High-order Accurate Difference Scheme Research Articles

A new high-order accurate difference scheme for the Benjamin-Bona-Mahoney-Burgers (BBMB) equation

In this article, a high-order linearized difference scheme is presented for the periodic initial value problem of the Benjamin-Bona-Mahoney-Burgers (BBMB) equation. It is proved that the proposed scheme is uniquely solvable and unconditionally convergent, with convergence order of O(h4 + k2) in the L?-norm. An application on the regularised long wave is thoroughly studied numerically. Furthermore, interaction of solitary waves with different amplitudes is shown. The three invariants of the motion are evaluated to determine the conservation properties of the system. Numerical experiments including the comparisons with other numerical methods are reported to demonstrate the accuracy and efficiency of our difference scheme and to confirm the theoretical analysis.

Convergence of two conservative high-order accurate difference schemes for the generalized Rosenau–Kawahara-RLW equation

In this paper, we present two high-order accurate difference schemes for the generalized Rosenau–Kawahara-RLW equation. The proposed schemes guarantee the conservation of the discrete energy. The unique solvability of the difference solution is proved. A priori error estimates for the numerical solution is derived. Convergence and stability of the difference schemes are proved. The convergence order is $$O(h^{4} + k^{2} )$$ in the uniform norm is discussed without any restrictions on the mesh sizes. Finally, numerical experiments are carried out to support the theoretical claims.

Modified method of splitting with respect to physical processes for solving radiation gas dynamics equations

An approach based on a modified splitting method is proposed for solving the radiation gas dynamics equations in the multigroup kinetic approximation. The idea of the approach is that the original system of equations is split using the thermal radiation transfer equation rather than the energy equation. As a result, analytical methods can be used to solve integrodifferential equations and problems can be computed in the multigroup kinetic approximation without iteration with respect to the collision integral or matrix inversion. Moreover, the approach can naturally be extended to multidimensional problems. A high-order accurate difference scheme is constructed using an approximate Godunov solver for the Riemann problem in two-temperature gas dynamics.

A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation

An initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. For this problem, a technique is developed for constructing higher order accurate difference schemes that converge ɛ-uniformly in the maximum norm (where ɛ is the perturbation parameter multiplying the highest order derivative, ɛ ∈ (0, 1]). A solution decomposition scheme is described in which the grid subproblems for the regular and singular solution components are considered on uniform meshes. The Richardson technique is used to construct a higher order accurate solution decomposition scheme whose solution converges ɛ-uniformly in the maximum norm at a rate of Open image in new window, where N + 1 and N0 + 1 are the numbers of nodes in uniform meshes in x and t, respectively. Also, a new numerical-analytical Richardson scheme for the solution decomposition method is developed. Relying on the approach proposed, improved difference schemes can be constructed by applying the solution decomposition method and the Richardson extrapolation method when the number of embedded grids is more than two. These schemes converge ɛ-uniformly with an order close to the sixth in x and equal to the third in t.

A new conservative high-order accurate difference scheme for the Rosenau equation

This article is devoted to the study of high-order conservative difference scheme for the Rosenau equation. The difference scheme is two-layered and a seven-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 scheme are also derived. 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 -norm. Some numerical examples are given in order to validate the theoretical results.

Efficiency of High-Order Accurate Difference Schemes for the Korteweg-de Vries Equation

Two numerical models to obtain the solution of the KdV equation are proposed. Numerical tools, compact fourth-order and standard fourth-order finite difference techniques, are applied to the KdV equation. The fundamental conservative properties of the equation are preserved by the finite difference methods. Linear stability analysis of two methods is presented by the Von Neumann analysis. The new methods give second- and fourth-order accuracy in time and space, respectively. The numerical experiments show that the proposed methods improve the accuracy of the solution significantly.

High-order accurate difference schemes for the Hodgkin–Huxley equations

A novel approach for simulating potential propagation in neuronal branches with high accuracy is developed. The method relies on high-order accurate difference schemes using the Summation-By-Parts operators with weak boundary and interface conditions applied to the Hodgkin–Huxley equations. This work is the first demonstrating high accuracy for that equation. Several boundary conditions are considered including the non-standard one accounting for the soma presence, which is characterized by its own partial differential equation. Well-posedness for the continuous problem as well as stability of the discrete approximation is proved for all the boundary conditions. Gains in terms of CPU times are observed when high-order operators are used, demonstrating the advantage of the high-order schemes for simulating potential propagation in large neuronal trees.

