Combined Compact Difference Research Articles

2D acoustic equation prestack reverse-time migration based on an optimized combined compact difference scheme

Abstract Reverse-time migration (RTM) is widely regarded as one of the most accurate migration methods available today. A crucial step in RTM involves extending seismic wavefields forward and backward. Compared to the conventional central finite-difference (CFD) scheme, the combined compact difference (CCD) scheme offers several advantages, including a shorter difference operator and the suppression of numerical dispersion under coarse grids. These attributes conserve memory and enhance effectiveness while maintaining the same level of differential precision. In this article, we begin with the five-point eighth-order CCD scheme and utilize the least squares method and Lagrange multiplier method to optimize the difference coefficients. This optimization is guided by the concept of dispersion-relation-preserving (DRP). The result is the acquisition of an optimized combined compact difference (OCCD) scheme, further enhancing the ability to suppress numerical dispersion. We thoroughly compare and analyze dispersion relationships and stability conditions. In addition, we examine several crucial steps in the RTM of the second-order acoustic wave equation. These steps include absorption boundary conditions, boundary storage strategy, and Poynting vector imaging conditions. Finally, we apply both the CCD and OCCD schemes in the RTM of the layered model, graben model, and SEG/EAGE salt model. We compare these results with those obtained from CFD's RTM. Numerical findings demonstrate that, in contrast to the CFD scheme, the CCD scheme effectively suppresses numerical dispersion and enhances imaging accuracy. Moreover, the optimized OCCD scheme further improves the ability to suppress numerical dispersion and can obtain better imaging results, which is an effective RTM method suitable for coarse grid conditions.

A combined compact finite difference scheme for solving the acoustic wave equation in heterogeneous media

AbstractIn this paper, we consider the development and analysis of a new explicit compact high‐order finite difference scheme for acoustic wave equation formulated in divergence form, which is widely used to describe seismic wave propagation through a heterogeneous media with variable media density and acoustic velocity. The new scheme is compact and of fourth‐order accuracy in space and second‐order accuracy in time. The compactness of the scheme is obtained by the so‐called combined finite difference method, which utilizes the boundary values of the spatial derivatives and those boundary values are obtained by one‐sided finite difference approximation. An empirical stability analysis has been conducted to obtain the Courant‐Friedrichs‐Levy (CFL) condition, which confirmed the conditional stability of the new scheme. Four numerical examples have been conducted to validate the convergence and effectiveness of the new scheme. The application of the new scheme to a realistic wave propagation problem with a Perfect Matched Layer is validated in this paper as well.

Reverse Time Migration Imaging Using SH Shear Wave Data

In this paper, we discussed the reverse time migration imaging of compressional wave (P-wave) and horizontally polarized shear wave (SH shear wave) seismic data, together with P- and SH shear wave constrained velocity model building. In the Sanhu area in Qaidam Basin, there are large areas of gas clouds, which leads to poor P-wave seismic imaging. The P and SH shear wave seismic data of a co-located survey line with the same acquisition geometry were used to access their imaging capability using reverse time migration. We first estimated the change in P-wave and SH shear wave velocity ratio using pre-stack time migration (PSTM) for constraining the overall depth domain velocity model. Additionally, we then used an eighth-order finite difference scheme for P-wave reverse time migration on a variable grid and used the sixth-order combined compact difference (CCD) wave field simulation method for SH shear wave reverse time migration on a regular grid. The results show that the constrained velocity model produces a good match in the overall geological structure shown in the P-wave and SH shear wave reverse time migration results. However, in the gas cloud areas, SH shear wave reverse time migration has obvious imaging advantages, which can clearly image the structure inside the gas clouds.

The Combined Compact Difference Scheme Applied to Shear-Wave Reverse-Time Migration

In this paper, the combined compact difference scheme (CCD) and the combined supercompact difference scheme (CSCD) are used in the numerical simulation of the shear-wave equation. According to the Taylor series expansion and shear-wave equation, the fourth-order discrete scheme of the displacement field is established; then, the CCD and CSCD schemes are used to calculate the spatial derivative of the displacement field. Additionally, the accuracy, dispersion, and stability of the CCD and CSCD are analyzed, and numerical simulation analyses are carried out using 1D uniform models. Lastly, based on the processing of artificial boundary reflection using PML boundary conditions, shear-wave reverse-time migrations are carried out using synthetic data. The results show that (1) CCD and CSCD have smaller truncation errors, higher simulation precision, and lower numerical dispersion than other normal difference schemes; (2) CCD and CSCD can use the coarse grid and larger time step to calculate, with less memory and high computational efficiency; (3) finally, the result of the shear-wave reverse-time migration of the 2D synthetic data model show that the reverse-time migration imaging is clear, and the proposed method for shear-wave reverse-time migration is practical and effective.

