Nonlinear Difference Scheme Research Articles

A new nonlinear compact difference scheme for a fourth-order nonlinear Burgers type equation with a weakly singular kernel

A new nonlinear compact difference scheme for a fourth-order nonlinear Burgers type equation with a weakly singular kernel

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.

A BDF3 and new nonlinear fourth-order difference scheme for the generalized viscous Burgers’ equation

This study explores a third-order backward differentiation formula (BDF3) and nonlinear fourth-order difference method (FODM) for solving the generalized viscous Burgers’ equation (GVBE). The BDF3 method is employed for discretizing the time derivative, while the nonlinear term uλux is handled using a newly constructed nonlinear fourth-order difference operator. The spatial second derivative is discretized using a linear fourth-order difference formula. We establish the convergence of the proposed BDF3-FODM by introducing a cut-off function (COF). The main contributions include the construction of a novel nonlinear fourth-order difference operator, addressing the limitation observed in Zhang et al. (2021), Guo et al. (2023), where the order of spatial convergence was restricted to second order. Finally, a numerical test is presented to validate our theoretical analysis.

A new $ \alpha $-robust nonlinear numerical algorithm for the time fractional nonlinear KdV equation

<abstract><p>In this work, we consider an $ \alpha $-robust high-order numerical method for the time fractional nonlinear Korteweg-de Vries (KdV) equation. The time fractional derivatives are discretized by the L1 formula based on the graded meshes. For the spatial derivative, the nonlinear operator is defined to approximate the $ uu_x $, and two coupling equations are obtained by processing the $ u_{xxx} $ with the order reduction method. Finally, the nonlinear difference schemes with order ($ 2-\alpha $) in time and order $ 2 $ precision in space are obtained. This means that we can get a higher precision solution and improve the computational efficiency. The existence and uniqueness of numerical solutions for the proposed nonlinear difference scheme are proved theoretically. It is worth noting the unconditional stability and $ \alpha $-robust stability are also derived. Moreover, the optimal convergence result in the $ L_2 $ norms is attained. Finally, two numerical examples are given, which is consistent with the theoretical analysis.</p></abstract>

Dynamic analysis of wave scenarios based on enhanced numerical models for the good Boussinesq equation

The good Boussinesq equation, a modification of the Boussinesq equation, aims to enhance predictions about shallow water wave behavior. This paper introduces two finite difference schemes for solving the good Boussinesq equation including linear and nonlinear implicit finite difference methods. Both schemes utilize the pseudo-compact difference approach, delivering second-order precision with an additional term to boost numerical simulation accuracy while maintaining the grid points of the standard scheme. These schemes rigorously preserve the critical physical characteristics of the good Boussinesq equation, ensuring more precise representation. We establish the existence of solutions with discrete differences and demonstrate, through the discrete energy method, their uniqueness, stability, and second-order convergence in the maximum norm. Furthermore, we propose an iterative algorithm tailored for the nonlinear implicit finite difference scheme, resulting in significant reductions in computational costs compared to the linear scheme. The results of our numerical experiments demonstrate that our methods are competitive and efficient when compared to difference schemes and previously used methods, while maintaining crucial physical qualities. Furthermore, we run relevant numerical simulations to demonstrate the accuracy of the current methods using evidence from the solitary wave interaction with the initial amplitudes of the wave. It is also suggested that the issue has a critical initial wave amplitude for the interaction of two solitary waves, where blow up occurs in a finite amount of time for initial wave amplitudes greater than the new blow-up criteria value.

Sixth-order finite difference schemes for nonlinear wave equations with variable coefficients in three dimensions

First, a nonlinear difference scheme is proposed to solve the three-dimensional (3D) nonlinear wave equation by combining the correction technique of truncation error remainder in time and a sixth-order finite difference operator in space, resulting in fourth-order accuracy in time and sixth-order accuracy in space. Then, the Richardson extrapolation method is applied to improve the temporal accuracy from the fourth-order to the sixth-order. To enhance computational efficiency, a linearized difference scheme is obtained by linear interpolation based on the nonlinear scheme. In addition, the stability of the linearized scheme is proved. Finally, the accuracy, stability and efficiency of the two proposed schemes are tested numerically.

