An explicit nonstandard finite difference scheme for the FitzHugh–Nagumo equations

In this work, we consider numerical solutions of the FitzHugh–Nagumo system of equations describing the propagation of electrical signals in nerve axons. The system consists of two coupled equations: a nonlinear partial differential equation and a linear ordinary differential equation. We begin with a review of the qualitative properties of the nonlinear space independent system of equations. The subequation approach is applied to derive dynamically consistent schemes for the submodels. This is followed by a consistent and systematic merging of the subschemes to give three explicit nonstandard finite difference schemes in the limit of fast extinction and slow recovery. A qualitative study of the schemes together with the error analysis is presented. Numerical simulations are given to support the theoretical results and verify the efficiency of the proposed schemes.     

Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection–diffusion equations

In this paper, we discuss the parameter-uniform finite difference method for a coupled system of singularly perturbed convection–diffusion equations. The leading term of each equation is multiplied by a small but different magnitude positive parameter, which leads to the overlap and interact boundary layer. We analyze the boundary layer and construct a piecewise-uniform mesh on the variant of the Shishkin mesh. We prove that our schemes converge almost first-order uniformly with respect to small parameters. We present some numerical experiments to support our theoretical analysis.     

Conservative compact difference scheme for the Zakharov–Rubenchik equations

In this article, a compact difference scheme is investigated to solve the Zakharov–Rubenchik equations in one dimension. The new scheme is proved to conserve the total mass and energy in the discrete sense. Rigorous error estimates are established for the new method with the help of an induction argument in energy space which show that the new scheme has second-order accuracy in time and fourth-order accuracy in space. Extensive numerical results are provided to verify our theoretical analysis, and show the accuracy and efficiency of the new scheme.     

A second-order linearized finite difference scheme for the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation

The purpose of this paper is to study the numerical simulation of the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation. After introducing a new variable, the integro-differential equation is transformed into an equivalent coupled system of first-order differential equations. A second-order accurate difference scheme is constructed for the new system of equations, which is proved to be local uncoupled by separation of variables. It is also proved that the scheme is uniquely solvable and second-order convergent in both time and space in L ₂-norm. A numerical example is given to demonstrate the theoretical results.     

An efficient split-step compact finite difference method for the coupled Gross–Pitaevskii equations

ABSTRACT

In this study, we propose an efficient split-step compact finite difference (SSCFD) method for computing the coupled Gross–Pitaevskii (CGP) equations. The coupled equations are divided into two parts, nonlinear subproblems and linear ones. Commonly, the nonlinear subproblems could be integrated directly and accurately, but it fails when the time-dependent potential cannot be integrated exactly. In this case, the midpoint and trapezoidal rules are applied approximately. At the same time, the split order is not reduced. For the linear ones, compact finite difference cannot be designed directly. To circumvent this problem, a linear transformation is introduced to decouple the system, which can make the split-step method be used again. Additionally, the proposed SSCFD method also holds for the coupled nonlinear Schrödinger (CNLS) system with time-dependent potential. Finally, numerical experiments for CGP equations and CNLS equations are well simulated, conservative properties and convergence rates are demonstrated as well. It is shown from the numerical tests that the present method is efficient and reliable.     

An unconditional stable compact fourth-order finite difference scheme for three dimensional Allen–Cahn equation

In this paper, we present an unconditional stable linear high-order finite difference scheme for three dimensional Allen–Cahn equation. This scheme, which is based on a backward differentiation scheme combined with a fourth-order compact finite difference formula, is second order accurate in time and fourth order accurate in space. A linearly stabilized splitting scheme is used to remove the restriction of time step. We prove the unconditional stability of our proposed method in analysis. A fast and efficient linear multigrid solver is employed to solve the resulting discrete system. We perform various numerical experiments to confirm the high-order accuracy, unconditional stability and efficiency of our proposed method. In particular, we show two applications of our proposed method: triply-periodic minimal surface and volume inpainting.     

A Galerkin finite element scheme for time–space fractional diffusion equation

In this paper, a Galerkin finite element scheme to approximate the time–space fractional diffusion equation is studied. Firstly, the fractional diffusion equation is transformed into a fractional Volterra integro-differential equation. And a second-order fractional trapezoidal formula is used to approach the time fractional integral. Then a Galerkin finite element method is introduced in space direction, where the semi-discretization scheme and fully discrete scheme are given separately. The stability analysis of semi-discretization scheme is discussed in detail. Furthermore, convergence analysis of semi-discretization scheme and fully discrete scheme are given in details. Finally, two numerical examples are displayed to demonstrate the effectiveness of the proposed method.     

A high-order compact finite difference method and its extrapolation for fractional mobile/immobile convection–diffusion equations

This paper is concerned with high-order numerical methods for a class of fractional mobile/immobile convection–diffusion equations. The convection coefficient of the equation may be spatially variable. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation for the spatial derivative and a second-order difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability and convergence of the method are proved using a discrete energy analysis method. A Richardson extrapolation algorithm is presented to enhance the temporal accuracy of the computed solution from the second-order to the third-order. Applications using two model problems give numerical results that demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm.     

Crank–Nicolson finite difference method based on a midpoint upwind scheme on a non-uniform mesh for time-dependent singularly perturbed convection–diffusion equations

A numerical approach is proposed to examine the singularly perturbed time-dependent convection–diffusion equation in one space dimension on a rectangular domain. The solution of the considered problem exhibits a boundary layer on the right side of the domain. We semi-discretize the continuous problem by means of the Crank–Nicolson finite difference method in the temporal direction. The semi-discretization yields a set of ordinary differential equations and the resulting set of ordinary differential equations is discretized by using a midpoint upwind finite difference scheme on a non-uniform mesh of Shishkin type. The resulting finite difference method is shown to be almost second-order accurate in a coarse mesh and almost first-order accurate in a fine mesh in the spatial direction. The accuracy achieved in the temporal direction is almost second order. An extensive amount of analysis has been carried out in order to prove the uniform convergence of the method. Finally we have found that the resulting method is uniformly convergent with respect to the singular perturbation parameter, i.e. ?-uniform. Some numerical experiments have been carried out to validate the proposed theoretical results.     

