Nonlinear Finite Difference Scheme Research Articles

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.

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.

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>

A lifting method for hyperbolic equations with δ-singularities

In this paper, we propose a method to lift the solution space of hyperbolic equations with δ-singularities such that dealing with the annoying δ-singularities directly is avoided. Thus, the easily implemented finite difference schemes can be employed conveniently for solving such problems with δ-singularities. In particular, we consider a fifth-order nonlinear finite difference scheme that can capture discontinuities robustly. To demonstrate the effectiveness of the proposed method, numerical examples are presented. Additionally, a novel splitting method is also proposed to extend the lifting method to high dimensions.

Trapping of two-component light bullets in a gradient waveguide with normal group dispersion.

In this paper we consider the process of the second harmonic generation in a gradient waveguide, taking into account diffraction and relatively weak temporal dispersion. Using the slowly varying envelope approximation and neglecting the dispersion of the nonlinear part of the response of the medium we obtain the system of parabolic equations for the envelopes of both harmonics. We also derive integrals of motion of this system. To solve it numerically we construct a nonlinear finite-difference scheme based on the Crank-Nicolson method preserving the integrals. Primarily, we focus our investigations on the processes of a two-component light bullets generation. We demonstrate that the generation of a coupled pair is possible in a planar waveguide even at normal group velocity dispersion.

A bounded and monotone finite-difference solution of a hyperbolic Burgers–Fisher equation

In this work, a nonlinear finite-difference scheme is provided to approximate the solutions of a hyperbolic generalization of the Burgers–Fisher equation from population dynamics. The model under study is a partial differential equation with nonlinear advection, reaction and damping terms. The existence of some traveling-wave solutions for this model has been established in the literature. In the present manuscript, we investigate the capability of our technique to preserve some of the most important features of those solutions, namely, the positivity, the boundedness and the monotonicity. The finite-difference approach followed in this work employs the exact solutions to prescribe the initial-boundary data. In addition to providing good approximations to the analytical solutions, our simulations suggest that the method is also capable of preserving the mathematical features of interest.

Solution of Optimal Harvesting Problem by Finite Difference Approximations of Size-Structured Population Model

We solve numerically a forest management optimization problem governed by a nonlinear partial differential equation (PDE), which is a size-structured population model. The formulated problem is supplemented with a natural constraint for a solution to be non-negative. PDE is approximated by an explicit or implicit in time finite difference scheme, whereas the cost function is taken from the very beginning in the finite-dimensional form used in practice. We prove the stability of the constructed nonlinear finite difference schemes on the set of non-negative vectors and the solvability of the formulated discrete optimal control problems. The gradient information is derived by constructing discrete adjoint state equations. The projected gradient method is used for finding the extremal points. The results of numerical testing for several real problems show good agreement with the known results and confirm the theoretical statements.

An efficient solving method for nonlinear convection diffusion equation

PurposeThis paper aims to provide a well-behaved nonlinear scheme and accelerating iteration for the nonlinear convection diffusion equation with fundamental properties illustrated.Design/methodology/approachA nonlinear finite difference scheme is studied with fully implicit (FI) discretization used to acquire accurate simulation. A Picard–Newton (PN) iteration with a quadratic convergent ratio is designed to realize fast solution. Theoretical analysis is performed using the discrete function analysis technique. By adopting a novel induction hypothesis reasoning technique, the L∞ (H1) convergence of the scheme is proved despite the difficulty because of the combination of conservative diffusion and convection operator. Other properties are established consequently. Furthermore, the algorithm is extended from first-order temporal accuracy to second-order temporal accuracy.FindingsTheoretical analysis shows that each of the two FI schemes is stable, its solution exists uniquely and has second-order spatial and first/second-order temporal accuracy. The corresponding PN iteration has the same order of accuracy and quadratic convergent speed. Numerical tests verify the conclusions and demonstrate the high accuracy and efficiency of the algorithms. Remarkable acceleration is gained.Practical implicationsThe numerical method provides theoretical and technical support to accelerate resolving convection diffusion, non-equilibrium radiation diffusion and radiation transport problems.Originality/valueThe FI schemes and iterations for the convection diffusion problem are proposed with their properties rigorously analyzed. The induction hypothesis reasoning method here differs with those for linearization schemes and is applicable to other nonlinear problems.

Energy Preserving Finite Difference Scheme for Sixth Order Boussinesq Equation

We study the general Boussinesq equation with a sixth order dispersion term. For the numerical solution of this problem a conservative, nonlinear finite difference scheme is constructed. The scheme is of second order of approximation. The implementation of the scheme is based on an appropriate splitting. The numerical experiments show that this scheme has second order of convergence in space and time.

