High-order Compact Finite Difference Scheme Research Articles

A high-order compact ADI finite difference scheme on uniform meshes for a weakly singular integro-differential equation in three space dimensions

A high-order compact ADI finite difference scheme on uniform meshes for a weakly singular integro-differential equation in three space dimensions

Analysis of the Dominant Structures of an Impinging Round Jet at M=0.9

Large-eddy simulation of subsonic round jet impinging on a flat circular plate is performed at Mach number 0.9 and Reynolds number 25,000, based on the diameter and centerline velocity of the jet for two impingement distances h of 6r0 and 8r0 using high-order compact finite difference scheme. The complex flow phenomena associated with the impinging jet flowfield are studied. The power spectral density (PSD) of instantaneous pressure reveals a discrete impingement tone. Dynamic mode decomposition (DMD) of the velocity and pressure is also done for both cases to extract the most dominant modes of the flow. The fundamental impingement tone frequency obtained from the PSD of pressure is identical to that of the first dominant pressure mode obtained from DMD. Both axisymmetric and semihelical modes are present for the h=8r0 case, but only axisymmetric modes are there in the h=6r0 case, and it is noted that the modes corresponding to the fundamental tone have an axisymmetric structure. Also, one dominant mode showing the wall jet structures denoting the liftoff of fluid from the wall after impingement at some radial distance away from the stagnation region is present in both impingement distance cases.

An efficient high-order compact approach for spiral wave dynamics by the FHN model

We simulate spiral waves through the famous Fitzhugh–Nagumo (FHN) model of excitable media by employing zero flux and periodic boundary conditions. This model has been solved numerically by modifying a recently developed implicit high-order compact (HOC) finite difference scheme. We also propose a novel approach for implementing the HOC scheme on boundaries of the domain for the periodic boundary conditions. Henceforth, we investigate the effects of the zero flux and periodic boundary conditions on the patterns in excitable media and the resulting dynamics. The introduction of periodic boundary conditions is seen to trigger the breakup of the spiral waves as opposed to the ones found by imposing no-flux boundary conditions. Our computed grid independent solutions are found to be extremely close to well-known numerical results. In the process, we also probe the unconditional stability of the HOC discretization for the FHN model.

Error estimates of high-order compact finite difference schemes for the nonlinear abcd Boussinesq 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 3D frequency-domain electromagnetic solver employing a high order compact finite-difference scheme

Three-dimensional (3D) electromagnetic (EM) modeling is an important tool for geophysical applications. In EM induction methods the modified Helmholtz equation is often used to describe scattered or residual electric fields in three dimensions. Throughout this paper, a high order compact finite difference scheme for the solution of that equation for vertical magnetic dipole source (VMD) is presented. The approximation of the residual electric field intensity using a fourth order compact finite difference (FD) discretizer is achieved with the solution of a block linear system, the coefficient matrices are large and sparse with a particular structure, implying the application of a matrix-free Krylov subspace method for an efficient numerical solution. The proposed solver is being examined using a number of test problems in uniform and non-uniform grid spacing where numerical and analytical solutions for the homogeneous half-space cases are being compared.

Numerical investigation of the cavitating flow and the cavitation-induced noise around one and two circular cylinders

In this work, we simulated a two-phase cavitating flow around one and two circular cylinders with the density based homogeneous mixture model. We discretized the governing equations with employing a high-order compact finite-difference scheme incorporating filtering scheme to account for the numerical instabilities and physical discontinuities. We applied a shock capturing sensor for discontinuities detection and switching between the second-order and high-order filtering and computed the far-field acoustic by the Ffowcs Williams-Hawkings surface integral method. We studied the flow features in the wake of a circular cylinder and two cylinders for noncavitating regime and cavitating conditions. In addition, we analyzed the cavitating flow for various gaps of two side-by-side cylinders. Results revealed that the shock waves due to the collapse of the cavity moving downstream of the cylinders is the main mechanism for the generation of the acoustic impulses. The results also showed that the noise peak can occur at lower frequencies for the smaller gap between the two cylinders. Furthermore, the wakes behind the side-by-side cylinders were merged together and a single vortex street was generated by the reduction of the gap between two cylinders.

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.

