High-order Finite Difference Scheme Research Articles

Error estimates of high-order compact finite difference schemes for the nonlinear abcd s systems

Abstract In this paper, some fourth-order compact finite difference schemes are derived and analyzed for the nonlinear $abcd$ Boussinesq systems. The optimal order error estimates for the semidiscrete compact finite difference schemes with different cases of dispersion coefficients $a,\ b,\ c,\ d$, are presented. The third-order and fourth-order linearized implicit multistep schemes are adopted for time discretization, and numerical experiments are conducted on the model problems. Numerical results show that the proposed schemes have high accuracy and are consistent with the theoretical analysis.

A high-order compact finite difference scheme and its analysis for the time-fractional diffusion equation

A high-order compact finite difference scheme and its analysis for the time-fractional diffusion equation

A new type of increasingly higher order finite difference and finite volume MR-WENO schemes with adaptive linear weights for hyperbolic conservation laws

In this paper, the new fifth-order, seventh-order, and ninth-order finite difference and finite volume multi-resolution weighted essentially non-oscillatory (MR-WENO) schemes with adaptive linear weights are presented for hyperbolic conservation laws on structured meshes. They are termed as high-order finite difference and finite volume ALW-WENO schemes. These ALW-WENO schemes only apply one small stencil and one large stencil in reconstruction processes, which could achieve the desired accuracy in the region of smoothness and non-oscillatory properties in the region of containing strong shocks. The linear weights that sum to one can be automatically adjusted to any positive numbers. This is the first time that arbitrary high-order finite difference and finite volume WENO schemes are designed by using only two unequal-sized central spatial stencils. The structure of these novel WENO schemes is simple, so it is easier for obtaining high-order accuracy and solving multi-dimensional problems in large scale engineering applications. Compared to traditional MR-WENO schemes with same order, the computational efficiency can be further improved. Some benchmark tests indicate that these new ALW-WENO schemes have good robustness and performance.

Solving Lane–Emden equations with boundary conditions of various types using high-order compact finite differences

In this study, a high-order compact finite difference method is used to solve Lane–Emden equations with various boundary conditions. The norm is to use a first-order finite difference scheme to approximate Neumann and Robin boundary conditions, but that compromises the accuracy of the entire scheme. As a result, new higher-order finite difference schemes for approximating Robin boundary conditions are developed in this work. We test the applicability and performance of the method using different examples of Lane–Emden equations. Convergence analysis is provided, and it is consistent with the numerical results. The results are compared with the exact solutions and published results from other methods. The method produces highly accurate results, which are displayed in tables and graphs.

Spurious currents suppression by accurate difference schemes in multiphase lattice Boltzmann method

Spurious currents are often observed near curved interfaces in multiphase simulations based on diffuse interface methods. These unphysical phenomena negatively affect both computational accuracy and stability. In this paper, the origin and suppression of spurious currents are investigated using the multiphase lattice Boltzmann method driven by a chemical potential. Both the difference error and insufficient isotropy of the discrete gradient operator give rise to directional deviations in the nonideal force, leading to spurious currents. Nevertheless, a high-order finite difference scheme produces far more accurate results than a high-order isotropic difference scheme. We compare several finite difference schemes that have different levels of accuracy and resolution. When a large proportional coefficient is used, the transition region is narrow and steep, and the resolution of the finite difference scheme provides a better indication of the computational accuracy than the formal accuracy. Conversely, for a small proportional coefficient, the transition region is wide and gentle, and the formal accuracy of the finite difference gives a better indication of the computational accuracy than the resolution. Numerical simulations show that the spurious currents calculated in the three-dimensional situation are highly consistent with those in two-dimensional simulations; in particular, the two-phase coexistence densities calculated by the high-order accuracy finite difference scheme are in excellent agreement with the theoretical predictions of the Maxwell equal-area construction down to a reduced temperature of 0.2.

High-order finite-difference entropy stable schemes for two-fluid relativistic plasma flow equations

In this article, we propose high-order finite-difference entropy stable schemes for the two-fluid relativistic plasma flow equations. This is achieved by exploiting the structure of the equations, which consists of three independent flux components. The first two components describe the ion and electron flows, which are modeled using the relativistic hydrodynamics equation and the third component is Maxwell's equations. The coupling of the ion and electron flows and electromagnetic fields is via source terms only, but the source terms do not affect the entropy evolution. To design semi-discrete entropy stable schemes, we extend the entropy stable schemes for relativistic hydrodynamics in [1] to three dimensions. This is then coupled with entropy stable discretization of the Maxwell's equations. Finally, we use SSP-RK schemes to discretize in time. We also propose ARK-IMEX schemes to treat the stiff source terms; the resulting nonlinear set of algebraic equations is local (at each discretization point) and hence can be solved cheaply using the Newton's Method. The proposed schemes are then tested using various test problems to demonstrate their stability, accuracy and efficiency.

