High-order Difference Scheme Research Articles

Variable-step [formula omitted] method combined with time two-grid algorithm for multi-singularity problems arising from two-dimensional nonlinear delay fractional equations

In this paper, we focus on the numerical simulation for two-dimensional nonlinear fractional sub-diffusion equations in the presence of time delay. Firstly, we investigate the existence, uniqueness and regularity of the solution for such problems. The theoretical result implies that the solution at τ+ is smoother than that at 0+, where τ is a constant time delay, and this is an improvement for the work (Tan et al., 2022). Secondly, a high-order difference scheme based on L1 method is constructed. For the sake of repairing the convergence order in temporal direction and improving the computational efficiency, an efficient time two-grid algorithm based on nonuniform meshes is first developed. The convergence order of the two-grid scheme reaches O(NF−min{rα,2−α}+NC−min{2rα,4−2α}+h12+h22), where NF and NC represent the number of the fine and coarse grids respectively, while h1 and h2 are the space-step sizes. Furthermore, stability and convergence analysis of the proposed scheme are carefully verified by energy method. Finally, numerical experiments are carried out to show the validity of theoretical statements.

Self-adaptive numerical dispersion suppressed method for shallow seismic simulation

Near-surface in seismic exploration generally refers to the low deceleration zone and a section of stratum below it, which can be described in geophysical language as “low velocity”, “low Q”, and “Free surface”, etc. Due to the presence of low-velocity layers in near-surface, shallow seismic simulation is plagued by numerical dispersion. Numerical dispersion affects the accuracy of seismic wave simulation, especially severe in shallow areas containing low-velocity layers. To improve the simulated quality, denser grids or higher-order difference schemes are used, but both of which would seriously reduce computational efficiency, especially for frequency-domain numerical modeling. Differentiation is replaced by difference in wave field simulation, which would inevitably result in numerical errors. These numerical errors are manifested as the difference between the phase velocity of the discretized wave field and the medium velocity. The waves with different wavenumber components have different phase velocities, and the larger the wavenumber, the more that its phase velocity lags the group velocity. Based on this theoretical analysis, a regularization factor is added to the frequency-domain acoustic wave equation to correct the phase velocity of high wavenumber components. According to the Von Neumann stability requirements, a regularization factor that adaptively changes with simulation parameters is derived. Compared to the fixed regularization factor, adaptive regularization factor can better match the velocity field and protect the effective wave field to the maximum extent while suppressing numerical dispersion. The improved acoustic wave equation can effectively suppress numerical dispersion without increasing the number of grids. Different numerical models demonstrate the effectiveness and efficiency of the improved acoustic equation with the adaptive regularization factor for suppressing numerical dispersion.

Simulation of fluid‐structure interaction using the boundary data immersion method with adaptive mesh refinement

SummaryThe fluid‐structure interaction is simulated using the boundary data immersion method. As the fluid‐structure interface is smeared in the smoothing region, deviations are incurred in fluid simulations. For compressible flow, high order difference schemes with more mesh cells for the stencils are usually employed to achieve high overall accuracy, but near interfaces it requires wider smoothing region of several mesh cells for computational stability and hence lowers its accuracy significantly. To address this issue, the proposed algorithm switches to lower order difference schemes near the interfaces and applies adaptive mesh refining there to compensate the accuracy loss. Implemented with Structured Adaptive Mesh Refinement Application Infrastructure (SAMRAI), the algorithm shows notable improvement in the overall accuracy and efficiency in cases such as channel flow and flow past a cylinder. The algorithm is used to simulate the shock wave past a fixed or free cylinder with Ma and Re , which reveals the relaxation process and the temporal evolution of the drag coefficient, it goes through a valley and maintains at relatively high value for the fixed cylinder, while that of the free cylinder tends to decrease in fluctuation which is found to be caused by the interaction between the forward moving cylinder and vortexes in the unsteady wake.

