Fourth-order In Space Research Articles

Numerical solution of time-dependent two-parameter singularly perturbed problems via trigonometric quintic B-spline collocation technique

Abstract This paper introduces a novel algorithm for solving time-dependent two-parameter singularly perturbed parabolic convection-diffusion-reaction equations with Dirichlet boundary conditions. The algorithm is formulated using the Crank-Nicolson (CN) scheme for the temporal derivative discretization. Then, the Trigonometric Quintic B-spline (TQBS) is applied to approximate the state variable and its spatial derivatives on nonuniform collocation points. We conducted a comprehensive convergence analysis and stability of the proposed method and proved that the scheme achieved a parameter uniform convergence of approximately fourth order in space and second order in time. To make additional evidence to support the theoretical findings and further assess the proposed method, we implemented the numerical algorithm to solve three test examples. Furthermore, using these test examples, we demonstrated the parameter-uniform convergence of the proposed numerical scheme.

Convergence analysis of a second order numerical scheme for the Flory–Huggins–Cahn–Hilliard–Navier–Stokes system

We present an optimal rate convergence analysis for a second order accurate in time, fully discrete finite difference scheme for the Cahn–Hilliard–Navier–Stokes (CHNS) system, combined with logarithmic Flory–Huggins energy potential. The numerical scheme has been recently proposed, and the positivity-preserving property of the logarithmic arguments, as well as the total energy stability, have been theoretically justified. In this paper, we rigorously prove second order convergence of the proposed numerical scheme, in both time and space. Since the CHNS is a coupled system, the standard ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh2) error estimate could not be easily derived, due to the lack of regularity to control the numerical error associated with the coupled terms. Instead, the ℓ∞(0,T;Hh1)∩ℓ2(0,T;Hh3) error analysis for the phase variable and the ℓ∞(0,T;ℓ2) analysis for the velocity vector, which shares the same regularity as the energy estimate, is more suitable to pass through the nonlinear analysis for the error terms associated with the coupled physical process. Furthermore, the highly nonlinear and singular nature of the logarithmic error terms makes the convergence analysis even more challenging, since a uniform distance between the numerical solution and the singular limit values of is needed for the associated error estimate. Many highly non-standard estimates, such as a higher order asymptotic expansion of the numerical solution (up to the third order accuracy in time and fourth order in space), combined with a rough error estimate (to establish the maximum norm bound for the phase variable), as well as a refined error estimate, have to be carried out to conclude the desired convergence result. To our knowledge, it will be the first work to establish an optimal rate convergence estimate for the Cahn–Hilliard–Navier–Stokes system with a singular energy potential.

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.

An alternating direction implicit compact finite difference scheme for the multi-term time-fractional mixed diffusion and diffusion–wave equation

An alternating direction implicit(ADI) compact finite difference scheme for the two-dimensional multi-term time-fractional mixed diffusion and diffusion–wave equation is given in this paper. By using the Riemann–Liouville fractional integral operator on both sides of the original equation, we obtain a time-fractional integro–differential equation with the order of the highest derivative being one. Using the weighted and shifted Grünwald formulas of Riemann–Liouville fractional derivative and fractional integral, and the Crank–Nicolson approximation, a temporal second-order approximation is obtained for the equivalent integrodifferential system. In the spatial direction, a fourth-order compact approximation is employed to give a fully discrete scheme. Using the splitting techniques for higher dimensional problems, we derive the fully discrete ADI Crank–Nicolson scheme. With the positive definiteness of the related time discrete coefficients and spatial operators, the stability and convergence (second-order in time and fourth-order in space) in the discrete L2-norm are proved by using energy method. Two numerical examples are given to demonstrate the efficiency and accuracy of the proposed method.

