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In this paper, two high-order compact difference schemes with graded meshes are proposed for solving the time-fractional Black-Scholes equation. We first eliminate the convection term in the equivalent form of the considered equation by using exponential transformation, then combine the sixth-order/eighth-order compact difference method with a temporal graded meshes-based trapezoidal formulation for the temporal integral term to obtain the fully discrete high-order compact difference schemes. The stability and convergence analysis of the two proposed schemes are studied by applying Fourier analysis. Finally, the effectiveness of the proposed schemes and the correctness of the theoretical results are verified by two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, high-order compact finite difference schemes for the one-dimensional regularized long wave equation are proposed. Firstly, the original regularized long-wave equation is deformed in two ways. The fourth-order compact difference scheme and the fourth-order Padé scheme are used for discretization of the second and first derivatives of the two deformed equations in space direction, respectively, and the θ -weighted scheme is used to discretize the time derivative of the first deformed, the fourth-order backward difference formula is used for the competition of the time derivative of the second deformed equation. Secondly, the nonlinear terms in the two schemes are linearized by Taylor series expansion method. Thirdly, the existence, uniqueness and conservation of energy of numerical solutions are proved by discrete energy method and mathematical induction method. And the two new schemes are shown to be conserved and unconditionally stable. Since each time level involves three grid points, a tridiagonal linear system is formed, which can be solved directly using the Thomas algorithm. Finally, the accuracy and reliability of the present method are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

The paper is concerned with development of an accurate and effective positivity-preserving high-order compact difference method for solving the Keller-Segel chemotaxis model, which is a kind of nonlinear parabolic-parabolic system in mathematical biology. Firstly, a stiffly-stable five-step fourth-order fully implicit compact difference scheme is proposed. The new scheme not only has fourth-order accuracy in the spatial direction, but also has fourth-order accuracy in the temporal direction, and the computational strategy for the nonlinear chemotaxis term is provided. Then, a positivity-preserving numerical algorithm is presented, which ensures the non-negativity of cell density at all time without accuracy loss. And a time advancement algorithm is established. Finally, the proposed method is applied to the numerical simulation for chemotaxis phenomena, and the accuracy, stability and positivity-preserving of the new scheme are validated with several numerical examples.

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 O (τ C 4 + τ F 2 + h 4) in the discrete L 2 -norm is obtained. Here, τ C and τ F 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. [ABSTRACT FROM AUTHOR]

In this paper, high-order compact difference method is used to solve the one-dimensional nonlinear advection diffusion reaction equation. The nonlinearity here is mainly reflected in the advection and reaction terms. Firstly, the diffusion term is discretized by using the fourth-order compact difference formula, the nonlinear advection term is approximated by using the fourth-order Padé formula of the first-order derivative, and the time derivative term is discretized by using the fourth-order backward differencing formula. An unconditionally stable five-step fourth-order fully implicit compact difference scheme is developed. This scheme has fourth-order accuracy in both time and space. Secondly, for the calculations of the start-up time steps, the time derivative term is discretized by the Crank-Nicolson method, and Richardson extrapolation formula is used to improve the accuracy in time direction from the second-order to the fourth-order. Thirdly, convergence and stability of the difference scheme in H 1 seminorm, L ∞ and L 2 norms, existence and uniqueness of the numerical solutions are proved, respectively. Fourthly, the Thomas algorithm is used to solve the nonlinear algebraic equations at each time step, and a time advancement algorithm with linearized iteration strategy is established. Finally, the accuracy, stability and efficiency of the present approach are verified by some numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, by using the local one-dimensional (LOD) method, Taylor series expansion and correction for the third derivatives in the truncation error remainder, two high-order compact LOD schemes are established for solving the two- and three- dimensional advection equations, respectively. They have the fourth-order accuracy in both time and space. By the von Neumann analysis method, it shows that the two schemes are unconditionally stable. Besides, the consistency and convergence of them are also proved. Finally, numerical experiments are given to confirm the accuracy and efficiency of the present schemes. [ABSTRACT FROM AUTHOR]

An efficient multiscale‐like multigrid (MSLMG) method combined with a high‐order compact (HOC) difference scheme on nonuniform grids is presented to solve the two‐dimensional (2D) convection‐diffusion equations. The discrete systems with given appropriate initial solutions on two finest grids are solved to obtain the MSLMG solutions with discretization‐level accuracy by performing fewer multigrid cycles; it is implemented with alternating line Gauss–Seidel smoother, interpolation, and restriction operators on the nonuniform grids. Numerical experiments of boundary layer or local singularity problems are conducted to show that the proposed algorithm with the HOC scheme on nonuniform grids is efficient and effective to decrease the computational cost and time, and the computed approximation on the nonuniform grids has fourth order accuracy. [ABSTRACT FROM AUTHOR]

A partial semi-coarsening multigrid method based on the high-order compact (HOC) difference scheme on nonuniform grids is developed to solve the 2D convection–diffusion problems with boundary or internal layers. The significance of this study is that the multigrid method allows different number of grid points along different coordinate directions on nonuniform grids. Numerical experiments on some convection–diffusion problems with boundary or internal layers are conducted. They demonstrate that the partial semi-coarsening multigrid method combined with the HOC scheme on nonuniform grids, without losing the high-order accuracy, is very efficient and effective to decrease the computational cost by reducing the number of grid points along the direction which does not contain boundary or internal layers. [ABSTRACT FROM PUBLISHER]

This paper provides a numerical investigation for European options under parabolic-ordinary system modeling markets to liquidity shocks. Our main results concern construction and analysis of fourth order in space compact finite difference schemes (CFDS). Numerical experiments using Richardson extrapolation in time are discussed. [ABSTRACT FROM AUTHOR]

A numerical simulation of the regularized long wave (RLW) equation is obtained using a high-order compact difference method, based on the fourth-order compact difference scheme in space and the fourth-order Runge–Kutta method in time integration. The method is tested on the problems of propagation of a solitary wave, interaction of two positive solitary waves, interaction of a positive and a negative solitary wave, the evaluation of Maxwellian pulse into stable solitary waves, the development of an undular bore and the solitary waves induced by boundary motion. The three invariants of the motion are calculated to determine the conservation properties of the algorithm. L2 and L∞ error norms are used to measure differences between the exact and numerical solutions. The results obtained by proposed method are compared with those of other recently published methods and shown to be more accurate and efficient. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The paper is concerned with development of an accurate and effective positivity-preserving high-order compact difference method for solving the Keller-Segel chemotaxis model, which is a kind of nonlinear parabolic-parabolic system in mathematical biology. Firstly, a stiffly-stable five-step fourth-order fully implicit compact difference scheme is proposed. The new scheme not only has fourth-order accuracy in the spatial direction, but also has fourth-order accuracy in the temporal direction, and the computational strategy for the nonlinear chemotaxis term is provided. Then, a positivity-preserving numerical algorithm is presented, which ensures the non-negativity of cell density at all time without accuracy loss. And a time advancement algorithm is established. Finally, the proposed method is applied to the numerical simulation for chemotaxis phenomena, and the accuracy, stability and positivity-preserving of the new scheme are validated with several numerical examples.