Higher-Order Difference Schemes in Heat- and Mass-Transfer Problems

Higher-Order Difference Schemes in Heat- and Mass-Transfer Problems

Higher-Order Difference Schemes in Heat and Mass Transfer Problems

Higher-Order Difference Schemes in Heat and Mass Transfer Problems

Three-Point Difference Schemes of High Order of Accuracy for the Sturm–Liouville Problem

Three-Point Difference Schemes of High Order of Accuracy for the Sturm–Liouville Problem

Fractal characteristics of surface roughness and their effects on laser shock waves

The Weierstrass–Mandelbrot function model was used to simulate the rough topography of a target surface, and the fractal theory was used to study the fractal characteristics of the target surface topography, which is the self-similarity of surface topography. The 3D fluid dynamics governing equations of the laser-supported detonation plasma flow field in Cartesian coordinates are given and discretized by a high-order difference scheme. The equation of state of plasma under different conditions is considered, and a new method of flux splitting is introduced to carry out numerical simulations. The numerical results of plasma pressure, density, temperature, and velocity at different times and different surface roughnesses are obtained, which show the evolvement of the ablative plasma. The simulated results show that a rough surface can obviously affect the propagation velocity of the detonation wave near the wall. In addition, the waveform is also destroyed, forming a broken waveform.

Revisiting the space-time gradient method: A time-clocking perspective, high order difference time discretization and comparison with the harmonic balance method

This paper revisits the Space-Time Gradient (STG) method which was developed for efficient analysis of unsteady flows due to rotor–stator interaction and presents the method from an alternative time-clocking perspective. The STG method requires reordering of blade passages according to their relative clocking positions with respect to blades of an adjacent blade row. As the space-clocking is linked to an equivalent time-clocking, the passage reordering can be performed according to the alternative time-clocking. With the time-clocking perspective, unsteady flow solutions from different passages of the same blade row are mapped to flow solutions of the same passage at different time instants or phase angles. Accordingly, the time derivative of the unsteady flow equation is discretized in time directly, which is more natural than transforming the time derivative to a spatial one as with the original STG method. To improve the solution accuracy, a ninth order difference scheme has been investigated for discretizing the time derivative. To achieve a stable solution for the high order scheme, the implicit solution method of Lower-Upper Symmetric Gauss-Seidel/Gauss-Seidel (LU-SGS/GS) has been employed. The NASA Stage 35 and its blade-count-reduced variant are used to demonstrate the validity of the time-clocking based passage reordering and the advantages of the high order difference scheme for the STG method. Results from an existing harmonic balance flow solver are also provided to contrast the two methods in terms of solution stability and computational cost.

On the importance of high-frequency damping in high-order conservative finite-difference schemes for viscous fluxes

This paper discusses the importance of high-frequency damping in high-order conservative finite-difference schemes for viscous terms in the Navier-Stokes equations. Investigating nonlinear instability encountered in a high-resolution viscous shock-tube simulation, we have discovered that a modification to the viscous scheme rather than the inviscid scheme resolves a problem with spurious oscillations around shocks. The modification introduces a term responsible for high-frequency damping that is missing in a conservative high-order viscous scheme. The importance of damping has been known for schemes designed for unstructured grids. However, it has not been recognized well in very high-order difference schemes, especially in conservative difference schemes. Here, we discuss how it is easily missed in a conservative scheme and how to improve such schemes by a suitably designed damping term.

The construction of higher-order numerical approximation formula for Riesz derivative and its application to nonlinear fractional differential equations (I)

The main goal of this paper is to construct high-order numerical differential formulas approximating the Riesz derivative and apply them to the numerical solution of the nonlinear space fractional Ginzburg–Landau equations. Firstly, we introduce a novel second-order fractional central difference operator for the approximation of the Riesz derivative with order α∈(1,2]. Moreover, based on the difference operator and the compact technique, a novel fourth-order fractional compact difference operator is also derived. Secondly, using the fourth-order difference operator in space and the Crank–Nicolson method in time, a high-order difference scheme is proposed for the nonlinear space fractional Ginzburg–Landau equations. Thirdly, besides the standard energy method, some new techniques and important lemmas are developed to prove the unique solvability, stability and convergence in the sense of different norms. It is proved that the difference scheme is unconditionally stable and convergent with order Oτ2+h4 for α∈(1,1.5), where τ and h are the temporal step size and spatial step size, respectively. Finally, some numerical examples are given to show the efficiency and accuracy of the numerical differential formulas and finite difference scheme.

Time two-grid technique combined with temporal second order difference method for two-dimensional semilinear fractional sub-diffusion equations

In this paper, we construct a high order difference scheme for two-dimensional semilinear fractional sub-diffusion equations at first. To reduce the computation time, an efficient time two-grid algorithm is then proposed. The global convergence order of the two-grid scheme reaches O(τF2+τC4+h12+h22), where τF and τC represent the time-step sizes on the fine and coarse grids respectively, while h1 and h2 are the space-step sizes. Furthermore, stability and convergence of the scheme are carefully studied by some skills. At last, the theoretical statements are verified by numerical experiments.

