FFT-based high order central difference schemes for three-dimensional Poisson's equation with various types of boundary conditions

An efficient time-splitting compact finite difference method for Gross–Pitaevskii equation

In consideration of the initial singularity of the solution, a temporally second-order fast compact difference scheme with unequal time-steps is presented and analyzed for simulating the subdiffusion problems in several spatial dimensions. On the basis of sum-of-exponentials technique, a fast Alikhanov formula is derived on general nonuniform meshes to approximate the Caputo’s time derivative. Meanwhile, the spatial derivatives are approximated by the fourth-order compact difference operator, which can be implemented by a fast discrete sine transform via the FFT algorithm. So the proposed algorithm is computationally efficient with the computational cost about $O(MN\log M\log N)$
 and the storage requirement $O(M\log N)$
 , where M and N are the total numbers of grids in space and time, respectively. With the aids of discrete fractional Gronwall inequality and global consistency analysis, the unconditional stability and sharp H1-norm error estimate reflecting the regularity of solution are established rigorously by the discrete energy approach. Three numerical experiments are included to confirm the sharpness of our analysis and the effectiveness of our fast algorithm.

A second-order fast compact scheme with unequal time-steps for subdiffusion problems

In this paper, the combination of efficient sixth-order compact finite difference scheme (E-CFDS6) based proper orthogonal decomposition and Strang splitting method (E-CFDS6-SSM) is constructed for the numerical solution of the multi-dimensional parabolic equation (MDPE). For this purpose, we first develop the CFDS6 to attain a high accuracy for the one-dimensional parabolic equation (ODPE). Then, by the Strang splitting method, we have converted the MDPE into a series of one-dimensional ODPEs successfully, which is easier to implement and program with CFDS6 than an alternating direction implicit method. Finally, we employ proper orthogonal decomposition techniques to improve the computational efficiency of CFDS6 and build the E-CFDS6-SSM with fewer unknowns and sufficiently high accuracy. Six numerical examples are presented to demonstrate that the E-CFDS6-SSM not only can largely alleviate the computational load but also hold a high accuracy and simplify the process of the program for the numerical solution of MDPE.

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An efficient high-order compact finite difference scheme based on proper orthogonal decomposition for the multi-dimensional parabolic equation

A graphics processing unit (GPU) is utilized to apply the direct-forcing immersed boundary method. The code running on the GPU is generated with the help of the Compute Unified Device Architecture (CUDA). The first and second spatial derivatives of the incompressible Navier-Stokes equations are discretized by the sixth-order central compact finite-difference schemes. Two flow fields are simulated. The first test case is the simulated flow around a square cylinder, with the results providing good estimations of the wake region mechanics and vortex shedding. The second test case is the simulated flow around a circular cylinder. This case was selected to better understand the effects of sharp corners on the force coefficients. It was observed that the estimation of the force coefficients did not result in any troubles in the case of a circular cylinder. Additionally, the performance values obtained for the calculation time for the solution of the Poisson equation are compared with the values for othe...

https://www.tandfonline.com/doi/pdf/10.1080/19942060.2016.1236749

An implementation of the direct-forcing immersed boundary method using GPU power

Fourth-order compact finite difference scheme has been proposed for solving the Poisson equation with Dirichlet boundary conditions for some time. An efficient implementation of such numerical scheme is often desired for practical usage. In this paper, based on fast discrete Sine transform, we design an efficient algorithm to implement this scheme. To do this, Poisson equation is first discretized by fourth-order compact finite difference method. The subsequent discretized system is not solved by the usual method-matrix inversion, instead it is solved with the fast discrete Sine transform. By doing this way, the computational cost of proposed algorithm for such scheme with large grid numbers can be greatly reduced. Detailed numerical algorithm of this fast solver for one-dimensional, two-dimensional and three dimensional Poisson equation has been presented. Numerical results in one dimension, two dimensions, three dimensions and four dimensions have shown that the applied compact finite difference scheme has fourth order accuracy and can be efficiently implemented.

Jun Qiu

Papers

An efficient implementation of fourth-order compact finite difference scheme for Poisson equation with Dirichlet boundary conditions

A B-Infinity Algebra Structure of Singular Hochschild Complex