High-order implicit finite difference schemes for the two-dimensional Poisson equation

A reduced high-order compact finite difference scheme based on proper orthogonal decomposition technique for KdV equation

One-dimensional heat equation was solved for different higher-order finite difference schemes, namely, forward time and fourth-order centered space explicit method, backward time and fourth-order centered space implicit method, and fourth-order implicit Crank-Nicolson finite difference method. Higher-order schemes have complexity in computing values at the neighboring points to the boundaries. It is required there a specification of the values of field variables at some points exterior to the domain. The complexity was incorporated using Hicks approximation. The convergence and stability analysis was also computed for those higher-order finite difference explicit and implicit methods in case of solving a one dimensional heat equation. The obtained numerical results were compared with exact solutions. It is found that backward time and fourth-order centered space implicit scheme along with Hicks approximation performed well over the other mentioned higher-order approaches.

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

Higher-order accurate two-step finite difference schemes for the many-dimensional wave equation

High-order cell-vertex finite difference schemes applied to multi-block structured grids are used widely in computational aeroacoustics for their low-dispersion and low-dissipation properties. Structured grids for complex geometries may contain discontinuous grid metrics at multi-block interfaces. In this work it is demonstrated that the grid-induced errors from such interfaces can be reduced by applying finite difference schemes in the cell-centered space. Further reduction of these grid-induced errors can be achieved by applying an additional finite volume method, which serves as an interface condition. In this paper, the development of a hybrid cell-centered finite difference and finite volume method is demonstrated. An interpolation scheme is derived from a high-order finite difference scheme to apply the finite volume method at interfaces. The order of accuracy of this hybrid method is demonstrated and the method is used to simulate the flow around a single cylinder, tandem cylinders, and a complex isolated wheel. Comparisons with experimental measurements and numerical predictions show that the hybrid method can provide accurate results at block interfaces and can be applied to high-order simulations of complex geometries.

Reverse-time migration (RTM) using the crosscorrelation imaging condition requires that the forward-propagated source wavefield and the backward-propagated receiver wavefield be accessible within the imaging domain at the same time step. There are two categories of methods to balance the computer memory requirement and the computational complexity of RTM: checkpointing methods and source-wavefield reconstruction methods. We have developed a new source-wavefield reconstruction method to improve the balance between the computer memory requirement and the computational complexity of RTM. During the forward simulation of the source wavefield, we stored boundary wavefields only at one or two layers of spatial grid points and reconstructed the back-propagated source wavefield at the same time step as that of the back-propagated receiver wavefield, using a high-order wave-equation extrapolation scheme. One conventional RTM method uses boundary wavefields stored at multiple layers of spatial grid points and a high-order finite-difference (FD) scheme to reconstruct the back-propagated source wavefield. For an FD scheme with the eighth or sixteenth order of accuracy in space, our new method used only 37.5% of the computer memory required by this conventional method to store the boundary wavefields. This reduction of computer memory usage is significant because storing the boundary wavefields consumes most of the computer memory required for 3D migration using reconstructed source wavefields. Moreover, our method maintained the spatial order of accuracy of the FD scheme for the entire imaging domain, whereas some conventional methods reduce the spatial-order accuracy of the FD scheme near the boundaries to back-propagate the source wavefield to decrease the computer memory requirement. We validated our method using synthetic seismic data. Our method produced 2D and 3D migration images of complex subsurface structures as accurate as those yielded using an RTM method without reducing the spatial order of accuracy near the boundaries.

Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier-Stokes equations

A direct numerical simulation of three-dimensional homogeneous isotropic turbulence by the spectral method and the higher-order finite difference scheme (FDS) has been conducted with 64×64×64 and 96×96×96 grids. The energy spectrum obtained by the FDS is compared with that obtained by the spectral method and that obtained by the experiment to investigate the effect of discretization. From these studies, the following conclusions are obtained. (1) As for the discretization of the nonlinear term with the 64×64×64 grids, the energy spectrum obtained by the 5th-order upwind FDS coincides with those obtained by the spectral method and the experiment up to the cutoff wave number, while the 3rd-order upwind FDS shows deviation from those obtained by the spectral method and the experiment. (2) With the 64×64×64 grids, the 6th-order central FDS with the skew-symmetric form is valid for DNS, while the 6th-order central FDS with the convective form overestimates the energy in the region around the cutoff wave number. (3) With the nonlinear term discretized by the 5th-order upwind FDS or 6th-order central FDS with the skew-symmetric from, the 4th-order FDS is suitable for the viscous term.