High-order accurate monotone difference schemes for solving gasdynamic problems by Godunov’s method with antidiffusion

An approach to the construction of high-order accurate monotone difference schemes for solving gasdynamic problems by Godunov’s method with antidiffusion is proposed. Godunov’s theorem on monotone schemes is used to construct a new antidiffusion flux limiter in high-order accurate difference schemes as applied to linear advection equations with constant coefficients. The efficiency of the approach is demonstrated by solving linear advection equations with constant coefficients and one-dimensional gasdynamic equations.

Factorized SM-stable two-level schemes

Additional requirements for unconditionally stable schemes were formulated by analyzing higher order accurate difference schemes in time as applied to boundary value problems for second-order parabolic equations. These requirements concern the inheritance of the basic properties of the differential problem and lead to the concept of an SM-stable difference scheme. An earlier distinguished class of SM-stable schemes consists of the schemes based on various Pade approximations. The computer implementation of such higher order accurate schemes deserves special consideration because certain matrix polynomials must be inverted at each new time level. Factorized SM-stable difference schemes are constructed that can be interpreted as diagonally implicit Runge-Kutta methods.

A version of essentially nonoscillatory high-order accurate difference schemes for systems of conservation laws

A version of an essentially nonoscillatory high-order accurate difference scheme for systems of conservation laws is proposed and based on norm minimization of the deviation of the inter-polation polynomial from the integral cell average. This can result in a more monotonous solution as compared to conventional ENO and WENO methods of the respective order, which is justified by the results of numerical calculations for model problems.

High-order accurate difference schemes for solving gasdynamic equations by the Godunov method with antidiffusion

A technique is proposed for improving the accuracy of the Godunov method as applied to gasdynamic simulations in one dimension. The underlying idea is the reconstruction of fluxes arsoss cell boundaries (“large” values) by using antidiffusion corrections, which are obtained by analyzing the differential approximation of the schemes. In contrast to other approaches, the reconstructed values are not the initial data but rather large values calculated by solving the Riemann problem. The approach is efficient and yields higher accuracy difference schemes with a high resolution.

Arbitrary-order difference schemes for solving linear advection equations with constant coefficients by the Godunov method with antidiffusion

An approach to the construction of second-and higher order accurate difference schemes in time and space is described for solving the linear one-and multidimensional advection equations with constant coefficients by the Godunov method with antidiffusion. The differential approximations for schemes of up to the fifth order are constructed and written. For multidimensional advection equations with constant coefficients, it is shown that Godunov schemes with splitting over spatial variables are preferable, since they have a smaller truncation error than schemes without splitting. The high resolution and efficiency of the difference schemes are demonstrated using test computations.

Direct Numerical Simulation of Supersonic Turbulent BoundaryLayer Flow

Direct numerical simulations of a spatially evolving supersonicflat-plate turbulent boundary layer flow with free Mach numberM∞ = 2.25 and Reynolds number Re = 365000/in are performed.The transition process from laminar to turbulent flow is obtained bysolving the three-dimensional compressible Navier–Stokes equations,using high-order accurate difference schemes. The obtained statisticalresults agree well with the experimental and theoretical data. From thenumerical results it can be seen that the transition process under theconsidered conditions is the process which skips theTollmien–Schlichting instability and the second instability throughthe instability of high gradient shear layer and becomes of laminarflow breakdown. This means that the transition process is a bypass-typetransition process. The spanwise asymmetry of the disturbance locallyupstream imposed is important to induce the bypass-type transition.Furthermore, with increasing the time disturbance frequency thetransition will delay. When the time disturbance frequency is largeenough, the transition will disappear.

