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We establish uniform error estimates of finite difference methods for the nonlinear Schrödinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter ε (ε Є (0,1]). When ε -> 0⁺, NLSW collapses to the standard NLS. In the small perturbation parameter regime, i. e., 0 < ε < 1, the solution of NLSW is perturbed from that of NLS with a function oscillating in time with O(ε²)-wavelength at O(ε⁴) and O(ε²) amplitudes for well-prepared and ill-prepared initial data, respectively. This high oscillation of the solution in time brings significant difficulties in establishing error estimates uniformly in ε of the standard finite difference methods for NLSW, such as the conservative Crank-Nicolson finite difference (CNFD) method, and the semi-implicit finite difference (SIFD) method. We obtain error bounds uniformly in ε, at the order of O(h² + r) and O(h² + r 2/3 ) with time step r and mesh size h for well-prepared and ill-prepared initial data, respectively, for both CNFD and SIFD in the l² -norm and discrete semi-H¹ norm. Our error bounds are valid for general nonlinearity in NLSW and for one, two, and three dimensions. To derive these uniform error bounds, we combine ε-dependent error estimates of NLSW, ε-dependent error bounds between the numerical approximate solutions of NLSW and the solution of NLS, together with error bounds between the solutions of NLSW and NLS. Other key techniques in the analysis include the energy method, cut-off of the nonlinearity, and a posterior bound of the numerical solutions by using the inverse inequality and discrete semi-H¹ norm estimate. Finally, numerical results are reported to confirm our error estimates of the numerical methods and show that the convergence rates are sharp in the respective parameter regimes.

In this paper, we construct four numerical methods to solve the Burgers–Huxley equation with specified initial and boundary conditions. The four methods are two novel versions of nonstandard finite difference schemes (NSFD1 and NSFD2), explicit exponential finite difference method (EEFDM) and fully implicit exponential finite difference method (FIEFDM). These two classes of numerical methods are popular in the mathematical biology community and it is the first time that such a comparison is made between nonstandard and exponential finite difference schemes. Moreover, the use of both nonstandard and exponential finite difference schemes are very new for the Burgers–Huxley equations. We considered eleven different combination for the parameters controlling diffusion, advection and reaction, which give rise to four different regimes. We obtained stability region or condition for positivity. The performances of the four methods are analysed by computing absolute errors, relative errors, L 1 and L ∞ errors and CPU time.

We present a hybridization technique for summation-by-parts finite difference methods with weak enforcement of interface and boundary conditions for second order, linear elliptic partial differential equations. The method is based on techniques from the hybridized discontinuous Galerkin literature where local and global problems are defined for the volume and trace grid points, respectively. By using a Schur complement technique the volume points can be eliminated, which drastically reduces the system size. We derive both the local and global problems, and show that the resulting linear systems are symmetric positive definite. The theoretical stability results are confirmed with numerical experiments as is the accuracy of the method.

The time-dependent mild-slope equation (MSE) is a second-order hyperbolic equation, which is adopted to consider the irregularity of waves. For the difficulty of directly solving the partial derivative terms and the second-order time derivative term, a novel mesh-free numerical scheme, based on the generalized finite difference method (GFDM) and the Houbolt finite difference method (HFDM), is developed to promote the precision and efficiency of the solution to time-dependent MSE. Based on the local characteristics of the GFDM, as a new domain-type meshless method, the linear combinations of nearby function values can be straightforwardly and efficiently implemented to compute the partial derivative term. It is worth noting that the application of the HFDM, an unconditionally stable finite difference time marching scheme, to solve the second-order time derivative term is critical. The results obtained from four examples show that the propagation of waves can be successfully simulated by the proposed numerical scheme in complex seabed terrain. In addition, the energy conversion of waves in long-distance wave propagation can be accurately captured using fast Fourier transform (FFT) analysis, which investigates the energy conservation in wave shoaling problems.

In this study, we develop the enhanced unconditionally positive finite difference method (EUPFD), and use it to solve linear and nonlinear advection–diffusion–reaction (ADR) equations. This method incorporates the proper orthogonal decomposition technique to the unconditionally positive finite difference method (UPFD) to reduce the degree of freedom of the ADR equations. We investigate the efficiency and effectiveness of the proposed method by checking the error, convergence rate, and computational time that the method takes to converge to the exact solution. Solutions obtained by the EUPFD were compared with the exact solutions for validation purposes. The agreement between the solutions means the proposed method effectively solved the ADR equations. The numerical results show that the proposed method greatly improves computational efficiency without a significant loss in accuracy for solving linear and nonlinear ADR equations.

This study introduces the higher-order unconditionally positive finite difference (HUPFD) methods to solve the linear, nonlinear, and system of advection–diffusion–reaction (ADR) equations. The stability and consistency of the developed methods are analyzed, which are necessary and sufficient for the numerical approach to converge to the exact solution. The problem under consideration is of the Cauchy type, and hence, Von Neumann stability analysis is used to analyze the stability of the proposed schemes. The HUPFD’s efficacy and efficiency are investigated by calculating the error, convergence rate, and computing time. For validation purposes, the higher-order unconditionally positive finite difference solutions are compared to analytical calculations. The numerical results demonstrate that the proposed methods produce accurate solutions to solve the advection diffusion reaction equations. The results also show that increasing the order of the unconditionally positive finite difference leads an implicit scheme that is conditionally stable and has a higher order of accuracy with respect to time and space.

In this paper, the enhanced higher-order unconditionally positive finite difference method is developed to solve the linear, non-linear and system advection diffusion reaction equations. Investigation into the effectiveness and efficiency of the proposed method is carried out by calculating the convergence rate, error and computational time. A comparison of the solutions obtained by the enhanced higher-order unconditionally positive finite difference and exact solution is conducted for validation purposes. The numerical results show that the developed method reduced the time taken to solve the linear and non-linear advection diffusion reaction equations as compared to the results obtained by the higher-order unconditionally positive finite difference method.

Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic partial differential equations (PDEs). These methods are best suited to regular rectangular grids, which leads to low accuracy near curved boundaries or singularities of solutions. In this article we combine monotone finite difference methods with an adaptive grid refinement technique to produce a PDE discretization and solver which is applied to a broad class of equations, in curved or unbounded domains which include free boundaries. The grid refinement is flexible and adaptive. The discretization is combined with a fast solution method, which incorporates asynchronous time stepping adapted to the spatial scale. The framework is validated on linear problems in curved and unbounded domains. Key applications include the obstacle problem and the one-phase Stefan free boundary problem.

We present two finite difference time domain methods for the biharmonic nonlinear Schrödinger equation (BNLS) by reformulating it into a system of second-order partial differential equations instead of a direct discretization, including a second-order conservative Crank–Nicolson finite difference (CNFD) method and a second-order semi-implicit finite difference (SIFD) method. The CNFD method conserves the mass and energy in the discretized level, and the SIFD method only needs to solve a linear system at each time step, which is more efficient. By energy method, we establish optimal error bounds at the order of O ( h 2 + τ 2 ) in both L 2 and H 2 norms for both CNFD and SIFD methods, with mesh size h and time step τ . The proof of the error bounds are mainly based on the discrete Gronwall’s inequality and mathematical induction. Finally, numerical results are reported to confirm our error bounds and to demonstrate the properties of our schemes.

We propose a family of mimetic discretization schemes for elliptic problems including convection and reaction terms. Our approach is an extension of the mimetic methodology for purely diffusive problems on unstructured polygonal and polyhedral meshes. The a priori error analysis relies on the connection between the mimetic formulation and the lowest order Raviart–Thomas mixed finite element method. The theoretical results are confirmed by numerical experiments.