Extensional flow of a free film of nematic liquid crystal with moderate elasticity

The human tear film is a multilayer structure in which the dynamics are often strongly affected by a floating lipid layer. That layer has liquid crystalline characteristics and plays important roles in the health of the tear film. Previous models have treated the lipid layer as a Newtonian fluid in extensional flow. Motivated to develop a more realistic treatment, we present a model for the extensional flow of thin sheets of nematic liquid crystal. The rod-like molecules of these substances impart an elastic contribution to the rheology. We rescale a weakly elastic model due to Cummings et al. [“Extensional flow of nematic liquid crystal with an applied electric field,” Eur. J. Appl. Math. 25, 397–423 (2014).] to describe a lipid layer of moderate elasticity. The resulting system of two nonlinear partial differential equations for sheet thickness and axial velocity is fourth order in space, but still represents a significant reduction of the full system. We analyze solutions arising from several different boundary conditions, motivated by the underlying application, with particular focus on dynamics and underlying mechanisms under stretching. We solve the system numerically, via collocation with either finite difference or Chebyshev spectral discretization in space, together with implicit time stepping. At early times, depending on the initial film shape, pressure either aids or opposes extensional flow, which changes the free surface dynamics of the sheet and can lead to patterns reminiscent of those observed in tear films. We contrast this finding with the cases of weak elasticity and Newtonian flow, where the sheet retains the same qualitative shape throughout time.

A new high-order compact and conservative numerical scheme for the generalized symmetric regularized long wave equations

ABSTRACT In this paper, a new three-level implicit and fourth-order compact conservative difference scheme is developed for the generalized symmetric regularized long wave equations. The proposed scheme adopts a new time discretization method. The conservation of the new compact scheme is proved, and the discrete conservation laws are obtained. The priori estimation in the maximum norm is discussed by the discrete energy method and mathematical induction. Based on the priori estimation and Brouwer's fixed point theorem, the unique existence of the difference solution is proved. It is proved that the compact difference scheme is unconditionally convergent and stable in the maximum norm, and its convergence order is the fourth order in space and the second order in time. A decoupled and linearized iterative algorithm is proposed to calculate the nonlinear algebraic system generated by the compact scheme, and its convergence is proved. Several numerical experiments are performed to prove the efficiency and reliability of the proposed numerical scheme and to confirm the results obtained from the theoretical analysis.

BICOMPACT SCHEMES FOR COMPRESSIBLE NAVIER–STOKES EQUATIONS

For the first time, bicompact schemes are generalized to non-stationary Navier–Stokes equations for a compressible heat-conducting fluid. The proposed schemes have an approximation of the fourth order in space and the second order in time, are absolutely stable (in the frozen-coefficients sense), conservative, and efficient. One of the new schemes is tested on several two-dimensional problems. It is shown that when the mesh is refined, the scheme converges with an increased third order. A comparison is made with the WENO5-MR scheme. The superiority of the chosen bicompact scheme in resolving vortices and shock waves, as well as their interaction, is demonstrated.

Split-step quintic B-spline collocation methods for nonlinear Schrödinger equations

<abstract><p>Split-step quintic B-spline collocation (SS5BC) methods are constructed for nonlinear Schrödinger equations in one, two and three dimensions in this paper. For high dimensions, new notations are introduced, which make the schemes more concise and achievable. The solvability, conservation and linear stability are discussed for the proposed methods. Numerical tests are carried out, and the present schemes are numerically verified to be convergent with second-order in time and fourth-order in space. The conserved quantity is also computed which agrees with the exact one. And solitary waves in one, two and three dimensions are simulated numerically which coincide with the exact ones. The SS5BC scheme is compared with the split-step cubic B-spline collocation (SS3BC) method in the numerical tests, and the former scheme is more efficient than the later one. Finally, the SS5BC scheme is also applied to compute Bose-Einstein condensates.</p></abstract>

Compact difference scheme for time-fractional nonlinear fourth-order diffusion equation with time delay

We construct a compact difference scheme for two-dimensional time-fractional nonlinear fourth-order diffusion equation with time delay. By choosing the second-order spatial derivative of the original unknown as an additional variable, the fourth-order problem is transformed into a second-order system. Then the linearized compact difference scheme is formulated by employing the L2−1σ formula and the compact difference operator to approximate temporal Caputo derivative and spatial second-order derivatives, respectively. The unique solvability of the proposed scheme is proved. Then the scheme is proved to be stable and convergent with the rate of second order in time and fourth order in space. The efficiency of the scheme is supported by some numerical experiments.

