time. Using this method, a set of linear elliptic equations are solved instead of the complicated and high-order nonlinear equations. We prove that our proposed scheme is uniquely solvable. We use a linear multigrid solver, which is fast and convergent, to solve the resulting discrete system. The numerical tests indicate that our scheme can use a large time step. The accuracy and other capability of the proposed algorithm are demonstrated by various computational results. [ABSTRACT FROM AUTHOR]

We introduce new numerical techniques for solving nonlinear unsteady Burgers equation. The numerical technique involves discretization of all variables except the time variable which converts nonlinear PDE into nonlinear ODE system. Stability of the nonlinear system is verified using Lyapunov’s stability criteria. Implicit stiff solvers backward differentiation formula of order one, two and three are used to solve the nonlinear ODE system. Four test problems are included to show the applicability of introduced numerical techniques. Numerical solutions so obtained are compared with solutions of existing schemes in literature. The proposed numerical schemes are found to be simple, accurate, fast, practical and superior to some existing methods.

Highlights • BDF is presented to solve the space-dependent neutron kinetics equations. • The backward differential formulas with adaptive step size is easily implemented. • The numerical tests show that the current method is accurate and efficient. Abstract Solving three-dimensional space-dependent neutron kinetics equations in hexagonal-z geometry is presented in this paper. The second order backward differentiation formula (BDF) is used to discretize the time derivative terms of the equations. The semi-discretized equations are the form of the fixed source problems, which can be solved by the function expansion nodal method in the hexagonal-z geometry (Cai et al., 2016). The second order accurate time integration BDF with an adaptive step size algorithm is applied. The backward Euler method and the precursor analytic integration method are also implemented and compared with each other. Some numerical tests are used to compare these methods. The numerical results show that BDF is rather more accurate than the backward Euler method and the precursor analytic integration method when the time step is same. The results also show that BDF with adaptive step size is very efficient. [ABSTRACT FROM AUTHOR]

In this paper, we develop two ensemble time-stepping algorithms to solve the convection-diffusion equation with random diffusion coefficients, forcing terms and initial conditions based on the pseudo-spectral stochastic collocation method. The key step of the pseudo-spectral stochastic collocation method is to solve a number of deterministic problems derived from the original stochastic convection-diffusion equation. In general, a common way to solve the set of deterministic problems is by using the backward differentiation formula, which requires us to store the coefficient matrix and right-hand-side vector multiple times, and solve them one by one. However, the proposed algorithm only need to solve a single linear system with one shared coefficient matrix and multiple right-hand-side vectors, reducing both storage required and computational cost of the solution process. The stability and error analysis of the first- and second-order ensemble time-stepping algorithms are provided. Several numerical experiments are presented to confirm the theoretical analyses and verify the feasibility and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We consider a second-order two-step time semi-discretization of the Cahn-Hilliard equation with an analytic nonlinearity. The time-step is chosen small enough so that the pseudo-energy associated withdiscretization is nonincreasing at every time iteration. We prove that the sequence generated by the scheme converges to a steady state as time tends to infinity. We also obtain convergence rates in the energy norm. The proof is based on the Łojasiewicz-Simon inequality.

A second derivative trigonometrically fitted block backward differentiation formula (SDTFBBDF) based on the collocation technique is proposed in this paper. The method is specifically considered for the numerical solution of oscillatory problems. The SDTFBBDF which has one main method and one complementary method as by-products of the continuous second derivative backward differentiation formula (CSDBDF) whose coefficients depend on the frequency and step size. The stability and the convergence of SDTFBBDF are established and the performance of the proposed method is verified on some numerical examples to show its accuracy and efficiency. [ABSTRACT FROM AUTHOR]

In this paper, the Black–Scholes PDE is solved numerically by using the high order numerical method. Fourth-order central scheme and fourth-order compact scheme in space are performed, respectively. The comparison of these two kinds of difference schemes shows that under the same computational accuracy, the compact scheme has simpler stencil, less computation and higher efficiency. The fourth-order backward differentiation formula (BDF4) in time is then applied. However, the overall convergence order of the scheme is less than O ( h 4 + δ 4 ) . The reason is, in option pricing, terminal conditions (also called pay-off function) is not able to be differentiated at the strike price and this problem will spread to the initial time, causing a second-order convergence solution. To tackle this problem, in this paper, the grid refinement method is performed, as a result, the overall rate of convergence could revert to fourth-order. The numerical experiments show that the method in this paper has high precision and high efficiency, thus it can be used as a practical guide for option pricing in financial markets. [ABSTRACT FROM AUTHOR]

A second‐order decoupled algorithm for the nonstationary Stokes‐Darcy system, which allows different time steps in two subregions, is proposed and analyzed in this paper. The algorithm, which is a combination of the second‐order backward differentiation formula and second‐order extrapolation method, uncouples the problem into two decoupled problems per time step. We prove the unconditional stability and long‐time stability of the decoupled scheme with different time steps and derive error estimates of this decoupled algorithm using finite element spatial discretization. Numerical experiments are provided to illustrate the accuracy, effectiveness, and stability of the decoupled algorithm and show its advantages of increasing accuracy and efficiency. [ABSTRACT FROM AUTHOR]

An adequate mathematical model of the dynamics of the nonequilibrium isothermal adsorption of carbon dioxide in the absorption cartridge of the model installation for the regeneration of the atmosphere of autonomous objects (space vehicles, ships, and stations, submarines, capsules, etc.) is proposed. An algorithm for the numerical solution of the problem is developed using the longitudinal variant of the method of straight lines, the method of continuation of the solution with respect to the best parameter, and implicit multi-step rear differentiation formulas with adaptive control of the order of the formulas and the magnitude of the integration step. Parametric calculations of the adsorption of carbon dioxide by sorbent on the basis of iron hydroxide in a wide range of determining parameters were performed: the inlet gas concentration, the gas filtration rate, and the kinetic adsorption coefficient. [ABSTRACT FROM AUTHOR]