Spectral analysis of nonlinear finite difference discretizations

The objective of the present work is to provide a theoretical analysis of the spectral properties of a selection of nonlinear finite difference schemes. The Nonlinear Spectral Analysis (NSA) of Fauconnier and Dick (2011) [6] is applied in order to study the statistical behavior of the modified wavenumber of TVD schemes in combination with five common limiter functions and also the 5th-order WENO scheme, for a large set of synthetic scalar fields with prescribed energy spectrum and random phase. The theoretical observations are verified numerically for the linear propagation of a broadband wave.

Nonlinear scheme with high accuracy for nonlinear coupled parabolic–hyperbolic system

A nonlinear finite difference scheme with high accuracy is studied for a class of two-dimensional nonlinear coupled parabolic–hyperbolic system. Rigorous theoretical analysis is made for the stability and convergence properties of the scheme, which shows it is unconditionally stable and convergent with second order rate for both spatial and temporal variables. In the argument of theoretical results, difficulties arising from the nonlinearity and coupling between parabolic and hyperbolic equations are overcome, by an ingenious use of the method of energy estimation and inductive hypothesis reasoning. The reasoning method here differs from those used for linear implicit schemes, and can be widely applied to the studies of stability and convergence for a variety of nonlinear schemes for nonlinear PDE problems. Numerical tests verify the results of the theoretical analysis. Particularly it is shown that the scheme is more accurate and faster than a previous two-level nonlinear scheme with first order temporal accuracy.

Convergence of a nonlinear finite difference scheme for the Kuramoto–Tsuzuki equation

A nonlinear finite difference scheme is studied for solving the Kuramoto–Tsuzuki equation. Because the maximum estimate of the numerical solution can not be obtained directly, it is difficult to prove the stability and convergence of the scheme. In this paper, we introduce the Brouwer-type fixed point theorem and induction argument to prove the unique existence and convergence of the nonlinear scheme. An iterative algorithm is proposed for solving the nonlinear scheme, and its convergence is proved. Based on the iterative algorithm, some linearized schemes are presented. Numerical examples are carried out to verify the correction of the theory analysis. The extrapolation technique is applied to improve the accuracy of the schemes, and some interesting results are obtained.

A modified numerical scheme for the cubic Schrödinger equation

A numerical scheme based on the use of rational approximants in a two-time level recurrence relation is applied to the cubic Schrodinger equation. The resulting nonlinear finite-difference scheme, which is analyzed for stability, is solved using an already known modified predictor-corrector (MPC) scheme. The efficiency of the proposed method is tested to the single and the double-soliton waves and the results arising from the experiments are compared with the relevant ones known in the available bibliography. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 608–620, 2011

ENO adaptive method for solving one-dimensional conservation laws

In this work an efficient third order non-linear finite difference scheme for solving adaptively hyperbolic systems of one-dimensional conservation laws is developed. The method is based on applying to the solution of the differential equation an interpolating wavelet transform at each time step, generating a multilevel representation for the solution, which is thresholded and a sparse point representation is generated. The numerical fluxes obtained by a Lax–Friedrichs flux splitting are evaluated on the sparse grid by an essentially non-oscillatory (ENO) approximation, which chooses the locally smoothest stencil among all the possibilities for each point of the sparse grid. The time evolution of the differential operator is done on this sparse representation by a total variation diminishing (TVD) Runge–Kutta method. Four classical examples of initial value problems for the Euler equations of gas dynamics are accurately solved and their sparse solutions are analyzed with respect to the threshold parameters, confirming the efficiency of the wavelet transform as an adaptive grid generation technique.

Solitary-wave propagation and interactions for the ‘good’ Boussinesq equation

A third-order rational approximant in a three-time level reccurence relation is applied successfully to the ‘good’ Boussinesq equation, already known in the literature. The resulting nonlinear finite-difference scheme, which is analysed for stability, is solved using a predictor–corrector (P–C) scheme, in which the predictor and corrector are both explicit schemes. This P–C scheme is accelerated by a modifed P–C (MPC) in which the already evaluated values are used for the corrector. The behaviour of both the P–C and MPC schemes is tested numerically on the single- and double-soliton waves, and the results from the experiments are compared with that in the literature.

A moving‐mesh splitting scheme for 2‐dimensional quenching problems

AbstractA novel moving‐mesh adaptive finite difference method is proposed for solving 2‐dimensional quenching differential equations of the form ut = uxx + uyy + (κ – u)–2. Splitting strategies equipped with proper arc‐length adaptations are utilized to achieve the computational efficiency as well as accuracy of the numerical solution of the potentially singular problems. Important results including the monotonicity of the nonlinear finite difference scheme are given. Simulation validations are provided to further demonstrate our conclusions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A second order numerical scheme for the solution of the one-dimensional Boussinesq equation

A predictor–corrector (P-C) scheme is applied successfully to a nonlinear method arising from the use of rational approximants to the matrix-exponential term in a three-time level recurrence relation. The resulting nonlinear finite-difference scheme, which is analyzed for local truncation error and stability, is solved using a P-C scheme, in which the predictor and the corrector are explicit schemes of order 2. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behaviour of the P-C/MPC schemes is tested numerically on the Boussinesq equation already known from the bibliography free of boundary conditions. The numerical results are derived for both the bad and the good Boussinesq equation and conclusions from the relevant known results are derived.