Об эффективности высокоточных разностных схем для уравнения Шрёдингера

Исследуется ряд двух- и трехслойных разностных схем, построенных на расширенных шаблонах, до восьмого порядка точности для уравнения Шрёдингера. Наряду с многоточечными схемами рассматривается метод коррекции Ричардсона в приложении к схеме четвертого порядка аппроксимации, повышающий порядок точности путем построения линейных комбинаций приближенных решений, полученных на различных вложенных сетках. Проведено сравнение методов по устойчивости, сложности реализации алгоритмов и объему вычислений, необходимых для достижения заданной точности. На основе теоретического анализа и численных экспериментов выявлены методы, наиболее эффективные для практического применения The efficiency of difference methods for solving problems of nonlinear wave optics is largely determined by the order of accuracy. Schemes up to the fourth order of accuracy have the traditional architecture of three-point stencils and standard conditions for the application of algorithms. However, a further increase in the order in the general case is associated with the need to expand the stencils using multipoint difference approximations of the derivatives. The use of such schemes forces formulating additional boundary conditions, which are not present in the differential problem, and leads to the need to invert the matrices of the strip structure, which are different from the traditional tridiagonal ones. An exception is the Richardson correction method, which is aimed at increasing the order of accuracy by constructing special linear combinations of approximate solutions obtained on various nested grids according to traditional structure schemes. This method does not require the formulation of additional boundary conditions and inversion of strip matrices. In this paper, we consider several explicit and implicit multipoint difference schemes up to the eighth order of accuracy for the Schr¨odinger equation. In addition, a simple and double Richardson correction method is also investigated in relation to the classical fourth-order scheme. A simple correction raises the order to sixth and a double correction to eighth. This large collection of schemes is theoretically compared in terms of their properties such as the order of approximation, stability, the complexity of the implementation of a numerical algorithm, and the amount of arithmetic operations required to achieve a given accuracy. The theoretical analysis is supplemented by numerical experiments on the selected test problem. The main conclusion drawn from the research results is that of all the considered schemes, the Richardson-corrected scheme is the most preferable in terms of the investigated properties

Valuation of the American put option as a free boundary problem through a high-order difference scheme

Abstract Valuation of the American options encountered commonly in finance is quite difficult due to the possibility of early exercise alternatives. Since an exact solution for the American options does not exist, effective numerical methods are needed to understand the behavior of option pricing models. Therefore, in this paper, a new approach based on a high-order difference scheme is proposed to discuss the valuation of an American put option as a free boundary problem. Using a front-fixing approach that transforms the unknown free boundary (optimal stopping) into a fixed one, a sixth-order finite difference scheme (FD6) in space and a third-order strong-stability preserving Runge–Kutta (SSPRK3) in time are applied to the model converted to a nonlinear partial differential equation. The computed results revealed that the combined method is seen to attempt to pull up the capacity of the algorithm to achieve higher accuracy. It is seen that the quantitative and qualitative results produced by the method proposed with minimal computational effort are sufficiently accurate and meaningful. Therefore, this article provides some new insights about the physical characteristics of financial problems and such realistic phenomena.

A new conservative fourth-order accurate difference scheme for the nonlinear Schrödinger equation with wave operator

This paper is devoted to the study of high-order finite difference scheme for the nonlinear Schrödinger equation with wave operator. The difference scheme is three level and a five point stencil is used for spatial variable. Existence of solutions is shown using a variant of Brouwer fixed point theorem. The unconditional stability as well as uniqueness of the difference scheme are also discussed in detail. The convergence of the difference scheme is proved by utilizing the energy method to be of fourth-order in space and second-order in time in the discrete maximum norm without any restrictions on the mesh sizes. Finally, some numerical experiments and comparisons with other existing methods in the literature are presented. The numerical results show that the high-order difference scheme of this article improves the accuracy of the space and time direction.

Solving the Sturm-Liouville problem by three-point difference schemes of high order of accuracy

For the solving Sturm-Liouville problem, three-point difference schemes of high order of accuracy on a nonuniform grid are constructed. It is shown that the coefficients of these schemes are expressed in terms of solutions of two auxiliary initial value problems. An estimate of the accuracy of three-point difference schemes is obtained and an iterative Newton method is proposed to determine their solution. Numerical experiments confirm theoretical conclusions.