Conservative second-order finite difference method for Camassa–Holm equation with periodic boundary condition

In this paper, we propose two momentum-preserving finite difference schemes for solving one-dimensional Camassa–Holm equation with periodic boundary conditions. A two-level nonlinear difference scheme and a three-level linearized difference scheme are constructed by using the method of order reduction. For nonlinear scheme, we combine mid-point rule and a specific difference operator, which ensures that our obtained scheme is of second-order convergence in both temporal and spatial directions. For linearized scheme, we apply a linear implicit Crank–Nicolson scheme in the temporal direction, then unique solvability and momentum conservation are analysed in detail. Numerical experiments are provided for Camassa–Holm equation admitting different types of solutions, which demonstrate the convergence order and accuracy of the proposed methods coincide with theoretical analysis. Moreover, numerical results show that the nonlinear scheme exhibits better accuracy for mass conservation, while the linearized scheme is more time-saving in computation.

A new high-accuracy difference method for nonhomogeneous time-fractional Schrödinger equation

The fractional Schrödinger equation is an important fractional nonlinear evolution equation, and the study of its numerical solution has profound scientific meaning and wide application prospects. This paper proposes a new high-accuracy difference method for nonhomogeneous time-fractional Schrödinger equation (TFSE). The Caputo time-fractional derivative is discretized by high-order formula and the fourth-order compact difference approximation is applied for spatial discretization. A new nonlinear compact difference scheme with temporal second-order and spatial fourth-order accuracy is constructed, which is solved by the efficient linearized iterative algorithm. The unconditional stability and convergence are analysed by the energy method. The unique existence and maximum-norm estimate of new compact difference scheme solution are obtained. Theoretical analysis shows that the convergence accuracy of new compact difference scheme is with the strong regularity assumption. Numerical experiments verify theoretical results and indicate that the proposed method is an efficient numerical method for solving TFSE.

A Time Two-Mesh Finite Difference Numerical Scheme for the Symmetric Regularized Long Wave Equation

The symmetric regularized long wave (SRLW) equation is a mathematical model used in many areas of physics; the solution of the SRLW equation can accurately describe the behavior of long waves in shallow water. To approximate the solutions of the equation, a time two-mesh (TT-M) decoupled finite difference numerical scheme is proposed in this paper to improve the efficiency of solving the SRLW equation. Based on the time two-mesh technique and two time-level finite difference method, the proposed scheme can calculate the velocity u(x,t) and density ρ(x,t) in the SRLW equation simultaneously. The linearization process involves a modification similar to the Gauss-Seidel method used for linear systems to improve the accuracy of the calculation to obtain solutions. By using the discrete energy and mathematical induction methods, the convergence results with O(τC2+τF+h2) in the discrete L∞-norm for u(x,t) and in the discrete L2-norm for ρ(x,t) are proved, respectively. The stability of the scheme was also analyzed. Finally, some numerical examples, including error estimate, computational time and preservation of conservation laws, are given to verify the efficiency of the scheme. The numerical results show that the new method preserves conservation laws of four quantities successfully. Furthermore, by comparing with the original two-level nonlinear finite difference scheme, the proposed scheme can save the CPU time.

An implicit nonlinear difference scheme for two-dimensional time-fractional Burgers’ equation with time delay

An implicit nonlinear difference scheme for two-dimensional time-fractional Burgers’ equation with time delay

A Preconditioned Iterative Method for a Multi-State Time-Fractional Linear Complementary Problem in Option Pricing

Fractional derivatives and regime-switching models are widely used in various fields of finance because they can describe the nonlocal properties of the solutions and the changes in the market status, respectively. The regime-switching time-fractional diffusion equations that combine both advantages are also used in European option pricing; however, to our knowledge, American option pricing based on such models and their numerical methods is yet to be studied. Hence, a fast algorithm for solving the multi-state time-fractional linear complementary problem arising from the regime-switching time-fractional American option pricing models is developed in this paper. To construct the solution strategy, the original problem has been converted into a Hamilton–Jacobi–Bellman equation, and a nonlinear finite difference scheme has been proposed to discretize the problem with stability analysis. A policy-Krylov subspace method is developed to solve the nonlinear scheme. Further, to accelerate the convergence rate of the Krylov method, a tri-diagonal preconditioner is proposed with condition number analysis. Numerical experiments are presented to demonstrate the validity of the proposed nonlinear scheme and the efficiency of the proposed preconditioned policy-Krylov subspace method.

