Combined Compact Difference Scheme Research Articles

2-D acoustic equation prestack reverse-time migration based on 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 reverse time migration 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.

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.

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 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.

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.

A dual communicator and dual grid-resolution algorithm for petascale simulations of turbulent mixing at high Schmidt number

A new dual-communicator algorithm with very favorable performance characteristics has been developed for direct numerical simulation (DNS) of turbulent mixing of a passive scalar governed by an advection–diffusion equation. We focus on the regime of high Schmidt number (Sc), where because of low molecular diffusivity the grid-resolution requirements for the scalar field are stricter than those for the velocity field by a factor Sc. Computational throughput is improved by simulating the velocity field on a coarse grid of Nv3 points with a Fourier pseudo-spectral (FPS) method, while the passive scalar is simulated on a fine grid of Nθ3 points with a combined compact finite difference (CCD) scheme which computes first and second derivatives at eighth-order accuracy. A static three-dimensional domain decomposition and a parallel solution algorithm for the CCD scheme are used to avoid the heavy communication cost of memory transposes. A kernel is used to evaluate several approaches to optimize the performance of the CCD routines, which account for 60% of the overall simulation cost. On the petascale supercomputer Blue Waters at the University of Illinois, Urbana–Champaign, scalability is improved substantially with a hybrid MPI-OpenMP approach in which a dedicated thread per NUMA domain overlaps communication calls with computational tasks performed by a separate team of threads spawned using OpenMP nested parallelism. At a target production problem size of 81923 (0.5 trillion) grid points on 262,144 cores, CCD timings are reduced by 34% compared to a pure-MPI implementation. Timings for 163843 (4 trillion) grid points on 524,288 cores encouragingly maintain scalability greater than 90%, although the wall clock time is too high for production runs at this size. Performance monitoring with CrayPat for problem sizes up to 40963 shows that the CCD routines can achieve nearly 6% of the peak flop rate. The new DNS code is built upon two existing FPS and CCD codes. With the grid ratio Nθ∕Nv=8, the disparity in the computational requirements for the velocity and scalar problems is addressed by splitting the global communicator MPI_COMM_WORLD into disjoint communicators for the velocity and scalar fields, respectively. Inter-communicator transfer of the velocity field from the velocity communicator to the scalar communicator is handled with discrete send and non-blocking receive calls, which are overlapped with other operations on the scalar communicator. For production simulations at Nθ=8192 and Nv=1024 on 262,144 cores for the scalar field, the DNS code achieves 94% strong scaling relative to 65,536 cores and 92% weak scaling relative to Nθ=1024 and Nv=128 on 512 cores.

Spectral analysis of finite difference schemes for convection diffusion equation

The paper presents numerical analysis of finite difference schemes for solving the linear convection-diffusion equation using a full domain spectral analysis method illustrated in Sengupta et al. (2003) [7]. Different numerical schemes ranging from simple central and upwind difference schemes to high accuracy schemes like compact and combined compact difference schemes are analyzed for their accuracy. Optimal values of simulation parameters are proposed for the analyzed schemes with a view to obtain accurate solutions.

Development of a Combined Compact Difference Scheme to Simulate Soliton Collision in a Shallow Water Equation

AbstractIn this paper a three-step scheme is applied to solve the Camassa-Holm (CH) shallow water equation. The differential order of the CH equation has been reduced in order to facilitate development of numerical scheme in a comparatively smaller grid stencil. Here a three-point seventh-order spatially accurate upwinding combined compact difference (CCD) scheme is proposed to approximate the firstorder derivative term. We conduct modified equation analysis on the CCD scheme and eliminate the leading discretization error terms for accurately predicting unidirectional wave propagation. The Fourier analysis is carried out as well on the proposed numerical scheme to minimize the dispersive error. For preserving Hamiltonians in Camassa- Holm equation, a symplecticity conserving time integrator has been employed. The other main emphasis of the present study is the use of u–P–α formulation to get nondissipative CH solution for peakon-antipeakon and soliton-anticuspon head-on wave collision problems.

Development of a High-Resolution Scheme for Solving the PNP-NS Equations in Curved Channels

AbstractA high-order finite difference scheme has been developed to approximate the spatial derivative terms present in the unsteady Poisson-Nernst-Planck (PNP) equations and incompressible Navier-Stokes (NS) equations. Near the wall the sharp solution profiles are resolved by using the combined compact difference (CCD) scheme developed in five-point stencil. This CCD scheme has a sixth-order accuracy for the second-order derivative terms while a seventh-order accuracy for the first-order derivative terms. PNP-NS equations have been also transformed to the curvilinear coordinate system to study the effects of channel shapes on the development of electroos-motic flow. In this study, the developed scheme has been analyzed rigorously through the modified equation analysis. In addition, the developed method has been computationally verified through four problems which are amenable to their own exact solutions. The electroosmotic flow details in planar and wavy channels have been explored with the emphasis on the formation of Coulomb force. Significance of different forces resulting from the pressure gradient, diffusion and Coulomb origins on the convective electroosmotic flow motion is also investigated in detail.

Simulation of Incompressible Free Surface Flow Using the Volume Preserving Level Set Method

AbstractThis study aims to develop a numerical scheme in collocated Cartesian grids to solve the level set equation together with the incompressible two-phase flow equations. A seventh-order accurate upwinding combined compact difference (UCCD7) scheme has been developed for the approximation of the first-order spatial derivative terms shown in the level set equation. Developed scheme has a higher accuracy with a three-point grid stencil to minimize phase error. To preserve the mass of each phase all the time, the temporal derivative term in the level set equation is approximated by the sixth-order accurate symplectic Runge-Kutta (SRK6) scheme. All the simulated results for the dam-break, Rayleigh-Taylor instability, bubble rising, two-bubble merging, and milkcrown problems in two and three dimensions agree well with the available numerical or experimental results.