Finite difference method

About: Finite difference method is a research topic. Over the lifetime, 21603 publications have been published within this topic receiving 468852 citations. The topic is also known as: Finite-difference methods & FDM.

Large-Eddy Simulation on Curvilinear Grids Using Compact Differencing and Filtering Schemes

Abstract: This work investigates the application of a high-order finite difference method for compressible large-eddy simulations on stretched, curvilinear and dynamic meshes. The solver utilizes 4th and 6th-order compact-differencing schemes for the spatial discretization, coupled with both explicit and implicit time-marching methods. Up to 10th order, Pade-type low-pass spatial filter operators are also incorporated to eliminate the spurious high-frequency modes which inevitably arise due to the lack of inherent dissipation in the spatial scheme. The solution procedure is evaluated for the case of decaying compressible isotropic turbulence and turbulent channel flow. The compact/filtering approach is found to be superior to standard second and fourth-order centered, as well as third-order upwind-biased approximations. For the case of isotropic turbulence, better results are obtained with the compact/filtering method (without an added subgrid-scale model) than with the constant-coefficient and dynamic Smagorinsky models. This is attributed to the fact that the SGS models, unlike the optimized low-pass filter, exert dissipation over a wide range of wave numbers including on some of the resolved scales

Steady and unsteady solutions of the incompressible Navier-Stokes equations

Abstract: An algorithm for the solution of the incompressible Navier-Stokes equations in three-dimensional generalized curvilinear coordinates is presented. The algorithm can be used to compute both steady-state and time-dependent flow problems. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The equations are solved with a line-relaxation scheme that allows the use of very large pseudotime steps leading to fast convergence for steady-state problems as well as for the subiterations of time-dependent problems. The steady-state solution of flow through a square duct with a 90-deg bend is computed, and the results are compared with experimental data. Good agreement is observed. Computations of unsteady flow over a circular cylinder are presented and compared to other experimental and computational results. Finally, the flow through an artificial heart configuration with moving boundaries is calculated and presented.

Computing the Ground State Solution of Bose--Einstein Condensates by a Normalized Gradient Flow

Abstract: In this paper, we present a continuous normalized gradient flow (CNGF) and prove its energy diminishing property, which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of Bose--Einstein condensates (BEC). We also investigate the energy diminishing property for the discretization of the CNGF. Two numerical methods are proposed for such discretizations: one is the backward Euler centered finite difference (BEFD) method, the other is an explicit time-splitting sine-spectral (TSSP) method. Energy diminishing for BEFD and TSSP for the linear case and monotonicity for BEFD for both linear and nonlinear cases are proven. Comparison between the two methods and existing methods, e.g., Crank--Nicolson finite difference (CNFD) or forward Euler finite difference (FEFD), shows that BEFD and TSSP are much better in terms of preserving the energy diminishing property of the CNGF. Numerical results in one, two, and three dimensions with magnetic trap confinement potential, as well as a potential of a stirrer corresponding to a far-blue detuned Gaussian laser beam, are reported to demonstrate the effectiveness of BEFD and TSSP methods. Furthermore we observe that the CNGF and its BEFD discretization can also be applied directly to compute the first excited state solution in BEC when the initial data is chosen as an odd function.

3-D ADI-FDTD method-unconditionally stable time-domain algorithm for solving full vector Maxwell's equations

Abstract: We previously introduced the alternating direction implicit finite-difference time domain (ADI-FDTD) method for a two-dimensional TE wave. We analytically and numerically verified that the algorithm of the method is unconditionally stable and free from the Courant-Friedrich-Levy condition restraint. In this paper, we extend this approach to a full three-dimensional (3-D) wave. Numerical formulations of the 3-D ADI-FDTD method are presented and simulation results are compared to those using the conventional 3-D finite-difference time-domain (FDTD) method. We numerically verify that the 3-D ADI-FDTD method is also unconditionally stable and it is more efficient than the conventional 3-D FDTD method in terms of the central processing unit time if the size of the local minimum cell in the computational domain is much smaller than the other cells and the wavelength.