High-Order Behaviors of Weighted Compact Fifth-Order Nonlinear Schemes

T HE typical theory for the finite difference method is that partial difference equations (PDEs) are discretized in space using a Taylor series expansion and solving it for the variables at discrete points. The derivatives are written as functions of variables on neighboring points. The approximations are said to be high-order accurate when the power p of the leading truncate error O h is greater than two. Because of the advantages of high-flow structure resolution and nice discontinuity capturing precision, high-order accurate schemes are playing important roles in computational fluid dynamics (CFD). Nonlinear high-order schemes for computing turbulence and aeroacoustic flows, which contain shock, have received more and more attention since the last decade. Although compact schemes have been derived by some researchers like Lele [1] and Gaitonde and Shang [2], theywere poorly used in capturing the shock owing to their linear property in construction. Cockburn and Shu [3] proposed fourth-order compact nonlinear schemes based on a total variation diminishing (TVD) and total variation bounded (TVB) concept. Hu et al. [4] developed a fifth-order upwind compact scheme to solve the acoustic phenomena arising from shock-vortex interactions. According to the principles from the physical consideration, Zhang et al. [5] constructed a mixing method, which degenerated into the second-order nonoscillatory, containing no free parameters and dissipative (NND) scheme in the shock region and retained highorder accuracy elsewhere. An appropriate weight technique treated in high-order scheme construction may also differentiate the vortex in the smooth region and shocks robustly. Levy et al. [6] applied fourth-order central weighted essentially nonoscillatory (WENO) to hypersonic flows. Shu [7] used fifth-order WENO for shock problems. Deng [8] and other researchers have developed a variety of compact high-order accurate schemes. A type of one-parameter linear dissipative compact scheme (DCS)was derived as to damp out the dispersive and parasite errors in the high-wave-number regions when central compact schemes were used. Although the linear compact schemes often caused numerical oscillations in the vicinity of discontinuities, a fourth-order cell-centered compact nonlinear finite difference scheme (CNS) [9] was then derived. An adaptive interpolation of variables at cell edges was designed, which automatically jumped to local one as discontinuities were encountered. It was a way to make the overall compact scheme capture discontinuities in a nonoscillatory manner. Afterward, using the weighted technique of Jiang and Shu [10] for interpolation at cell edge, fourthand fifth-order weighted compact nonlinear schemes (WCNS) were developed [11,12]. They were more efficient than CNS, for two tridiagonal inversions were cancelled and no logical algorithms were needed. To eliminate the unphysical oscillations in the vortex field near the shock wave when inviscid flow was computed, Deng constructed high-order dissipative weighted compact nonlinear schemes (DWCNS) [13], which improved the ability of restraining nonphysical oscillation in the vorticity field. For viscous flow computations, as proposed in [13], WCNS-5 and WCNS-E-5 would better be used. This Note shows the high-order features of WCNS including WCNS-E-5 and WCNS-5 with such typical high-order schemes as the explicit upwind biased fifth-order scheme (EUW-5) [14], Pade [1], and other special schemes like WENOs [6,7], and the explicit fifth-order shock-fitting upwind scheme [15]. We do the Fourier analysis to discussWCNS-E-5 andWCNS-5 with EUW-5 and Pade. Owing to nontridiagonal inversion,WCNS-E-5 acts more efficiently than WCNS-5 during computations. Therefore, in Sec. IV, we use WCNS-E-5 to solve the multidimensional inviscid/viscous flows. WCNS-E-5 obtains several numerical results to show the good performances with those by WENOs and the explicit fifth-order shock-fitting upwind scheme.

A third order numerical scheme for the two-dimensional sine-Gordon equation

A rational approximant of third order, which is applied to a three-time level recurrence relation, is used to transform the two-dimensional sine-Gordon (SG) equation into a second-order initial-value problem. The resulting nonlinear finite-difference scheme, which is analyzed for stability, is solved by an appropriate predictor–corrector (P–C) scheme, in which the predictor is an explicit one of second order. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behavior of the proposed P–C/MPC schemes is tested numerically on the line and ring solitons known from the bibliography, regarding SG equation and conclusions for both the mentioned schemes regarding the undamped and the damped problem are derived.