A multiple‐relaxation‐time lattice Boltzmann model for Burgers equation

A multiple‐relaxation‐time (MRT) lattice Boltzmann (LB) model is developed to solve Burgers equation. The general MRT‐LB model can be thought as a three‐level nonlinear finite difference scheme. Maximum value principle has been proved, and the existence, uniqueness, and stability parameters are discussed for the three‐level finite difference scheme of MRT‐LB model. Numerical stability analysis shows that the MRT‐LB model is not only a three‐level nonlinear finite difference one but also has the good numerical stability. Numerical solutions have been compared with the exact solutions and other LB models reported in previous studies. The experiments results show that the scheme is accurate and effective for Burgers equation.

Explicit high-order compact difference method for solving nonlinear hyperbolic equations with three types of boundary conditions

In this paper, an explicit compact difference method is constructed to solve nonlinear hyperbolic equations with initial and three types of boundary conditions. Firstly, a nonlinear explicit fourth-order compact difference scheme is derived by truncation error residual correction and the fourth-order Padé approximation. The corresponding scheme for the linear hyperbolic equation is proven to be conditionally stable by the Fourier analysis method. Then, to improve the computational efficiency, the nonlinear scheme is linearized to obtain a linear explicit fourth-order compact difference scheme. Finally, numerical experiments are conducted to verify the stability and accuracy of the presented method.

High-order compact difference methods for solving two-dimensional nonlinear wave equations

<abstract><p>Nonlinear wave equations are widely used in many areas of science and engineering. This paper proposes two high-order compact (HOC) difference schemes with convergence orders of $ O\left({{\tau ^4} + h_x^4 + h_y^4} \right) $ that can be used to solve nonlinear wave equations. The first scheme is a nonlinear compact difference scheme with three temporal levels. After calculating the second-order spatial derivatives of the previous time level using the Padé scheme, numerical solutions of the next time level are obtained through repeated iterations. The second scheme is a three-level linearized compact difference scheme. Unlike the first scheme, iterations are not required and it obtains numerical solutions through an explicit calculation. The two proposed schemes are applied to solutions of the coupled sine-Gordon equations. Finally, some numerical experiments are presented to confirm the effectiveness and accuracy of the proposed schemes.</p></abstract>

Conservative finite difference methods for the Boussinesq paradigm equation

In this paper, we concentrate on developing and analyzing two types of finite difference schemes with energy conservation properties for solving the Boussinesq paradigm equation. Firstly, by introducing the auxiliary potential function ∂u∂t=Δv and lim|x|→∞v=0, the BPE is reformulated as an equivalent system of coupled equations. Then a class of efficient difference schemes are proposed for solving the resulting system, where one proposed scheme is a two-level nonlinear difference scheme and the other is a three-level linearized difference scheme using the invariant energy quadratization technique. Subsequently, we present the theoretical analysis of the proposed energy conservative finite difference schemes, which involves the discrete energy conservation properties, unique solvability and optimal error estimates. By using the discrete energy method, it is proven that the proposed schemes can achieve the optimal convergence rates of O(Δt2+hx2+hy2) in discrete L2-, H1- and L∞-norms. At last, numerical experiments illustrate the physical behaviors and efficiency of the proposed schemes.

An improved fifth-order nonlinear spectral difference scheme for hyperbolic conservation laws

In previous work (Lin et al., 2021), a fifth-order nonlinear spectral difference scheme was proposed for hyperbolic conservation laws. Since the mentioned nonlinear shock-capturing scheme is cell-based, it keeps the compactness of the original linear one and hence the advantage in terms of parallel efficiency. However, it was shown that the scheme encountered defects in stability and computational cost. The current work attempts to address these two defects by employing a positivity preserving method originally developed for discontinuous Galerkin schemes and a trouble cell detecting method based on edge detection. Some numerical results are also presented to demonstrate the effectiveness of the positivity preserving method and the shock detecting method. To be more specific, this work focuses on one- and two-dimensional inviscid Euler equations to validate the algorithms. It is shown that the two aforementioned defects are effectively solved by the current improved scheme.