Sixth-Order Combined Compact Finite Difference Scheme for the Numerical Solution of One-Dimensional Advection-Diffusion Equation with Variable Parameters

A high-accuracy numerical method based on a sixth-order combined compact difference scheme and the method of lines approach is proposed for the advection–diffusion transport equation with variable parameters. In this approach, the partial differential equation representing the advection-diffusion equation is converted into many ordinary differential equations. These time-dependent ordinary differential equations are then solved using an explicit fourth order Runge–Kutta method. Three test problems are studied to demonstrate the accuracy of the present methods. Numerical solutions obtained by the proposed method are compared with the analytical solutions and the available numerical solutions given in the literature. In addition to requiring less CPU time, the proposed method produces more accurate and more stable results than the numerical methods given in the literature.

Numerical study of incompressible interfacial flows by an one-step level set method

We develop a two-phase flow model for solving the nonlinear level set (LS) equation and Navier-Stokes equations in two dimension. In the proposed one-step LS method, the sharp interface is captured by implicitly representing the zero LS contour. The nonlinear LS equation is solved by upwinding Combined Compact Difference (CCD) scheme for the convection terms and center difference scheme for the other terms in space for predicting propagation of interface. Within the framework of Navier-Stokes solver, the semi-implicit Gear algorithm on a semi-staggered fixed grid is used for velocity-pressure coupling. The interface normal and curvature are calculated in terms of a hyperbolic tangent profile across the interface. To verify the proposed LS method, we consider reversed single vortex and Zalesak’s cases which are amenable to exact solutions. The proposed two-phase model is also investigated by solving the dam-break, Rayleigh-Taylor instability and bubble rising problems.

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 spatial sixth-order CCD-TVD method for solving multidimensional coupled Burgers’ equation

In this paper, a high-order compact difference scheme is proposed for solving multidimensional nonlinear Burgers’ equation. The three-stage third-order total variation diminishing (TVD) Runge–Kutta scheme is employed in time, and the three-point combined compact difference (CCD) scheme is used for spatial discretization. The proposed TVD-CCD method is free of using Hopf–Cole transformation, and treats the nonlinear term explicitly. Thus it is very efficient and easy to implement. Our method is effective to capture shock wave, third-order accurate in time, and sixth-order accurate in space. In addition, we show the unique solvability of the CCD system under non-periodic boundary conditions. Numerical experiments are given to demonstrate the high efficiency and accuracy of the proposed method.

A combined compact difference scheme for option pricing in the exponential jump-diffusion models

In the present paper, starting with the Black–Scholes equations, whose solutions are the values of European options, we describe the exponential jump-diffusion model of Levy process type. Here, a jump-diffusion model for a single-asset market is considered. Under this assumption the value of a European contingency claim satisfies a general “partial integro-differential equation” (PIDE). With a combined compact difference (CCD) scheme for the spatial discretization, a high-order method is proposed for solving exponential jump-diffusion models. The method is sixth-order accurate in space and second-order accurate in time. A known analytical solution to the model is used to evaluate the performance of the numerical scheme.

A Blended Compact Difference (BCD) Method for Solving 3D Convection–Diffusion Problems with Variable Coefficients

In this study, we present a fourth-order and a sixth-order blended compact difference (BCD) schemes for approximating the three-dimensional (3D) convection–diffusion equation with variable convective coefficients. The proposed schemes, where transport variable, its first and second derivatives are carried as the unknowns, combine virtues of compact discretization, fourth-order Padé scheme and sixth-order combined compact difference (CCD) scheme for spatial derivatives and can efficiently capture numerical solutions of linear and nonlinear convection–diffusion equations with Dirichlet boundary conditions. The fourth-order scheme requires only 7 grid points and the sixth-order scheme requires 19 grid points. The distinguishing feature of the present method is that methodologies of explicit compact difference and implicit compact difference are blended together. The truncation errors of the two difference schemes are analyzed for the interior grid points, respectively. Simultaneously, a sixth-order accuracy scheme is proposed to compute the first and second derivatives of the grid points on boundaries. Finally, the presented methods are applied to several test problems from the literature including linear and nonlinear problems. It is found that the presented schemes exhibit good performance.