An efficient extrapolation multigrid method based on a HOC scheme on nonuniform rectilinear grids for solving 3D anisotropic convection–diffusion problems

We develop an efficient multigrid method combined with a high-order compact (HOC) finite difference scheme on nonuniform rectilinear grids for solving 3D diagonal anisotropic convection–diffusion problems with boundary/interior layers. Firstly, we derive a fourth-order compact finite difference scheme to discretize the 3D anisotropic convection–diffusion equation on a rectilinear grid. Then, the resulting large-scale asymmetric linear system of equations is solved by a generalized extrapolation cascadic multigrid (gEXCMG) method based on two novel multigrid (MG) prolongation operators. The highlight of this paper is the application of the quintic Lagrange interpolation and the completed Richardson extrapolation in the design of the new MG prolongation operator on nonuniform rectilinear grids, which can produce a good initial guess (sixth-order approximation to the finite difference solution) for the SSOR-preconditioned biconjugate gradient stabilized (BiCGStab) smoother. In the end, numerical experiments show that the gEXCMG method combined with the HOC scheme can achieve fourth-order accuracy for 3D anisotropic convection–diffusion problems with few smoothing steps on the finest grid. Moreover, the proposed gEXCMG method can offer substantially better efficiency than the state-of-the-art algebraic MG solver, namely, aggregation-based algebraic multigrid (AGMG) method, for large linear systems arising from the discretization of second order elliptic PDEs.

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.

High-order compact finite difference schemes for the time-fractional Black-Scholes model governing European options

Based on the price fluctuations of the underlying fractal transmission system, the α-order time-fractional Black-Scholes model is obtained. In this paper, we introduce two compact finite difference schemes for the solution of the time-fractional Black-Scholes equation governing European option pricing. In proposed schemes, in order to gain sixth-order and eighth-order accuracy in space, we first use exponential transformation to eliminate the convection term of the Black-Scholes equation. Then, the time-fractional derivative discretizes by a 3−αth order numerical formula (called the L1–2 formula here) which is constructed by a linear interpolating polynomial on the first subinterval and the quadratic interpolating polynomials on the other subintervals. We investigate stability and convergence of proposed schemes by Fourier method. Finally, some numerical examples perform to demonstrate the theoretical order of accuracy and illustrate the effectiveness of proposed methods. We also discuss the influence of different parameters on the option price in the time-fractional Black-Scholes model.

High-order compact finite difference scheme with two conserving invariants for the coupled nonlinear Schrödinger–KdV equations

In this paper, a new high-order accurate compact finite difference scheme for the coupled nonlinear Schrödinger–KdV (CNLS–KdV) equations is proposed. Conservation of the discrete number of plasmon and the discrete number of particle are given in detail. Convergence with second-order in time and fourth-order in space of the present scheme are proved by using “cut-off” function technique and discrete energy method. Numerical experiments are given to support the theoretical analysis.

A class of HOC finite difference method for elliptic interface problems with imperfect contact

<abstract><p>The elliptic interface problems with imperfect contact have found applications in numerical solutions of the Stefan problem of the solidification process and crystal growth, composite materials, multi-phase flows, etc. In this paper a 1D elliptic interface problem with imperfect contact is considered. A class of high-order compact finite difference schemes are constructed on body-fitted and non-body-fitted mesh, respectively. For each case, the second-, third- and fourth-order approximations of implicit jump conditions are provided by using the jump conditions and its high-order derivatives. Numerical examples are provided to verify the performance of the schemes. The numerical results demonstrate that the schemes have theoretical accuracy for elliptic interface problems with imperfect contact.</p></abstract>

Fast high-accuracy compact conservative difference schemes for solving the nonlinear Schrödinger equation

Fast high-order compact finite difference schemes are investigated for solving the two-dimensional nonlinear Schrödinger equation with periodic boundary conditions. These schemes are convergent of order s in space, where () with second-order temporal accuracy. The discrete conservation laws and convergence of the finite difference schemes are rigorously demonstrated. Thanks to the circulant matrices resulting from spatial discretization, we significantly reduce the computation complexity and storage requirement of the proposed schemes via fast Fourier transform. Numerical examples are presented to show the accuracy and efficiency of these schemes and verify the theoretical analysis. Moreover, we extend the tenth-order scheme to solve the generalized nonlinear Schrödinger equation.

Time-adaptive high-order compact finite difference schemes for option pricing in a family of stochastic volatility models

We propose a time-adaptive high-order compact finite difference scheme for option pricing in a family of stochastic volatility models. We employ a semi-discrete high-order compact finite difference method for the spatial discretisation, and combine this with an adaptive time discretisation, extending ideas from [LSRHF02] to fourth-order multistep methods in time.