A Time Two-Mesh Compact Difference Method for the One-Dimensional Nonlinear Schrödinger Equation.

The nonlinear Schrödinger equation is an important model equation in the study of quantum states of physical systems. To improve the computing efficiency, a fast algorithm based on the time two-mesh high-order compact difference scheme for solving the nonlinear Schrödinger equation is studied. The fourth-order compact difference scheme is used to approximate the spatial derivatives and the time two-mesh method is designed for efficiently solving the resulting nonlinear system. Comparing to the existing time two-mesh algorithm, the novelty of the new algorithm is that the fine mesh solution, which becomes available, is also used as the initial guess of the linear system, which can improve the calculation accuracy of fine mesh solutions. Compared to the two-grid finite element methods (or finite difference methods) for nonlinear Schrödinger equations, the numerical calculation of this method is relatively simple, and its two-mesh algorithm is implemented in the temporal direction. Taking advantage of the discrete energy, the result with in the discrete -norm is obtained. Here, and are the temporal parameters on the coarse and fine mesh, respectively, and h is the space step size. Finally, some numerical experiments are conducted to demonstrate its efficiency and accuracy. The numerical results show that the new algorithm gives highly accurate results and preserves conservation laws of charge and energy. Furthermore, by comparing with the standard nonlinear implicit compact difference scheme, it can reduce the CPU time without loss of accuracy.

Numerical solution of nonlinear parabolic systems by block monotone methods

This paper deals with numerical methods for solving systems of nonlinear parabolic problems. Block monotone iterative methods, based on the Jacobi and Gauss–Seidel methods, are in use for solving nonlinear difference schemes which approximate the systems of nonlinear parabolic problems. In the view of the method of upper and lower solutions, two monotone upper and lower sequences of solutions are constructed, where the monotone property ensures the theorem on existence and uniqueness of solutions. Constructions of initial upper and lower solutions are discussed. Numerical experiments are presented.

Numerical Solution of In-Viscid Burger Equation in the Application of Physical Phenomena: The Comparison between Three Numerical Methods

In this paper, upwind approach, Lax–Friedrichs, and Lax–Wendroff schemes are applied for working solution of In-thick Burger equation in the application of physical phenomena and comparing their error norms. First, the given solution sphere is discretized by using an invariant discretization grid point. Next, by using Taylor series expansion, we gain discretized nonlinear difference scheme of given model problem. By rearranging this scheme, we gain three proposed schemes. To verify validity and applicability of proposed techniques, one model illustration with subordinated to three different original conditions that satisfy entropy condition are considered, and solved it at each specific interior grid points of solution interval, by applying all of the techniques. The stability and convergent analysis of present three techniques are also worked by supporting both theoretical and numerical fine statements. The accuracy of present techniques has been measured in the sense of average absolute error, root mean square error, and maximum absolute error norms. Comparisons of numerical gets crimes attained by these three methods are presented in table. Physical behaviors of numerical results are also presented in terms of graphs. As we can see from numerical results given in both tables and graphs, the approximate solution is good agreement with exact solutions. Therefore, the present systems approaches are relatively effective and virtually well suited to approximate the solution of in-viscous Burger equation.

Numerical simulation of a generalized nonlinear derivative Schrödinger equation

<abstract><p>In this paper, three nonlinear finite difference schemes are proposed for solving a generalized nonlinear derivative Schrödinger equation which exposits the propagation of ultrashort pulse through optical fiber and has been illustrated to admit exact soliton-solutions. Two of the three schemes are two-level ones and the third scheme is a three-level one. It is proved that the two-level schemes only preserve the total mass or the total energy in the discrete sense and the three-level scheme preserves both the total mass and total energy. Furthermore, many numerical results are presented to test the conservative properties and convergence rates of the proposed schemes. Several dynamical behaviors including solitary-wave collisions and the first-order rogue wave solution are also simulated, which further illustrates the effectiveness of the proposed method for the generalized nonlinear derivative Schrödinger equation.</p></abstract>