High-order conservative scheme for the coupled space fractional nonlinear Schrödinger equations

In this paper, an efficient finite difference scheme is proposed for one dimension and two dimension coupled space fractional nonlinear Schrödinger equations. First, the high-order difference scheme and Crank–Nicolson scheme are used to one dimension coupled space fractional nonlinear Schrödinger equations. second, we show that the high-order conservative difference scheme satisfies the mass and energy conservation laws respectively, and convergence and unconditional stability of the scheme are also proved. Next, we give the high-order conservative scheme for two dimension coupled space fractional nonlinear Schrödinger equations. Finally, some numerical results are reported to verify our theoretical analysis.

Characteristic features of error in high-order difference calculation of 1D Poisson equation and unlimited high-accurate calculation under multi-precision calculation

In a previous paper based on the interpolation finite difference method, a calculation system was shown for calculating 1D (one-dimensional) Laplace’s equation and Poisson’s equation using high-order difference schemes. Finite difference schemes, from the usual second-order to tenth-order differences, including odd number order differences, were systematically and instantaneously derived over equally/unequally spaced grid points based on the Lagrange interpolation function. Using the direct method with the band diagonal matrix algorithm, 1D Poisson equations were numerically calculated under double precision floating arithmetic, but it became clear that high accurate calculations could not be secured in high-order differences because “digit-loss errors” caused by the finite precision of computations occurred in the calculations when using the high-order differences. The double precision calculation corresponds to 15 (significant) digit calculation. In this paper, we systematically investigate how the calculation accuracy changes by high precision calculations (30-digit, and 45-digit calculations). Under 45-digit calculation, where the digit-loss error can be almost ignored, the high-order differences enable extremely high-accurate calculations.

An advanced meshless approach for the high-dimensional multi-term time-space-fractional PDEs on convex domains

In this article, an advanced differential quadrature (DQ) approach is proposed for the high-dimensional multi-term time-space-fractional partial differential equations (TSFPDEs) on convex domains. Firstly, a family of high-order difference schemes is introduced to discretize the time-fractional derivative, and a semi-discrete scheme for the considered problems is presented. We strictly prove its unconditional stability and error estimate. Further, we derive a class of DQ formulas to evaluate the fractional derivatives, which employs radial basis functions (RBFs) as test functions. Using these DQ formulas in spatial discretization, a fully discrete DQ scheme is then proposed. Our approach provides a flexible and high accurate alternative to solve the high-dimensional multi-term TSFPDEs on convex domains, and its actual performance is illustrated by contrast to the other methods available in the open literature. The numerical results confirm the theoretical analysis and the capability of our proposed method finally.

The pointwise error estimates of two energy-preserving fourth-order compact schemes for viscous Burgers’ equation

A novel fourth-order three-point compact operator for the nonlinear convection term uux is provided in this paper. The operator makes the numerical analysis of higher-order difference schemes become possible for a wide class of nonlinear evolutionary equations under the unified framework. We take the classical viscous Burgers’ equation as an example and establish a new conservative fourth-order implicit compact difference scheme based on the method of order reduction. A detailed theoretical analysis is carried out by the discrete energy argument and mathematical induction. It is rigorously proved that the difference scheme is conservative, uniquely solvable, stable, and unconditionally convergent in discrete $L^{\infty }$ -norm. The convergence order is two in time and four in space, respectively. Furthermore, we derive a three-level linearized compact difference scheme for viscous Burgers’ equation based on the proposed operator. All numerically theoretical results similar to that of the nonlinear numerical scheme are inherited completely; meanwhile, it is more time saving. Applying the compact operator to other more complex and higher-order nonlinear evolutionary equations is feasible, including Benjamin-Bona-Mahony-Burgers’ equation, Korteweg-de Vries equation, Kuramoto-Sivashinsky equation, and classification to name a few. Numerical results demonstrate that the presented schemes for Burgers’ equation can achieve second-order accuracy in time and fourth-order accuracy in space in $L^{\infty }$ -norm.

High-order conservative schemes for the space fractional nonlinear Schrödinger equation

In the paper, the high-order conservative schemes are presented for space fractional nonlinear Schrödinger equation. First, we give two class high-order difference schemes for fractional Risze derivative by compact difference method and extrapolating method, and show the convergence analysis of the two methods. Then, we apply high-order conservative difference schemes in space direction, and Crank-Nicolson, linearly implicit and relaxation schemes in time direction to solve fractional nonlinear Schrödinger equation. Moreover, we show that the arising schemes are uniquely solvable and approximate solutions converge to the exact solution at the rate O(τ2+h4), and preserve the mass and energy conservation laws. Finally, we given numerical experiments to show the efficiency of the conservative finite difference schemes.