High-order efficient numerical method for solving a generalized fractional Oldroyd-B fluid model

This paper investigates the high-order efficient numerical method with the corresponding stability and convergence analysis for the generalized fractional Oldroyd-B fluid model. Firstly, a high-order compact finite difference scheme is derived with accuracy $$O\left( \tau ^{\min {\{3-\gamma ,2-\alpha }\}}+h^{4}\right) $$ , where $$\gamma \in (1,2)$$ and $$\alpha \in (0,1)$$ are the orders of the time fractional derivatives. Then, by means of a new inner product, the unconditional stability and convergence in the maximum norm of the derived high-order numerical method have been discussed rigorously using the energy method. Finally, numerical experiments are presented to test the convergence order in the temporal and spatial direction, respectively. To precisely demonstrate the computational efficiency of the derived high-order numerical method, the maximum norm error and the CPU time are measured in contrast with the second-order finite difference scheme for the same temporal grid size. Additionally, the derived high-order numerical method has been applied to solve and analyze the flow problem of an incompressible Oldroyd-B fluid with fractional derivative model bounded by two infinite parallel rigid plates.

On high-order compact schemes for the multidimensional time-fractional Schr\xf6dinger equation

In this article, some high-order compact finite difference schemes are presented and analyzed to numerically solve one- and two-dimensional time fractional Schrödinger equations. The time Caputo fractional derivative is evaluated by the L1 and L1-2 approximation. The space discretization is based on the fourth-order compact finite difference method. For the one-dimensional problem, the rates of the presented schemes are of order O(tau ^{2-alpha }+h^{4}) and O(tau ^{3-alpha }+h^{4}), respectively, with the temporal step size τ and the spatial step size h, and alpha in (0,1). For the two-dimensional problem, the high-order compact alternating direction implicit method is used. Moreover, unconditional stability of the proposed schemes is discussed by using the Fourier analysis method. Numerical tests are performed to support the theoretical results, and these show the accuracy and efficiency of the proposed schemes.

Acceleration of high-order combined compact finite difference scheme for simulating three-dimensional flow and heat transfer problems in GPUs

In this article, the high-order upwinding combined compact difference scheme developed in a three-point grid stencil is applied to solve the incompressible Navier-Stokes (NS) and energy equations in three dimensions. The time integrator with symplectic property is employed to approximate the temporal derivative term in inviscid Euler equation so as to numerically retain the embedded Hamiltions and Casimir to get long-time accurate solutions. For the sake of computational efficiency in solving the three-dimensional NS equations, all the calculations will be accelerated using the hybrid CUDA and OpenAcc GPU programing models. The parallel speedup performance compared to the multicore of an Intel Xeon E5-2690V5 CPU is reported.

A high-order compact finite difference scheme and precise integration method based on modified Hopf-Cole transformation for numerical simulation of n-dimensional Burgers’ system

This paper modifies a n-dimensional Hopf-Cole transformation to the n-dimensional Burgers’ system. We obtain the n-dimensional heat conduction equation through the modification of the Hopf-Cole transformation. Then the fourth-order precise integration method (PIM) in combination with a spatially global sixth-order compact finite difference (CFD) scheme is presented to solve the equation with high accuracy. Moreover, coupling with the Strang splitting method, the scheme is extended to multi-dimensional (two, three-dimensional) Burgers’ system. Numerical results show that the proposed method appreciably improves the computational accuracy compared with the existing numerical method. Moreover, the two-dimensional and three-dimensional examples demonstrate excellent adaptability, and the numerical simulation results also have very high accuracy in medium Reynolds numbers.

A new high-order compact finite difference scheme based on precise integration method for the numerical simulation of parabolic equations

This paper presents two high-order exponential time differencing precise integration methods (PIMs) in combination with a spatially global sixth-order compact finite difference scheme (CFDS) for solving parabolic equations with high accuracy. One scheme is a modification of the compact finite difference scheme of precise integration method (CFDS-PIM) based on the fourth-order Taylor approximation and the other is a modification of the CFDS-PIM based on a (4,4)-Padé approximation. Moreover, on coupling with the Strang splitting method, these schemes are extended to multi-dimensional problems, which also have fast computational efficiency and high computational accuracy. Several numerical examples are carried out in order to demonstrate the performance and ability of the proposed schemes. Numerical results indicate that the proposed schemes improve remarkably the computational accuracy rather than the empirical finite difference scheme. Moreover, these examples show that the CFDS-PIM based on the fourth-order Taylor approximation yields more accurate results than the CFDS-PIM based on the (4,4)-Padé approximation.