Numerical modeling of lock-exchange gravity/turbidity currents by a high-order upwinding combined compact difference scheme

This study presents two-dimensional direct numerical simulations for sediment-laden current with higher density propagating forward through a lighter ambient water. The incompressible NavierStokes equations including the buoyancy force for the density difference between the light and heavy fluids are solved by a finite difference scheme based on a structured mesh. The concentration transport equations are used to explore such rich transport phenomena as gravity and turbidity currents. Within the framework of an Upwinding Combined Compact finite Difference (UCCD) scheme, rigorous determination of weighting coefficients underlies the modified equation analysis and the minimization of the numerical modified wavenumber. This sixth-order UCCD scheme is implemented in a four-point grid stencil to approximate advection and diffusion terms in the concentration transport equations and the first-order derivative terms in the Navier-Stokes equations, which can greatly enhance convective stability and increase dispersive accuracy at the same time. The initial discontinuous concentration field is smoothed by solving a newly proposed Heaviside function to prevent numerical instabilities and unreasonable concentration values. A two-step projection method is then applied to obtain the velocity field. The numerical algorithm shows a satisfying ability to capture the generation, development, and dissipation of the Kelvin-Helmholz instabilities and turbulent billows at the interface between the current and the ambient fluid. The simulation results also are compared with the data in published literatures and good agreements are found to prove that the present numerical model can well reproduce the propagation, particle deposition, and mixing processes of lock-exchange gravity and turbidity currents.

A time marching strategy for solving parabolic and elliptic equations with Neumann boundary conditions

In the community of computational fluid dynamics, pressure Poisson equation with Neumann boundary condition is usually encountered when solving the incompressible Navier–Stokes equations in a segregated approach such as SIMPLE, PISO, and projection methods. To deal with Neumann boundary conditions more naturally and to retain high order spatial accuracy as well, a sixth-order accurate combined compact difference scheme developed on staggered grids (NSCCD6) is adopted to solve the parabolic and elliptic equations subject to Neumann boundary conditions. The staggered grid system is usually used when solving the incompressible Navier–Stokes equations. By adopting the combined compact difference concept, there is no need to discretize Neumann boundary conditions with one-sided discretization scheme which is of lower accuracy order. The conventional Crank–Nicolson scheme is applied in this study for temporal discretization. For two-dimensional cases, D’yakonov alternating direction implicit scheme is adopted. A newly proposed time step changing strategy is adopted to improve convergence rate when solving the steady state solutions of the parabolic equation. High accuracy order of the currently proposed NSCCD6 scheme for one- and two-dimensional cases are shown in this article.

A CCD-ADI method for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients

ABSTRACTIn this paper, a combined compact finite difference method (CCD) together with alternating direction implicit (ADI) scheme is developed for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients. The proposed CCD-ADI method is second-order accurate in time variable and sixth-order accurate in space variable. For the linear hyperbolic equation, the CCD-ADI method is shown to be unconditionally stable by using the Von Neumann stability analysis. Numerical results for both linear and nonlinear hyperbolic equations are presented to illustrate the high accuracy of the proposed method.

A combined compact finite difference scheme for predicting the evolution of a mean curvature driven interface

This study is aimed to develop a high-order finite difference scheme to simulate the mean curvature equation for accurately predicting the time-evolving mean curvature driven interface. Within the semi-discretization framework, the optimal third-order accurate temporal scheme is applied for the approximation of the time derivative term. In a three-point grid stencil, a combined compact difference scheme has been developed for offering a fifth-order spatial accuracy for the first-order and second-order derivative terms shown in the level set equation for simulating mean curvature flow. In the simulation of the mean curvature equation, reinitialization procedure has been performed to improve the prediction of curve motion. The aim of this study is to give insights into the issue about “how a mean curvature driven curve is varied with time”. Two mechanisms leading to the change of curve slope are attributed partly to the damping Laplacian operator and partly to the embedded nonlinear differential operator. These mechanisms have been studied in detail with respect to the curvature of curve. The effect of performing reinitialization is also studied numerically.

Unique solvability of the CCD scheme for convection\u2013diffusion equations with variable convection coefficients

The combined compact difference (CCD) scheme has better spectral resolution than many other existing compact or noncompact high-order schemes, and is widely used to solve many differential equations. However, due to its implicit nature, very little theoretical results on the CCD method are known. In this paper, we provide a rigorous theoretical proof for the unique solvability of the CCD scheme for solving the convection-diffusion equation with variable convection coefficients subject to periodic boundary conditions.

Sixth-order non-uniform combined compact difference scheme for multi-term time fractional diffusion-wave equation

In this paper, we propose a high-order scheme for the numerical solution of multi-term time fractional diffusion-wave (FDW) equation in one and two-dimensional on non-uniform grids. Based on the sixth-order non-uniform combined compact difference (NCCD) scheme in the space directions on non-uniform grids, an alternating direction implicit (ADI) method is proposed to split the equation into two separate one dimensional equations. The multi-term time fractional derivation is described in the Caputo's sense with scheme of order O(τ3−α), 1<α<2. A numerical analysis of Fourier analysis completed by stability calculations in terms of semi-discrete eigenvalue problems are proposed. The advantage of the non-uniform combined compact difference (NCCD) scheme is that it can decrease the CPU time in comparison with the uniform combined compact difference (CCD) scheme. The sixth-order accuracy in the space directions on non-uniform grids has not been achieved in previous studies.

