Temporal Step Sizes Research Articles

Compact ADI Difference Scheme for the 2D Time Fractional Nonlinear Schrödinger Equation

In this paper, we will introduce a compact alternating direction implicit (ADI) difference scheme for solving the two-dimensional (2D) time fractional nonlinear Schrödinger equation. The difference scheme is constructed by using the L1−2−3 formula to approximate the Caputo fractional derivative in time and the fourth-order compact difference scheme is adopted in the space direction. The proposed difference scheme with a convergence accuracy of O(τ1+α+hx4+hy4)(α∈(0,1)) is obtained by adding a small term, where τ, hx, hy are the temporal and spatial step sizes, respectively. The convergence and unconditional stability of the difference scheme are obtained. Moreover, numerical experiments are given to verify the accuracy and efficiency of the difference scheme.

Higher order method based on the coupling of local discontinuous Galerkin method with multistep implicit-explicit time-marching for the micropolar Navier–Stokes equations

In this paper, the spatial local discontinuous Galerkin (LDG) approximation coupled with the temporal multistep implicit-explicit (IMEX) evolution for the micropolar Navier–Stokes equations (MNSE) is adopted to construct an efficient fully discrete method. First, the multistep IMEX-LDG methods are constructed up to the third order, which have small computational scale and facilitate implementation to higher orders. Second, the unconditional stability of the fully discrete method on Cartesian grids is derived theoretically, meaning that the temporal step size τ is upper bounded by a positive constant independent of the spatial mesh size h. Last, numerical results for nonlinear MNSE with different boundary conditions are presented, confirming the theoretical validity and effectiveness of the method.

Soliton solutions of derivative nonlinear Schrödinger equations: Conservative schemes and numerical simulation

In this paper, we numerically study soliton solutions of derivative nonlinear Schrödinger equations based on several conservative finite difference methods. All schemes own second-order accuracy with the convergence order O(τ2+h2) in the discrete L∞-norm, where h denotes the spatial step size and τ denotes the temporal step size. We show that difference schemes preserve some discrete counterparts of continuous conservation laws, and all these schemes are solvable. Extensive numerical examples with soliton solutions are carried out to verify the theoretical results. These results manifest that our schemes have potential application to soliton propagation in optical fibers.

High-order approximation of Caputo–Prabhakar derivative with applications to linear and nonlinear fractional diffusion models

Abstract In this study, we devise a high-order numerical scheme to approximate the Caputo–Prabhakar derivative of order α ∈ (0, 1), using an rth-order time stepping Lagrange interpolation polynomial, where 3 ≤ r ∈ N $3\le r\in \mathbb{N}$ . The devised scheme is a generalization of the existing schemes developed earlier. Further, we adopt the discussed scheme for solving a linear time fractional advection–diffusion equation and a nonlinear time fractional reaction–diffusion equation with Dirichlet type boundary conditions. We show that the discussed method is unconditionally stable, uniquely solvable and convergent with convergence order O(τ r+1−α , h 2), where τ and h are the temporal and spatial step sizes, respectively. Without loss of generality, applicability of the discussed method is established by illustrative examples for r = 4, 5.

A spatial sixth-order numerical scheme for solving fractional partial differential equation

In this paper, a spatial sixth-order numerical scheme for solving the time-fractional diffusion equation (TFDE) is proposed. The convergence order of the constructed numerical scheme is O(τ2+h6), where τ and h are the temporal and spatial step sizes, respectively. The stability and error estimation of proposed scheme are given by using Fourier method. Some numerical examples are studied to demonstrate the correctness and effectiveness of the scheme and validate the theoretical analysis.

A linear second-order maximum bound principle preserving finite difference scheme for the generalized Allen–Cahn equation

In this paper, we consider the numerical method for generalized Allen–Cahn equation with nonlinear mobility and convection term. We propose a linear second-order finite difference scheme which preserves the discrete maximum bound principle (MBP). The scheme is discretized by stabilized Crank–Nicolson in time, upwind scheme for convection term and central-difference scheme for diffusion term. We show that the proposed scheme preserves the discrete MBP under some constraints on temporal step size and stabilizing parameter. Optimal L∞-error estimate is obtained for our scheme. Several numerical experiments are performed to validate our theoretical results.