A second-order energy stable and nonuniform time-stepping scheme for time fractional Burgers' equation

In this paper, an effective finite difference scheme of high order accuracy is proposed for the nonlinear time fractional Burgers' equation. Specifically, we apply the Alikhanov's scheme on graded mesh in the temporal direction and a novel fourth-order compact scheme in the spatial discretization. The proposed scheme resolves initial weak singularity of the solution and preserves high resolution in the space direction. It is rigorously proved that the finite difference scheme is uniquely solvable, discrete variational energy dissipation law and unconditionally stable and convergent in sense of discrete L 2 -norm. With appropriate choice of the grading parameter, the convergence accuracy is min ⁡ { r α , 2 } order in time and fourth order in space, where r is the mesh grading. In the numerical implementation procedure, fast convolution technique and adaptive time-stepping strategy are adopted to accelerate the presented solver and to capture evolution of the solution. Numerical experiments are carried out to verify the validity and effectiveness of the proposed scheme for solving nonlinear time-fractional Burgers' equation.

Efficient and conservative compact difference scheme for the coupled Schrödinger-Boussinesq equations

In this paper, we construct an efficient and conservative compact difference scheme based on the scalar auxiliary variable (SAV) approach for the coupled Schrödinger-Boussinesq (CSB) equations. The presented scheme preserves the discrete modified energy. We prove the convergent rates of second-order in time and fourth-order in space by using the discrete energy method in detail. Some numerical experiments are given to verify our theoretical analysis.

A high‐order and fast scheme with variable time steps for the time‐fractional Black‐Scholes equation

In this paper, a high‐order and fast numerical method is investigated for the time‐fractional Black‐Scholes equation. In order to deal with the typical weak initial singularity of the solution, we construct a finite difference scheme with variable time steps, where the fractional derivative is approximated by the nonuniform Alikhanov formula and the sum‐of‐exponentials (SOE) technique. In the spatial direction, an average approximation with fourth‐order accuracy is employed. The stability and the convergence with second order in time and fourth order in space of the proposed scheme are religiously derived by the energy method. Numerical examples are given to demonstrate the theoretical statement.

High-order compact finite difference scheme with two conserving invariants for the coupled nonlinear Schrödinger–KdV equations

In this paper, a new high-order accurate compact finite difference scheme for the coupled nonlinear Schrödinger–KdV (CNLS–KdV) equations is proposed. Conservation of the discrete number of plasmon and the discrete number of particle are given in detail. Convergence with second-order in time and fourth-order in space of the present scheme are proved by using “cut-off” function technique and discrete energy method. Numerical experiments are given to support the theoretical analysis.

Numerical analysis for compact difference scheme of fractional viscoelastic beam vibration models

In this article, a compact difference method is proposed for fractional viscoelastic beam vibration in stress-displacement form. The solvability, the unconditional stability and the convergence rates of second-order in time and fourth-order in space are rigorously proved for the fractional stress v and the displacement u, respectively, under a mild assumption on the loading f. Numerical experiments are given to verify the theoretic findings. One of the main contributions of this article is to evaluate the positive lower- and upper-bound of the eigenvalues of the Toeplitz matrix Λ generated from the weighted Grünwald difference operator for fractional integral operators, and thus prove that the matrix Λ is positive definite and can induce a norm in a vector space. This finding improves significantly the existing semi-positive definiteness theory of the matrix Λ for fractional differential operators and facilitates the proof of the stability and convergence for the stress v.