Solving singular boundary value problems using higher-order compact finite difference schemes with a novel higher-order implementation of Robin boundary conditions

This study introduces a novel and very accurate method for implementing Robin boundary conditions while solving boundary value problems with compact finite difference schemes. Most studies numerically approximate Neumann and Robin boundary conditions with a first-order approximation, however, this reduces the accuracy in general. Our implementation produces highly accurate results in solving nonlinear singular boundary value problems. Other reported results achieved utilizing various techniques are used for comparison. Tables and graphs showing the results are given.

The construction of an optimal fourth-order fractional-compact-type numerical differential formula of the Riesz derivative and its application

In this paper, a novel optimal fourth-order fractional-compact-type numerical differential formula for the Riesz derivatives is derived by constructing the appropriate generating function. Meanwhile, some interesting and important properties of the proposed formula are also presented and proved. Then, we take the nonlinear space fractional Ginzburg–Landau equations as an example and establish a high-order finite difference scheme by combining the compact integral factor method with Padé approximation. Furthermore, based on the discrete energy analysis method, the proposed difference scheme is proved to be uniquely solvable and convergent with order O(τ2+h4) under different norms, where τ and h denote the temporal and spatial mesh step sizes, respectively. Finally, some numerical results are provided to illustrate the accuracy of numerical differential formula and the effectiveness of numerical method, and further demonstrate the correctness of theoretical analysis.

An efficient fourth-order accurate conservative scheme for Riesz space fractional Schrödinger equation with wave operator

In this study, we investigate a high-order accurate conservative finite difference scheme by utilizing a fourth-order fractional central finite difference method for the two-dimensional Riesz space-fractional nonlinear Schrödinger wave equation. The conservation laws of the discrete difference scheme are shown. Meanwhile, the exactness, uniqueness, and prior estimate of the numerical solution are rigorously established. Then, it is proved that the proposed scheme is unconditionally convergent in the discrete L2 and Hγ/2 norm, where γ is a fractional order. Furthermore, we demonstrate that when the fractional order γ and the spatial grid number J increase, the block-Toeplitz coefficient matrix generated by the spatial discretization becomes ill-conditioned. As a result, we adopt an effective linearized iteration method for the nonlinear system, allowing it to be solved efficiently by the Krylov subspace solver with an appropriate circulant preconditioner, in which the fast Fourier transform is applied to speed up the computational cost at each iterative step. Finally, numerical experiments are presented to validate the theoretical findings and the efficiency of the fast algorithm.

A high-order finite difference method with immersed-boundary treatment for fully-nonlinear wave–structure interaction

In order to predict nonlinear wave loading on marine structures, a fully nonlinear higher-order finite difference based potential flow solver with all boundary conditions treated by an immersed boundary method has been developed in this paper. The solver adopts high-order finite difference schemes for the spatial derivatives and a 4th order Runge–Kutta method for time stepping. Test cases of forced oscillation of a cylinder in an infinite fluid domain are first studied, which reveal the advantage of the acceleration potential method in terms of wave load computation. Then a wave generation problem using a piston type wave maker is tested. Special attention is paid to the intersection point between the free surface and the body surface, and a scheme which best meets the accuracy and stability requirements is suggested from several proposals. A novel hyperviscosity filter, which works for both uniform and non-uniform grids, is introduced to stabilize the time-domain solution of the wave maker problem. Finally, a forced heaving cylinder on the free surface is considered, and the nonlinear wave loads on the cylinder are analyzed and compared to benchmark results.

Efficient temporal high-order staggered-grid scheme with a dispersion-relation-preserving method for the scalar wave modeling

Staggered-grid finite-difference (FD) method is widely used to solve the wave equation for the numerical seismic modeling, and it is a key step of the advanced seismic imaging and inversion problem. However, the conventional FD method is prone to instability and dispersion error due to the insufficient approximation accuracy. In this work, we propose an efficient temporal high-order finite-difference (FD) scheme with the cross-rhombus stencil. Compared with the standard cross-rhombus method, the new method has less computational cost due to we simplify the FD scheme. Moreover, the dispersion relation of the new method is easy to obtain the dispersion-relation-preserving (DRP) FD coefficients, which can significantly alleviate the spatial and temporal dispersion errors. Dispersion and stability analyses indicate that the new scheme has better performance in seismic modeling than the conventional method, and numerical experiments also indicate that the new scheme can effectively mitigate dispersion error and improve the numerical accuracy.