Unconditional optimal [formula omitted]-norm error estimate and superconvergence analysis of a linearized nonconforming finite element variable-time-step BDF2 method for the nonlinear complex Ginzburg–Landau equation

In this paper, a linearized fully discrete scheme combining the variable-time-step two-step backward differentiation formula (VSBDF2) in time and the nonconforming finite element methods (FEMs) in space is constructed and analyzed for the nonlinear complex Ginzburg–Landau equation. A novel convergence analysis approach is proposed, which shows that the H1-norm error of this scheme can reach optimal order in space and sharp second-order convergence in time. The key ingredients of our analysis are temporal–spatial error splitting approach, a new positive definiteness property of discrete orthogonal convolution (DOC) kernels with respect to complex sequences, an energy projection operator and a discrete Laplace operator Δh related to the nonconforming element space, and a relationship between Δh and classical L2 projection operator in complex space. Furthermore, through a specific treatment of the error equation and the combination technique of interpolating operator and projection operator, we prove that the proposed scheme has the global superconvergence result of order O(τ2+h2) in H1-norm for the first time, which further improves the efficiency of numerical computation. Here τ and h are the maximum time and spatial step sizes, respectively. It should be pointed out that all these error results are rigorously established under mild assumptions of the time step, and get rid of the usual grid ratio constraints between temporal and spatial step sizes in linearized scheme. At last, numerical experiments are provided to verify the theoretical analysis.

Nonstandard finite difference methods for a convective predator-prey pursuit and evasion model

One standard and two nonstandard finite difference methods are used to solve a convective predator-prey model consisting of cross-diffusion terms for which no exact solution is known. The model involves a system of strongly coupled nonlinear parabolic partial differential equations. Through some numerical experiments, we observe that reasonable numerical solutions are obtained when the temporal step size satisfies k ≤ 0.0005 with spatial step size h = 0.1 . We construct a nonstandard method using ideas from Anguelov et al. and the scheme is denoted by NSFD1. We obtain four conditions for NSFD1 to replicate the positivity property of the continuous model, but these conditions consist of many parameters. Therefore, an unconditionally positivity-preserving scheme is considered which is denoted as NSFD2. However, we found that NSFD2 is in general not consistent. We obtain conditions under which NSFD2 is both consistent and bounded and some numerical results are presented when these conditions are satisfied.

Time two-grid fitted scheme for the nonlinear time fractional Schrödinger equation with nonsmooth solutions

In this paper, we delve into the regularity properties and the development of an efficient numerical strategy for addressing the nonlinear time-fractional Schrödinger equation. Initially, we embark on an examination of the solution’s regularity, followed by an investigation into regularity enhancement through the application of the Laplace and finite Fourier sine transforms. Subsequently, after the decomposition of u-u0, we introduce an innovative time two-grid fitted scheme designed for the equation’s resolution, which demonstrates unconditional stability and convergence in the L2 norm, achieving the global convergence order of O(τF2+τC4+h2). Here, τF and τC denote the temporal step sizes on the fine and coarse grids, respectively, while h represents the spatial step size. Notably, this is the inaugural application of such an approach for solving the nonlinear time fractional Schrödinger equation. Ultimately, our numerical experiments validate that this method not only retains computational efficiency but also significantly enhances convergence accuracy.

Some novel exact and non-standard finite difference schemes for a class of diffusion–advection–reaction equation

In this paper, a generalized diffusion–advection–reaction equation (GDARE) is considered and by using the solitary wave solution two new class of exact finite difference (EFD) schemes are proposed under certain functional relationship between the temporal and space step sizes. Suitable denominator functions are identified from these EFD schemes and a new non-standard finite difference (NSFD) scheme is developed. This NSFD scheme preserves positivity and boundedness properties of the solution of GDARE. The L 2 , L ∞ errors and CPU times are obtained for the proposed NSFD and EFD schemes to show the proficiency and accuracy of the schemes.