A fourth-order compact finite difference scheme for the quantum Zakharov system that perfectly inherits both mass and energy conservation

In this paper, we propose and analyze a new fourth-order compact finite difference scheme for solving the quantum Zakharov system (QZS). The new scheme perfectly inherits the mass and energy conservation possessed by the QZS, while the energy preserved by the existing schemes expressed by two-level's solution at each time step. By using the energy method to analyze an equivalent form of the difference scheme, without any requirement on the grid ratio, we establish rigorously the optimal error estimate of the proposed scheme in the energy norm. The accuracy is shown to be fourth order in space and second order in time. Numerical examples are presented to verify the theoretical results and show the effectiveness of the proposed scheme.

Discrete Maximum Principle and Energy Stability of the Compact Difference Scheme for Two-Dimensional Allen-Cahn Equation

The Allen-Cahn model is discussed mainly in the phase field simulation. The compact difference method will be used to numerically approximate the two-dimensional nonlinear Allen-Cahn equation with initial and boundary value conditions, and then, a fully discrete compact difference scheme with second-order accuracy in time and fourth-order in space is established. And its numerical solution satisfies the discrete maximum principle under the constraints of reasonable space and time steps. On this basis, the energy stability of the scheme is investigated. Finally, numerical examples are given to illustrate the theoretical results.

High-order bicompact schemes for the quasilinear multidimensional diffusion equation

High-order bicompact schemes are designed for the quasilinear multidimensional diffusion equation. They are constructed using the method of lines, the finite volume method, the bicubic (or tricubic) Hermite interpolation. Implicit-explicit and diagonally-implicit Runge-Kutta methods are applied to time integration. The resulting schemes are stable for any ratios between grid steps, are conservative, have approximation of the fourth order in space and the third order in time. To implement the implicit-explicit schemes, an iteration method is proposed based on the approximate factorization of their multidimensional difference operators. This method is modified to implement the schemes with diagonally-implicit Runge-Kutta time stepping. High-order grid convergence and implementation efficiency of the new bicompact schemes are demonstrated on numerical examples.

Compact Difference Method for Time-Fractional Neutral Delay Nonlinear Fourth-Order Equation

In this paper, we present a compact finite difference method for a class of fourth-order nonlinear neutral delay sub-diffusion equations in two-dimensional space. The fourth-order problem is first transformed into a second-order system by a reduced-order method. Next by using compact operator to approximate the second order space derivatives and L2-1σ formula to approximate the time fractional derivative, the difference scheme which is fourth order in space and second order in time is obtained. Then, the existence and uniqueness of solution, the convergence rate of and the stability of the scheme are proved. Finally, numerical results are given to verify the accuracy and validity of the scheme.

Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay

The purpose of this paper is to develop a numerical scheme for the two-dimensional fourth-order fractional subdiffusion equation with variable coefficients and delay. Using the L2−1σ approximation of the time Caputo derivative, a finite difference method with second-order accuracy in the temporal direction is achieved. The novelty of this paper is to introduce a numerical scheme for the problem under consideration with variable coefficients, nonlinear source term, and delay time constant. The numerical results show that the global convergence orders for spatial and time dimensions are approximately fourth order in space and second-order in time.

On the nonlinear wave transmission in a nonlinear continuous hyperbolic regime with Caputo-type temporal fractional derivative

This present manuscript studies a nonlinear hyperbolic model in fractional form which generalizes the nonlinear Klein–Gordon system. The equation under investigation includes the presence of a time-fractional operator of the Caputo type. A space-fractional form of that equation with integer-order temporal derivative has been previously investigated to elucidate the existence of localized wave transmission in relativistic wave equations. Here, we employ numerical techniques to estimate the solution of the fractional equation. The method has consistency of fourth order in space. Meanwhile, the temporal order of consistency is equal to 3−α. We considered herein a sinusoidal perturbation of the medium. The simulations show the existence of the transmission of localized nonlinear modes in some complex fractional media governed by hyperbolic models. Physically, the present work investigates the phenomenon of nonlinear supratransmission in a continuous generalization of linear chains of harmonic oscillators with memory effects. In particular, the present work corroborates the presence of this nonlinear phenomenon in chains of pendula with memory and arrays of Josephson junctions attached through superconducting wires and memory effects.