GPU acceleration of a petascale application for turbulent mixing at high Schmidt number using OpenMP 4.5

This paper reports on the successful implementation of a massively parallel GPU-accelerated algorithm for the direct numerical simulation of turbulent mixing at high Schmidt number. The work stems from a recent development (Comput. Phys. Commun., vol. 219, 2017, 313–328), in which a low-communication algorithm was shown to attain high degrees of scalability on the Cray XE6 architecture when overlapping communication and computation via dedicated communication threads. An even higher level of performance has now been achieved using OpenMP 4.5 on the Cray XK7 architecture, where on each node the 16 integer cores of an AMD Interlagos processor share a single Nvidia K20X GPU accelerator. In the new algorithm, data movements are minimized by performing virtually all of the intensive scalar field computations in the form of combined compact finite difference (CCD) operations on the GPUs. A memory layout in departure from usual practices is found to provide much better performance for a specific kernel required to apply the CCD scheme. Asynchronous execution enabled by adding the OpenMP 4.5 NOWAIT clause to TARGET constructs improves scalability when used to overlap computation on the GPUs with computation and communication on the CPUs. On the 27-petaflops supercomputer Titan at Oak Ridge National Laboratory, USA, a GPU-to-CPU speedup factor of approximately 5 is consistently observed at the largest problem size of 81923 grid points for the scalar field computed with 8192 XK7 nodes.

Efficient compact finite difference methods for a class of time-fractional convection–reaction–diffusion equations with variable coefficients

ABSTRACTThis paper is devoted to the construction and analysis of compact finite difference methods for a class of time-fractional convection–reaction–diffusion equations with variable coefficients. Based on some new techniques coupled with the - approximation formula of the time-fractional derivative and a fourth-order compact finite difference approximation to the spatial derivative, a compact finite difference method is proposed for the equations with spatially variable convection and reaction coefficients. The local truncation error and the solvability of the method are discussed in detail. The unconditional stability of the resulting scheme and also its convergence of second order in time and fourth order in space are rigorously proved using a discrete energy analysis method. The proposed method is further extended to the more general case when the convection and reaction coefficients are variable both spatially and temporally. A high-order combined compact finite difference method is also proposed. Numerical results demonstrate the effectiveness of the methods.

Numerical solutions to a two-component Camassa–Holm equation

Abstract In the present paper, a three-step iterative algorithm for solving a two-component Camassa–Holm (2CH) equation is presented. In the first step, the time-dependent equation for the horizontal fluid velocity with nonlinear convection is solved. Then an inhomogeneous Helmholtz equation is solved. Finally, the equation for modeling the transport of density is solved in the third step. The differential order of 2CH equation has been reduced in order to facilitate numerical scheme development in a comparatively smaller grid stencil. In this study, a fifth-order spatially accurate upwinding combined compact difference scheme (UCCD5) which differs from that in Sheu et al. (2011) is developed in a four-point grid stencil for approximating the first-order derivative term. For the purpose of retaining long-time Hamiltonians in the 2CH equation, the time integrator (or time-stepping scheme) chosen is symplectic. Various numerical experiments such as the single peakon, peakon–antipeakon interaction and dam-break problems are conducted to illustrate the effectiveness of the proposed numerical method. It is shown that both the Hamiltonians and Casimir functions are conserved well for all problems.

Development of a coupled level set and immersed boundary method for predicting dam break flows

Dam-break flow over an immersed stationary object is investigated using a coupled level set (LS)/immersed boundary (IB) method developed in Cartesian grids. This approach adopts an improved interface preserving level set method which includes three solution steps and the differential-based interpolation immersed boundary method to treat fluid–fluid and solid–fluid interfaces, respectively. In the first step of this level set method, the level set function ϕ is advected by a pure advection equation. The intermediate step is performed to obtain a new level set value through a new smoothed Heaviside function. In the final solution step, a mass correction term is added to the re-initialization equation to ensure the new level set is a distance function and to conserve the mass bounded by the interface. For accurately calculating the level set value, the four-point upwinding combined compact difference (UCCD) scheme with three-point boundary combined compact difference scheme is applied to approximate the first-order derivative term shown in the level set equation. For the immersed boundary method, application of the artificial momentum forcing term at points in cells consisting of both fluid and solid allows an imposition of velocity condition to account for the presence of solid object. The incompressible Navier–Stokes solutions are calculated using the projection method. Numerical results show that the coupled LS/IB method can not only predict interface accurately but also preserve the mass conservation excellently for the dam-break flow.