A stabilized SAV difference scheme and its accelerated solver for spatial fractional Cahn–Hilliard equations

A novel energy-stable scheme is proposed to solve the spatial fractional Cahn–Hilliard equations, using the idea of scalar auxiliary variable (SAV) approach and stabilization technique. Thanks to the stabilization technique, it is shown that larger temporal stepsizes can be applied in numerical simulations. Moreover, the proposed SAV finite difference scheme is non-coupled and linearly implicit, which can be efficiently solved by the preconditioned conjugate gradient (PCG) method with a sine transform based preconditioner. Optimal error estimates of the fully-discrete scheme are obtained rigorously. Numerical examples are given to confirm the theoretical results and show the higher efficiency of the proposed scheme than the previous SAV schemes.

Temporal High-Order Accurate Numerical Scheme for the Landau–Lifshitz–Gilbert Equation

In this paper, a family of temporal high-order accurate numerical schemes for the Landau–Lifshitz–Gilbert (LLG) equation is proposed. The proposed schemes are developed utilizing the Gauss–Legendre quadrature method, enabling them to achieve arbitrary high-order time discretization. Furthermore, the geometrical properties of the LLG equation, such as the preservation of constant magnetization magnitude and the Lyapunov structure, are investigated based on the proposed discrete schemes. It is demonstrated that the magnetization magnitude remains constant with an error of (2p+3) order in time when utilizing a (2p+2)th-order discrete scheme. Additionally, the preservation of the Lyapunov structure is achieved with a second-order error in the temporal step size. Numerical experiments and simulations effectively verify the performance of our proposed algorithm and validate our theoretical analysis.

Accounting intra-annual variability in mortality and recruitment in stock status evaluation of Parapenaeopsis stylifera fishery of north-eastern Arabian Sea under moderate data scenario

Length-based integrated mixed effect (LIME) approach was applied for the assessment of Parapenaeopsis stylifera stock, the most prominent penaeid resource along Gujarat coast. Month was selected as a temporal step size to improve estimate precision and capture intra-annual variation in population dynamics. Catch was provided as an additional input to create a moderate data scenario and reduce bias in estimates of reference indices. The values of key life history priors Linf, K, M, t0, Lm50, Lm95 were 160.59 mm total length (TL), 1.68 yr−1, 2.406 yr−1, −0.01 yr, 89.1 mm TL, 95.06 mm TL, respectively. The LWR was established as W = 0.0117*TL2.652. The estimated SPR ranged between 0.33 and 0.70. The monthly fishing mortality was estimated between 0.06 and 0.31. In 37.5% of the observed month, the fishing mortality was above optimum, which by and large coincides with the peak fishing months. The recruitment is continuous, with a coefficient of variation of only 18.5%. Based on these indices, the stock can be considered as healthy or sustainable. A continuous monitoring and collection of abundance data through experimental surveys are recommended as a future course of action.

Shape transformation on curved surfaces using a phase-field model

Shape transformation on evolving curved surfaces is essential for its diverse applications across various scientific disciplines and facilitates the deeper understanding of natural phenomena, the development of new materials, and engineering design optimization. In this study, we develop a phase-field model and its numerical methods for shape transformation on curved surfaces. A modified surface Allen–Cahn (AC) equation with a fidelity term is proposed to simulate shape transformation on curved surfaces. To numerically solve the modified surface AC equation on curved surfaces, we propose a fully explicit scheme and an unconditionally stable method. The proposed stable approach is not only simple and efficient to implement numerically but is also unconditionally stable and eliminates the restrictive temporal time step size constraints. Through numerical experiments using the proposed approach, we demonstrate that shape transformation on evolving curved surfaces can be implemented on both simple and complex curved surfaces.

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies rk := τk/τk−1 < 4.8645. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.