Multiscale extensions for enhancing coarse grid computations

Many phenomena in Physics and Engineering are modeled by Partial Differential Equations. Typically, analytical solutions of such problems are unknown. Hence, numerical schemes are applied for approximating the exact solutions. In order to accurately approximate the exact solutions, very fine grids are required. However, the computations become too costly. In this work, a multiscale iterative approach is utilized for enhancing the accuracy of coarse grid computations which results from a high-order finite difference scheme. The main idea is to run the finite difference computations on a coarse grid until some intermediate time level. Then, the coarse grid results are interpolated and extended to a fine grid using Laplacian Pyramids, a multiscale iterative approach. Finally, the finite difference scheme is employed on the finer grid until a prescribed final time. Comparing the results to those obtained on a fine grid, we see that the convergence rates obtained from the two methods are comparable, while the computational time is significantly reduced.A modified multimodal Laplacian Pyramids algorithm for predicting future values of the solution is also suggested. The method approximates and extends a function based on two or more input modalities coded by a series of multiscale kernels, which are averaged as a convex combination. In this work, the modalities are the numerical model’s approximations of the solution of the differential equation and its derivative at previous time steps, and the goal is to predict the solution at a proceeding time step. It can be seen that, by adapting the convex combination of the kernels to local regions of the solution with stronger or weaker gradients, the predicted results are improved.

New computational technique based on Bernstein polynomials and higher order finite difference schemes for temporal and spatial reactor calculations

In this research, two main objectives about solving the space time dependent diffusion equations are addressed. The first is to present higher order finite difference schemes at the core interior meshes and also at the points adjacent to the core boundaries, for spatial discretization. The second is to provide an accurate iteratively numerical method for temporal calculations based on convergent Bernstein polynomials. Bernstein series coefficients are estimated by using an easily computable recurrence relation. Estimation of the Bernstein recurrence relation requires nothing except the previous estimation of the series coefficients. Stability and convergence of the proposed method is discussed. It has proved that the method is conditionally stable and it has an exponential rate of convergence. Recently, we provided fourth and sixth order schemes to approximate the neutron leakage terms. These schemes provided highly accurate results but a large amount of these gained accuracies are lost due to using the classical second order finite difference formula at the core boundaries which uses three point stencils with order of accuracy OΔ2. Here, Lagrange’ polynomials are used to construct another second order scheme using five point stencils instead of three points at the core edges. It is proved that using central fourth order scheme at the interior meshes associated with the new scheme at the boundaries significantly improves the accuracy. Also, it is proved that using such finite difference schemes do not affect the system complexity and the system size remains unchanged. Homogenous and heterogeneous problems are considered to test the efficiency and the validity of the proposed method. In addition to the simplicity of the method algorithm, the results are so accurate that they can be compared with the recently methods that divided the same cores into thousands of fuel assemblies using mesh generator tools like GAMBIT or ICEM-CFD.

Model-based Estimator Design for the Curing Process of a Concrete Structure

Concrete is one of the most important building materials and is used as a collective term for different dispersions of cement and aggregates. High performance concrete (HPC) is a rather new type of concrete mixture. A topic of current research is the behavior of fresh concrete during heat treatment. It is known that the heat treatment of fresh concrete accelerates the reaction of the cement and thus the chemically complex process of hydration. The mathematical representation of the curing concrete is a further step towards the systematic investigation. The objective of this work is the development of a state estimator to determine the temperature behavior of the fresh HPC mixture under heat treatment. Using a continuum representation describing the spatial-temporal evolution of temperature, moisture, and maturity of fresh HPC during heat treatment a finite-dimensional approximation is derived using a high order finite difference scheme. Experimental data is included for model parameterization by optimization. Based on this, three different nonlinear extensions of the Kalman filter (KF) techniques are realized and compared combining simulated and experimental data.