An efficient split-step quasi-compact finite difference method for the nonlinear fractional Ginzburg–Landau equations

In this paper, we propose a split-step quasi-compact finite difference method to solve the nonlinear fractional Ginzburg–Landau equations both in one and two dimensions. The original equations are split into linear and nonlinear subproblems. The Riesz space fractional derivative is approximated by a fourth-order fractional quasi-compact method. Furthermore, an alternating direction implicit scheme is constructed for the two dimensional linear subproblem. The unconditional stability and convergence of the schemes are proved rigorously in the linear case. Numerical experiments are performed to confirm our theoretical findings and the efficiency of the proposed method.     

A stabilized finite element method for stochastic incompressible Navier–Stokes equations

The major emphasis of this work is the development of a stabilized finite element method for solving incompressible Navier–Stokes equations with stochastic input data. The polynomial chaos expansion is used to represent stochastic processes in the variational problem, resulting in a set of deterministic variational problems to be solved for each Wiener polynomial chaos. To obtain the chaos coefficients in the corresponding deterministic incompressible Navier–Stokes equations, we combine the modified method of characteristics with the finite element discretization. The obtained Stokes problem is solved using a robust conjugate-gradient algorithm. This algorithm avoids projection procedures and any special correction for the pressure. These numerical techniques associate the geometrical flexibility of the finite element method with the ability offered by the modified method of characteristics to solve convection-dominated problems using time steps larger than its Eulerian counterpart. Numerical results are shown for the benchmark problems of driven cavity flow and backward-facing step flow. We also present numerical results for a problem of stochastic natural convection. It is found that the proposed stabilized finite element method offers a robust and accurate approach for solving the stochastic incompressible Navier–Stokes equations, even when high Reynolds and Rayleigh numbers are used in the simulations.     

A class of linearized energy-conserved finite difference schemes for nonlinear space-fractional Schrödinger equations

In this work, we propose a class of linearized energy-conserved finite difference schemes for nonlinear space-fractional Schrödinger equations. We prove the energy conservation, stability, and convergence of our schemes. In the proposed schemes, we only need to solve linear algebraic systems to obtain the numerical solutions. Numerical examples are presented to verify the accuracy, energy conservation, and stability of these schemes.     

Delay optimal control and viscosity solutions to associated Hamilton–Jacobi–Bellman equations

In this article, optimal control problems of differential equations with delays are investigated for which the associated Hamilton–Jacobi–Bellman (HJB) equations are nonlinear partial differential equations with delays. This type of HJB equation has not been previously studied and is difficult to solve because the state equations do not possess smoothing properties. We introduce a new notion of viscosity solutions and identify the value functional of the optimal control problems as the unique solution to the associated HJB equations. An analytical example is given as application.     

A multigrid scheme for 3D Monge–Ampère equations*

The elliptic Monge–Ampère equation is a fully nonlinear partial differential equation which has been the focus of increasing attention from the scientific computing community. Fast three-dimensional solvers are needed, for example in medical image registration but are not yet available. We build fast solvers for smooth solutions in three dimensions using a nonlinear full-approximation storage multigrid method. Starting from a second-order accurate centred finite difference approximation, we present a nonlinear Gauss–Seidel iterative method which has a mechanism for selecting the convex solution of the equation. The iterative method is used as an effective smoother, combined with the full-approximation storage multigrid method. Numerical experiments are provided to validate the accuracy of the finite difference scheme and illustrate the computational efficiency of the proposed multigrid solver.     

A highly accurate adaptive finite difference solver for the Black–Scholes equation

In this paper, we develop a highly accurate adaptive finite difference (FD) discretization for the Black–Scholes equation. The final condition is discontinuous in the first derivative yielding that the effective rate of convergence in space is two, both for low-order and high-order standard FD schemes. To obtain a method that gives higher accuracy, we use an extra grid in a limited space- and time-domain. This new method is called FD6G2. The FD6G2 method is combined with space- and time-adaptivity to further enhance the method. To obtain solutions of high accuracy, the adaptive FD6G2 method is superior to both a standard and an adaptive second-order FD method.     

A two-level linearized compact ADI scheme for two-dimensional nonlinear reaction–diffusion equations

A novel two-level linearized compact alternating direction implicit (ADI) scheme is proposed for solving two-dimensional nonlinear reaction–diffusion equations. The computational cost is reduced by use of the Newton linearized method and the ADI method. The existence and uniqueness of the numerical solutions are proved. Moreover, the error estimates in $H^1$ and $L^{\infty }$ norms are presented. Numerical examples are given to illustrate our theoretical results.     

A second-order TVD implicit–explicit finite volume method for time-dependent convection-reaction equations

A class of conservative methods is developed in the more general framework of cell-centered upwind differences to approximate numerically the solution of one-dimensional non-linear conservation laws with (possibly) stiff reaction source terms. These methods are based on a non-oscillatory piecewise linear polynomial representation of the discrete solution within any mesh interval to compute pointwise solution values. The piecewise linear approximate solution is obtained by approximating the cell average of the analytical solution and the solution slope in every mesh cell. These two quantities are evolved in time by solving a set of discrete equations that are suitably designed to ensure formal second-order consistency. Several numerical tests which are taken from literature illustrate the performance of the method in solving non-stiff and stiff convection-reaction equations in conservative form.