Unconditional analysis of the linearized second-order time-stepping scheme combined with a mixed element method for a nonlinear time fractional fourth-order wave equation

In this article, we present a numerical algorithm for solving a two-dimensional nonlinear fourth-order time fractional wave model, where a Galerkin mixed finite element method yielded by introducing two auxiliary variables v=ut and σ=Δu−f(u) is used in the spatial direction and a second-order θ scheme with the weighted shifted Grünwald difference (WSGD) formula is applied in the time direction. Utilizing the space-time splitting technique without limiting the relationship between spatial mesh size and temporal step size, we derive the unconditional optimal error estimate in L2-norm and the stability result. Further, for verifying the feasibility and effectiveness of our algorithm with or without starting parts, we implement numerical calculations by choosing nonsmooth solutions.

Non-Iterative Calculation of Quasi-Dynamic Energy Flow in the Heat and Electricity Integrated Energy Systems

Quasi-dynamic energy flow calculation is an indispensable tool for the heat and electricity integrated energy system (HE-IES) analysis. One solves the nonlinear partial differential algebraic equations to obtain thermal, hydraulic and electric variations. However, mainstream iteration solvers face the challenges of inefficiency and bad robustness. For one thing, the frequent update and factorization of Jacobian matrices utilize high CPU time. For another, the per-step iteration numbers grow exponentially as the system loading level creeps up. This paper presents a novel non-iterative algorithm for the quasi-dynamic energy flow calculation. The kernel of the proposed algorithm is to transform these nonlinear equations into linear recursive ones, by solving which, we obtain explicit closed-form solutions of unknown variables. In each step, the proposed algorithm requires only one matrix factorization and fixed times of arithmetic operations regardless of the loading levels, so that it achieves small and consistent per-step time costs. A semi-discrete scheme is used in PDE solution to avoid dissipative and dispersive errors that are often overlooked in previous literature. To ensure convergence, we also propose to control the temporal step sizes adaptively by estimating the simulation errors. Case studies showed that the proposed method manifested efficient and robust time performance compared with the iterative algorithms, and meanwhile preserved high accuracy.

Analysis of a High-Accuracy Numerical Method for Time-Fractional Integro-Differential Equations

A high-order finite difference numerical scheme based on the compact difference operator is proposed in this paper for time-fractional partial integro-differential equations with a weakly singular kernel, where the time-fractional derivative term is defined in the Riemann-Liouville sense. Here, the stability and convergence of the constructed compact finite difference scheme are proved in L∞ norm, with the accuracy order O(τ2+h4), where τ and h are temporal and spatial step sizes, respectively. The advantage of this numerical scheme is that arbitrary parameters can be applied to achieve the desired accuracy. Some numerical examples are presented to support the theoretical analysis.

On Error-Based Step Size Control for Discontinuous Galerkin Methods for Compressible Fluid Dynamics

AbstractWe study a temporal step size control of explicit Runge-Kutta (RK) methods for compressible computational fluid dynamics (CFD), including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler equations. We demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy (CFL) number. Our numerical examples show that the error-based step size control is easy to use, robust, and efficient, e.g., for (initial) transient periods, complex geometries, nonlinear shock capturing approaches, and schemes that use nonlinear entropy projections. We demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases, the open source Julia packages Trixi.jl with OrdinaryDiffEq.jl and the C/Fortran code SSDC based on PETSc.

Analysis of Numerical Method for Diffusion Equation with Time-Fractional Caputo–Fabrizio Derivative

In this paper, we propose a high-precision discrete scheme for the time-fractional diffusion equation (TFDE) with Caputo-Fabrizio type. First, a special discrete scheme of C-F derivative is used in time direction and a compact difference operator is used in space direction. Second, we discuss the convergence of the proposed method in discrete L 1 -norm and L 2 -norm. The convergence order of our discrete scheme is O τ 2 + h 4 , where τ and h are the temporal and spatial step sizes, respectively. The aim of this paper is to show that fractional operator without singular term is very useful for improving the accuracy of discrete scheme.