High-order finite difference approximation of the Keller-Segel model with additional self- and cross-diffusion terms and a logistic source

<abstract><p>In this paper, we consider the Keller-Segel chemotaxis model with self- and cross-diffusion terms and a logistic source. This system consists of a fully nonlinear reaction-diffusion equation with additional cross-diffusion. We establish some high-order finite difference schemes for solving one- and two-dimensional problems. The truncation error remainder correction method and fourth-order Padé compact schemes are employed to approximate the spatial and temporal derivatives, respectively. It is shown that the numerical schemes yield second-order accuracy in time and fourth-order accuracy in space. Some numerical experiments are demonstrated to verify the accuracy and reliability of the proposed schemes. Furthermore, the blow-up phenomenon and bacterial pattern formation are numerically simulated.</p></abstract>

High-order compact finite difference schemes for solving the regularized long-wave equation

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.

A high order moving boundary treatment for convection-diffusion equations

In this paper, a new type of inverse Lax-Wendroff boundary treatment is designed for high order finite difference schemes for solving general convection-diffusion equations on time-varying domain. This new method can achieve high order accuracy on Dirichlet boundary conditions with moving boundary. To ensure stability of the boundary treatment, we give a convex combination of the boundary treatments for the diffusion-dominated and the convection-dominated cases. A group convex combination of weights is carefully designed to avoid zero denominator, resulting in a unified algorithm for pure convection, convection-dominated, convection-diffusion, diffusion-dominated and pure diffusion cases. In order to match the time levels when constructing values of ghost points in the two intermediate stages of the third order Runge-Kutta method, we propose a new approximation to the mixed derivatives at the boundaries to ensure high order accuracy and to improve computational efficiency. In particular, we extend the boundary treatment to the compressible Navier-Stokes equations, which satisfies the isothermal no-slip wall boundary condition at any Reynolds number. We provide numerical tests for one- and two-dimensional problems involving both scalar equations and systems, demonstrating that our boundary treatment is high order accurate for problems with smooth solutions and also performs well for problems involving interactions between viscous shocks and moving rigid bodies.

Efficacy of line-based explicit and compact high-order finite difference schemes for hybrid unstructured grids

This study enables the use of explicit and compact high-order finite-difference schemes with a line-based solver to solve conservation laws on unstructured grids. A quadrilateral subdivision process is used to identify unique line structures (also known as Hamiltonian loops) before formulating stencil-based discretizations. To demonstrate the methodology for canonical flows represented by the Navier–Stokes equations, up to sixth-order spatial discretization and a maximum of tenth-order low-pass filters are implemented along with the Hamiltonian loops. The filter restores the benefits of the high-order approach even in the presence of abrupt grid discontinuities that cause abrupt changes in loop curvature. Isentropic vortex convection, lid-driven cavity, double periodic shear layer, and inviscid and viscous flow past a cylinder and NACA 0012 airfoil are all used to demonstrate the formulation. The ability of some schemes, particularly the fourth-order explicit, to nearly achieve the formal order of accuracy and successfully predict the flow following available results in the literature is one of the key findings. Compact schemes have been found to be more sensitive to loop curvature than explicit schemes.

An explicit high-order compact finite difference scheme for the three-dimensional acoustic wave equation with variable speed of sound

In this paper, the correction for the remainder of the truncation error of the second-order central difference scheme is employed to discretize temporal derivative, while the fourth-order Padé schemes are directly used to compute spatial derivatives, an explicit high-order compact finite difference scheme to solve the three-dimensional acoustic wave equation with variable speed of sound is proposed. This new scheme has the fourth-order accuracy in both temporal and spatial directions. It has high computational efficiency since the Thomas algorithm is employed to solve three tridiagonal linear systems formed by the Padé schemes on the (n)th time step and then an explicit time advancement process is conducted for the th time step. The stability and convergence condition of the proposed scheme are proved. We extend the proposed method to solve problems with nonlinear source terms and systems. Numerical experiments are conducted to demonstrate theoretical analysis results of the proposed scheme.

Highly Accurate Compact Finite Difference Schemes for Two-Point Boundary Value Problems with Robin Boundary Conditions

In this study, a high-order compact finite difference method is used to solve boundary value problems with Robin boundary conditions. The norm is to use a first-order finite difference scheme to approximate Neumann and Robin boundary conditions, but that compromises the accuracy of the entire scheme. As a result, new higher-order finite difference schemes for approximating Robin boundary conditions are developed in this work. Six examples for testing the applicability and performance of the method are considered. Convergence analysis is provided, and it is consistent with the numerical results. The results are compared with the exact solutions and published results from other methods. The method produces highly accurate results, which are displayed in tables and graphs.