Compact difference approximation with consistent boundary condition*

For simulating multi-scale complex flow fields it should be noted that all the physical quantities we are interested in must be simulated well. With limitation of the computer resources it is preferred to use high order accurate difference schemes. Because of their high accuracy and small stencil of grid points computational fluid dynamics (CFD) workers pay more attention to compact schemes recently. For simulating the complex flow fields the treatment of boundary conditions at the far field boundary points and near far field boundary points is very important. According to authors' experience and published results some aspects of boundary condition treatment for far field boundary are presented, and the emphasis is on treatment of boundary conditions for the upwind compact schemes. The consistent treatment of boundary conditions at the near boundary points is also discussed. At the end of the paper are given some numerical examples. The computed results with presented method are satisfactory.

Simulating disinfection processes in chlorine contact tanks using various turbulence models and high-order accurate difference schemes

The numerical simulation of the disinfection processes in a contact tank involves determining both the flow and solute transport distributions. In this study it has been established that the reliability of the solute transport predictions depends strongly upon the accuracy of the hydrodynamic predictions. In the hydrodynamic computations, it is common practice to combine various turbulence models and numerical schemes to approximate the shear stress and advection terms, e.g. the depth mean eddy viscosity model with either a first-order or third-order upwind difference scheme; the k- ε model or the Smagorinsky model in Large Eddy Simulation (LES) with a first-order upwind scheme, etc. A question which this study also sought to answer, is to what extent the accuracy of the discretization of the advection terms in the solute transport equations affects the discrepancy between the concentration prediction and the physical model, or field measured results, whilst the flow is simulated using various difference and turbulence models. In this study the advection terms in the solute transport equation were evaluated using the QUICK and the third-order upwind difference schemes in separate simulation runs. The numerical predictions of the dye transport within a contact tank for a pulse injection, were compared with experimental measurements carried out in a physical model. It has been demonstrated that the reliability of some predicted bulk characteristics of the flow in a contact tank, such as the depth of flow and the cross-sectional averaged velocity, are of crucial importance in the computation of solute transport processes.

A High Order Accurate Difference Scheme for Complex Flow Fields

A high order accurate finite difference method for direct numerical simulation of coherent structure in the mixing layers is presented. The reason for oscillation production in numerical solutions is analyzed. It is caused by a nonuniform group velocity of wavepackets. A method of group velocity control for the improvement of the shock resolution is presented. In numerical simulation the fifth-order accurate upwind compact difference relation is used to approximate the derivatives in the convection terms of the compressible N–S equations, a sixth-order accurate symmetric compact difference relation is used to approximate the viscous terms, and a three-stage R–K method is used to advance in time. In order to improve the shock resolution the scheme is reconstructed with the method of diffusion analogy which is used to control the group velocity of wavepackets.

Convergence of Finite Difference Schemes for Conservation Laws in Several Space Dimensions: A General Theory

A general framework is proposed for proving convergence of high-order accurate difference schemes for the approximation of conservation laws with several space variables. The standard approach deduces compactness from a BV (bounded variation) stability estimate and Helly's theorem. In this paper, it is proved that an a priori estimate weaker than a BV estimate is sufficient. The method of proof is based on the result of uniqueness given by Di Perna in the class of measure-valued solutions. Several general theorems of convergence are given in the spirit of the Lax-Wendroff theorem. This general method is then applied to the high-order schemes constructed with the modified-flux approach.

Numerical Analysis and Steam-Tunnel Tests of Transonic Flows throught a SteamTurbine Cascade.

An implicit time-marching finite-difference method for solving two-dimensional compressible Navier-Stokes equations is applied to the numerical analysis of transonic viscous flows through a steam turbine cascade. The distinctive feature of this method is its use of the momentum equations of contravariant velocity components. A high-order accurate difference scheme modified from the Chakravarthy-Osher TVD scheme is used in order to clearly capture shock waves which occur at the blade trailing edges. Steam-tunnel tests to measure losses of a linear cascade of transonic steam turbine blades are carried out in order to verify the effectiveness of the numerical method. Computed shock wave structures agree well with Schlieren photographs of the cascade tests. The agreement between the computed results and the measurements of total pressure loss coefficients and those of total pressure profiles in the blade wakes is also found to be reasonable.

Stability of High Order Accurate Difference Methods for Parabolic Equations with Boundary Conditions

We consider high order accurate difference schemes for second order parabolic equations in the quarter-plane $x \geqq 0,\, t \geqq 0$. We discuss the stability of these schemes for the mixed initial boundary value problem when particular high order accurate discrete boundary conditions are used.