A new linear optimized time–space domain spatial implicit and temporal high-order finite-difference scheme for scalar wave modeling

The present thesis describes the development of a computational method for the numerical simulation of compressible flows in aerodynamics. The main goal is to study high-order discretization schemes that can potentially deliver a better computational efficiency. The spatial part of the conservation equations is discretized with the finite difference method. In particular, Summation By Parts (SBP) finite difference operators and treatment of boundary conditions via Simultaneous Approximation Terms (SAT) have been investigated. An SBP finite difference operator is essentially a (high-order) centered finite difference scheme with a specific closure at the boundary. The SAT are penalty-like terms that enforce the boundary conditions weakly and are used to augment the SBP schemes. The combination of SBP operators with SAT boundary terms constitutes schemes that, for the linearized equations governing a smooth flow, are provably stable (thus convergent) schemes. The discretization in time is carried out with high-order schemes as well. Both implicit and explicit time integration schemes have been investigated. The system of non-linear equations that arises when using the implicit time integration schemes is solved via (a damped) Newton's method. On the target grid, the initial guess required for Newton's method is obtained by performing grid sequencing combined a series of explicit relaxation iterations on each grid level. The flow equations and the equation of the turbulence model (if present) are solved in a fully-coupled manner. Numerical simulations for inviscid flow, laminar flow, as well as simulations based on DNS, LES and RANS have been performed. Results show that the described computational method is particularly well-suited for smooth external flows about relatively simple geometries, solved with the DNS and LES approach. For these types of flows the gain in efficiency obtained by using high-order schemes is clear (more than a factor of 5). For RANS, however, the comparison with a state-of-the-art second order finite volume code suggests that solving the turbulence and the flow equations fully coupled with implicit schemes may not be the most efficient iteration strategy. On the other hand, the numerical simulations of non-smooth/internal flows highlighted the need of further development of the SBP-SAT framework.

In the present paper, high order boundary conditions for high order finite difference schemes on curvilinear coordinates are implemented and analyzed. The finite difference scheme employed in the present paper is a conservative scheme on a finite volume type grid arrangement for which the implementation of the boundary conditions is characterized by a proper evaluation of the numerical fluxes at the boundaries. Simple extrapolation procedures dealing with both the inviscid and viscous fluxes at the boundary are proposed. Specifically, for the inviscid fluxes, a characteristic-variable-based extrapolation procedure is adopted, which effectively reduces the numerical reflection at the far-field boundaries. The emphasis of the present paper is to design straightforward and efficient numerical boundary closure procedures which are capable of achieving high order of accuracy near the boundaries. Special attention is paid to the influence of the extrapolation accuracy on the overall accuracy of the scheme. Numerical results show that the finite difference scheme can achieve its nominal order accuracy when the extrapolation accuracy is sufficiently high.

A direct numerical simulation (DNS) of flow around a two-dimensional circular cylinder by the higher-order finite difference scheme (FDS) is conducted. Inflow and outflow boundary conditions constructed and successfully tested originally on a spatially developing mixing layer are applied to the above mentioned flow with Re=226. Comparisons of the general flowfield and the statistics with those obtained in an extended computational domain arc made. In addition, the appropriateness of the outflow boundary condition imposing the zero gradient of the velocity and pressure is investigated. From these studies, the following conclusions are obtained. (1) For the newly proposed outflow boundary condition, the velocity and pressure fields are highly correlated to the results obtained in the extended computational domain. The correlation coefficient of the pressure with that obtained in the extended computational domain is higher than 0.999. (2) For the conventional zero gradient condition, a significant discrepancy from results obtained in the extended domain is observed for both velocity and pressure fields, and the correlation coefficient of the pressure is as low as 0.966. (3) The conventional zero gradient condition overestimates the drag coefficient.

High-order accurate entropy stable adaptive moving mesh finite difference schemes for special relativistic (magneto)hydrodynamics

A staggered-grid finite-difference scheme optimized in the time–space domain for modeling scalar-wave propagation in geophysical problems

Analyses were performed on the dispersion overshoot and inverse dissipation of the high-order finite difference scheme using Fourier and precision analysis. Schemes under discussion included the pointwise- and staggered-grid type, and were presented in weighted form using candidate schemes with third-order accuracy and three-point stencil. All of these were commonly used in the construction of difference schemes. Criteria for the dispersion overshoot were presented and their critical states were discussed. Two kinds of instabilities were studied due to inverse dissipation, especially those that occur at lower wave numbers. Criteria for the occurrence were presented and the relationship of the two instabilities was discussed. Comparisons were made between the analytical results and the dispersion/dissipation relations by Fourier transformation of typical schemes. As an example, an application of the criteria was given for the remedy of inverse dissipation in Weirs & Martin’s third-order scheme.